Separation of Variables and Exact Solutions to Nonlinear PDEs (Advances in Applied Mathematics) [1 ed.] 036748689X, 9780367486891

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Separation of Variables and Exact Solutions to Nonlinear PDEs

Advances in Applied Mathematics Series Editors: Daniel Zwillinger, H. T. Banks Handbook of Mellin Transforms Yu. A. Brychkov, O. I. Marichev, N. V. Savischenko Advanced Mathematical Modeling with Technology William P. Fox, Robert E. Burks Introduction to Quantum Control and Dynamics Domenico D’Alessandro Handbook of Radar Signal Analysis Bassem R. Mahafza, Scott C. Winton, Atef Z. Elsherbeni Separation of Variables and Exact Solutions to Nonlinear PDEs Andrei D. Polyanin, Alexei I. Zhurov Boundary Value Problems on Time Scales, Volume I Svetlin Georgiev, Khaled Zennir Boundary Value Problems on Time Scales, Volume II Svetlin Georgiev, Khaled Zennir Observability and Mathematics Fluid Mechanics, Solutions of Navier-Stokes Equations, and Modeling Boris Khots Handbook of Differential Equations, 4th Edition Daniel Zwillinger, Vladimir Dobrushkin Experimental Statistics and Data Analysis for Mechanical and Aerospace Engineers James Middleton Advanced Engineering Mathematics with MATLAB Dean G. Duffy Handbook of Fractional Calculus for Engineering and Science Harendra Singh, H. M. Srivastava, Juan J Nieto https://www.routledge.com/Advances-in-Applied-Mathematics/book-series/CRCADVAPPMTH?pd=published,fort hcoming&pg=1&pp=12&so=pub&view=list

Separation of Variables and Exact Solutions to Nonlinear PDEs

Andrei D. Polyanin Alexei I. Zhurov

First edition published 2022 by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2022 Taylor and Francis Group, LLC CRC Press is an imprint of Informa UK Limited Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf. co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-367-48689-1 (hbk) ISBN: 978-1-032-11524-5 (pbk) ISBN: 978-1-003-04229-7 (ebk) DOI: 10.1201/9781003042297 Typeset in TimesNewRoman by KnowledgeWorks Global Ltd.

Dedicated to the memory of Valentin Fedorovich Zaitsev, our good friend and co-author

Contents Preface

xi

Authors

xv

Some Notations and Remarks

xvii

1. Methods of Generalized Separation of Variables 1.1. Simple Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Multiplicative and Additive Separable Solutions . . . . . . . . 1.1.2. Simple Cases of Separation of Variables in Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . 1.1.3. Examples of Nontrivial Separation of Variables in Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . 1.2.1. General Form of Solutions. The Classes of Nonlinear Differential Equations of Interest . . . . . . . . . . . . . . . . . . . . . 1.2.2. Functional Differential Equations Arising in Generalized Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Simplified Method for Constructing Generalized Separable Solutions . . 1.3.1. Description of a Simplified Method Based on a Priori Setting of a System of Coordinate Functions . . . . . . . . . . . . . . . . 1.3.2. Examples of Constructing Exact Solutions to Nonlinear Equations in Two Independent Variables . . . . . . . . . . . . . . . 1.3.3. Equations in Three or More Independent Variables. Exact Solutions to the Navier–Stokes Equations . . . . . . . . . . . . . . 1.4. Solution of Functional Differential Equations by the Method of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Description of the Method of Differentiation . . . . . . . . . . 1.4.2. Examples of Constructing Generalized Separable Solutions by the Differentiation Method . . . . . . . . . . . . . . . . . . . . 1.5. Solution of Functional Differential Equations by the Splitting Method . 1.5.1. Preliminary Remarks. Description of the Method. The Splitting Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2. Solutions of Bilinear Functional Equations . . . . . . . . . . . 1.5.3. Examples of Constructing Generalized Separable Solutions by the Splitting Method . . . . . . . . . . . . . . . . . . . . . . . 1.6. Method of Invariant Subspaces (Titov–Galaktionov Method) . . . . . . 1.6.1. Subspaces Invariant under a Nonlinear Differential Operator. Description of the Method . . . . . . . . . . . . . . . . . . . . . . vii

1 1 1 3 10 12 12 14 15 15 15 26 35 35 36 46 46 47 50 58 58

viii

C ONTENTS

1.6.2. Some Modifications and Generalizations . . . . . . . . . . . . 1.6.3. Finding Linear Subspaces Invariant under a Given Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Other Nonlinear Equations Having Generalized Separable Solutions . . 1.7.1. Nonlinear Partial Differential Equations with Delay . . . . . . . 1.7.2. Nonlinear Integro-Differential Equations . . . . . . . . . . . . 1.7.3. Nonlinear Equations with a Fractional Derivative . . . . . . . . 1.7.4. Pseudo-Differential Equations . . . . . . . . . . . . . . . . . . 2. Methods of Functional Separation of Variables 2.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Structure of Functional Separable Solutions . . . . . . . . . . . 2.1.2. Direct and Indirect Functional Separation of Variables . . . . . 2.2. Simplified Method for Constructing Functional Separable Solutions . . 2.2.1. Description of the Simplified Method Based on Transformations of the Unknown Function . . . . . . . . . . . . . . . . . . . . 2.2.2. Examples of Constructing Exact Solutions to Nonlinear PDEs . 2.3. Functional Separable Solutions of Special Form . . . . . . . . . . . . . 2.3.1. Generalized Traveling Wave Solutions and Other Solutions of Special Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Examples of Constructing Generalized Traveling Wave Solutions 2.3.3. Construction of Other Functional Separable Solutions of Special Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Method of Differentiation. Using Nonlinear Functional Equations . . . 2.4.1. Brief Description of the Method of Differentiation . . . . . . . 2.4.2. Examples of Constructing Functional Separable Solutions by the Method of Differentiation . . . . . . . . . . . . . . . . . . . . 2.4.3. Using Nonlinear Functional Equations to Construct Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Construction of Functional Separable Solutions in Implicit Form . . . . 2.5.1. Preliminary Remarks. Traveling Wave Solutions in Implicit Form 2.5.2. Direct Method for Constructing Functional Separable Solutions in Implicit Form. The Splitting Principle . . . . . . . . . . . . 2.5.3. Nonlinear Reaction–Diffusion Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4. Nonlinear Convection–Diffusion Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5. Nonlinear Klein–Gordon Type Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6. Nonlinear Equations with Three or More Independent Variables 2.5.7. Nonlinear Third- and Higher-Order Equations . . . . . . . . . . 2.5.8. Nonlinear Schr¨odinger Type Equation . . . . . . . . . . . . . . 2.6. General Functional Separation of Variables. Explicit Representation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. General Form of Functional Separable Solutions . . . . . . . .

62 67 71 71 80 81 85 91 91 91 93 93 93 94 96 96 99 105 110 110 111 117 125 125 127 129 145 155 170 173 177 179 179

C ONTENTS 2.6.2. Nonlinear Reaction–Diffusion Type Equations . . . . . . . . . 2.6.3. Nonlinear Convection–Diffusion Type Equations . . . . . . . . 2.6.4. Nonlinear Klein–Gordon Type Equations and Nonlinear Telegraph Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5. Anisotropic Heat and Wave Equations with Three or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. General Functional Separation of Variables. Implicit Representation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. Description of the Method. The Generalized Splitting Principle 2.7.2. Usage of Equivalent Equations. Simplification of Equations . . 2.7.3. Nonlinear Reaction–Convection–Diffusion Equations . . . . . . 2.7.4. Generalized Porous Medium Equations with a Nonlinear Source

ix 179 190 200 208 211 211 214 216 242

3. Direct Method of Symmetry Reductions. Weak Symmetries 3.1. Direct Method of Symmetry Reductions . . . . . . . . . . . . . . . . . 3.1.1. Simplified Scheme. Generalized Burgers–Korteweg–de Vries Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Special Form of Reductions. The Boussinesq Equation . . . . . 3.1.3. General Form of Reductions. The Harry Dym Equation . . . . . 3.2. Direct Method of Weak Symmetry Reductions . . . . . . . . . . . . . . 3.2.1. General Description of the Method. Steady-State Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Burgers–Huxley Equation (Diffusion Type Equation with a Cubic Nonlinearity) . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Unsteady Plane and Axisymmetric Boundary Layer Equations . 3.2.4. Axisymmetric Boundary Layer Equations for an Extended Body of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Plane and Axisymmetric Boundary Layer Equations for NonNewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . .

251 251

4. Method of Differential Constraints 4.1. Method of Differential Constraints for Ordinary Differential Equations . 4.1.1. Description of the Method. First-Order Differential Constraints 4.1.2. Differential Constraints of Arbitrary Order. General Compatibility Method for two Equations . . . . . . . . . . . . . . . . . 4.1.3. Using Point Transformations in Combination with Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Using Several Differential Constraints . . . . . . . . . . . . . . 4.2. Description of the Method of Differential Constraints for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Preliminary Remarks. A Simple Example . . . . . . . . . . . . 4.2.2. General Description of the Method of Differential Constraints . 4.3. First-Order Differential Constraints for Partial Differential Equations . . 4.3.1. Second-Order Evolution Equations . . . . . . . . . . . . . . . 4.3.2. Second-Order Hyperbolic Equations . . . . . . . . . . . . . . . 4.3.3. Second-Order Equations of General Form . . . . . . . . . . . .

303 303 303

251 254 258 260 260 262 265 276 285

309 313 316 317 317 319 321 321 328 331

x

C ONTENTS

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations331 4.4.1. Second-Order Differential Constraints . . . . . . . . . . . . . . 331 4.4.2. Higher-Order Differential Constraints. Determining Equations . 334 4.4.3. Utilizing Several Differential Constraints. Systems of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 4.5. Connection Between the Method of Differential Constraints and Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 4.5.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 342 4.5.2. Generalized Separation of Variables and Differential Constraints 343 4.5.3. Functional Separation of Variables and Differential Constraints . 344 4.5.4. Direct Method of Symmetry Reductions and Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 4.5.5. Nonclassical Method of Symmetry Reductions . . . . . . . . . 350 References

353

Index

375

Preface Nonlinear partial differential equations (PDEs) of the second and higher orders (nonlinear equations of mathematical physics) often arise in various fields of mathematics, physics, mechanics, chemistry, biology, and in numerous applications. The general solution of nonlinear equations of mathematical physics can only be obtained in exceptional cases. Therefore, one usually has to confine themselves to the search and analysis of particular solutions, which are usually called exact solutions. Exact solutions to equations of mathematical physics have always played and continue to play a massive role in the formation of a correct understanding of the qualitative features of many phenomena and processes in various fields of natural science. Exact solutions to nonlinear equations facilitate a better understanding of the mechanisms of complex nonlinear effects such as the spatial localization of transfer processes, the multiplicity or absence of stationary states under certain conditions, the existence of peaking modes, the possible non-smoothness or discontinuity of the desired quantities, among others. Simple solutions to linear and nonlinear differential equations are widely used to illustrate theoretical material and some applications in educational courses at universities and colleges (in applied and computational mathematics, asymptotic methods, theoretical physics, heat and mass transfer theory, fluid dynamics, gas dynamics, wave theory, nonlinear optics, etc.). Exact traveling wave solutions and self-similar solutions often represent asymptotics of substantially broader classes of solutions corresponding to different initial and boundary conditions. This property allows drawing general conclusions and predicting the dynamics of various nonlinear phenomena and processes. Even those particular exact solutions to differential equations that do not have a clear physical meaning can serve as a basis for formulating test problems designed to check the correctness and assess the accuracy of various numerical, asymptotic, and approximate analytical methods. Besides, model equations and problems admitting exact solutions serve as a basis for developing new numerical, asymptotic, and approximate methods, which, in turn, make it possible to study more complex problems that do not have an exact analytical solution. Exact methods and solutions are also needed to develop and improve the corresponding sections of computer programs intended for analytical calculations (with computer algebra systems such as Mathematica, Maple, Maxima, among others). It is important to note that many equations of applied and theoretical physics, chemistry, and biology contain empirical parameters or empirical functions. Exact solutions allow one to plan experiments to determine these parameters or functions by artificially creating suitable (boundary and initial) conditions. Throughout the book, exact solutions to nonlinear PDEs refer to the following: (i) Solutions that are expressed in terms of elementary functions. xi

xii

P REFACE

(ii) Solutions that are expressed by quadrature.∗ (iii) Solutions that are expressed through solutions to ordinary differential equations (ODEs) or systems of such equations. Combinations of cases (i) and (iii) as well as (ii) and (iii) are also allowed. The simplest case (i) is purposely isolated from the more general case (ii), since some authors limit themselves to seeking only such exact solutions. In cases (i) and (ii), an exact solution can also be represented in explicit, implicit, or parametric form. In all cases, the solutions can involve arbitrary functions apart from constants of integration. Exact methods for solving nonlinear PDEs are understood as methods that allow obtaining exact solutions. The most common and highly effective exact methods for solving nonlinear partial differential equations are listed below in the summary table. These methods have a wide range of applicability, allowing one to construct exact solutions to various types of nonlinear PDE of different orders. There are numerous publications in which a large number of exact solutions have been obtained using these methods. Remark 1. The most commonly used methods are the methods of group analysis and inverse scattering problem (according to Internet keyword searches). An extensive literature is devoted to the description of these methods; see, for example, the books [34, 35, 150, 225, 231] (group analysis) and [2, 3, 51, 84, 98, 222, 238] (inverse scattering problem). Remark 2. In fluid dynamics and the theory of heat and mass transfer,∗∗ only the first six methods shown in the table work effectively.

The book focuses on describing and utilizing the methods of generalized (nonlinear) and functional separation of variables. These are among the most effective methods for constructing exact solutions to nonlinear PDEs of a reasonably general form dependent on one or more arbitrary functions. Notably, these nonlinear PDEs are most challenging to analyze and construct exact solutions. The book also details the direct method of symmetry reductions (which is akin to the methods of functional separation of variables) and its more general version, based on using the splitting principle. Besides, the method of differential constraints is presented, which generalizes many other exact methods. The effectiveness of the methods is compared. Remark 3. Exact solutions obtained by the methods of generalized and functional separation of variables, as a rule, cannot be obtained using the classical method of symmetry reductions, method of inverse scattering problem, and the method of truncated Painlev´e expansions.

The presentation is accompanied by numerous specific examples, in which the authors tried to give informal comments and express the considerations that we used ∗ Integration of differential equations in closed form is a representation of solutions to differential equations by analytical formulas, which are written using a set of admissible functions specified a priori and a set of mathematical operations listed in advance. A solution is expressed by quadrature (or in quadratures) if elementary functions and the functions included in the equation are used as admissible functions (this is necessary when the equation depends on arbitrary or special functions), and the admissible operations are a finite set of arithmetic operations, composition operations (to make composite functions), operations of differentiation, and operations of taking an indefinite integral. ∗∗ Here we mean the search for exact solutions to the Navier–Stokes equations and hydrodynamic boundary layer equations.

P REFACE

xiii

Table Basic methods for seeking exact solutions to nonlinear partial differential equations No. Name of method

Characteristic features

1

Classical method of symmetry reductions (a Lie group analysis method)

Bases on seeking one-parameter Lie groups of continuous transformations that preserve the form of the PDE. Allows one to obtain self-similar and other invariant solutions

2

Nonclassical method of symmetry reductions (allows various modifications)

Generalizes the classical method of symmetry reductions on the basis of an invariant surface condition. Allows one to describe a wide class of exact solutions, but is more complex in practical use

3

Direct method of symmetry reductions (Clarkson–Kruskal method)

Starts with setting the general form of solutions with several free functions. Employs special techniques to determine these functions. One of the functions must satisfy a single ODE

4

Method of differential constraints

Bases on a compatibility analysis of the PDE in question and one or more auxiliary (usually simpler) differential equations, called differential constraints

5

Methods of generalized separation of variables

Starts with seeking solutions as the sum of pairwise products of unknown functions with different arguments. Employs a few different methods to determine the unknown functions

6

Methods of functional separation of variables

Starts with setting the (explicit or implicit) form of solutions with several free functions. Employs a few different methods to determine the unknown functions

7

Method of inverse scattering problem (soliton theory)

Bases on a special representation of the equation (using a Lax pair of linear operators) or on a compatibility condition of two linear differential equations

8

Method of truncated Painlev´e expansions

Bases on seeking solutions in the form of truncated Painlev´e expansions containing a movable pole singularity. The position of the pole is defined by an arbitrary function

to construct specific solutions. To illustrate the wide range of applicability of the methods, we consider many second-order and higher-order nonlinear PDEs. When selecting suitable practical material, the authors gave preference to the following two essential types of PDE: • nonlinear equations that arise in various applications (theory of heat and mass transfer, wave theory, fluid dynamics, gas dynamics, combustion theory, nonlinear optics, biology, chemical engineering science, etc.) and

xiv

P REFACE • nonlinear equations of a reasonably general form that involve arbitrary functions (exact solutions to such equations are of significant interest for testing numerical and approximate analytical methods).

Notably, the overwhelming majority of known general solutions to nonlinear ODEs are presented in an implicit or parametric form (a similar conclusion follows from the statistical processing of most comprehensive reference books on exact solutions of ODEs [273, 276]). This circumstance allows us to state a plausible hypothesis that nonlinear PDEs also admit exact solutions (by quadrature) in an implicit or parametric form more often than in explicit form.∗ Therefore, this book includes direct methods for constructing functional separable solutions in an implicit form developed in the last few years; a characteristic qualitative feature of these methods is that they usually allow one to obtain solutions in closed form. Overall, this book provides a lot of new material previously unpublished in monographs. To maximize the potential readership with different mathematical backgrounds, the authors tried, whenever possible, to avoid specialized terminology. Therefore, the presentation of some results is schematic and simplified; however, this should suffice for their use in most applications. Many sections can be read independently of each other, making it easier to work with the material. The authors hope that the book will be useful for a broad audience of researchers, university professors, engineers, graduate students, and students specializing in applied and computational mathematics, theoretical physics, mechanics, control theory, biology, and chemical engineering science. Individual sections of the book and examples can help prepare lecture courses on equations of mathematical physics, methods of mathematical physics, or PDEs and organize special courses and practical exercises.

Andrei D. Polyanin Alexei I. Zhurov

∗ In particular, the currently known nonlinear PDEs involving one or more arbitrary functions of the dependent variable do not have nondegenerate solutions that allow an explicit representation.

Authors Andrei D. Polyanin received his Ph.D. in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences where he has been working since 1975. He is also professor of Applied Mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. He is an author of more than 30 books and over 270 articles and holds three patents. His books include A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 and 2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998 and 2008; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004 and 2012; A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2007; A. D. Polyanin and V. F. Zaitsev, Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, CRC Press, 2018. Alexei I. Zhurov is a senior research scientist at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, where he received his Ph.D. in theoretical and fluid mechanics in 1995. Since 2001, he has joined Cardiff University as a research scientist in the area of Biomechanics and Morphometrics. Dr. Zhurov has published over 120 research articles and five books, including Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics by A. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov (Fizmatlit, 2005; in Russian).

xv

Some Notations and Remarks Latin Characters C1 , C2 , . . . t u x, y, z x1 , . . . , xn x

arbitrary constants; time (t ≥ 0); unknown function (dependent variable); space variables (Cartesian coordinates); Cartesian coordinates in n-dimensional space; n-dimensional vector, x = (x1 , . . . , xn ).

Greek Characters ∆

Laplace operator: ∂2 ∂2 ∆ = ∂x two-dimensional case, 2 + ∂y 2 ∂2 ∂2 ∂2 + + three-dimensional case, ∆ = ∂x 2 ∂y 2 ∂z 2 n P 2 ∂ n-dimensional case. ∆= ∂x2 k=1

∆∆

k

biharmonic operator: ∂4 ∂4 ∆∆ = ∂x 4 + 2 ∂x2 ∂y 2 +

∂4 ∂y 4

two-dimensional case.

Short Notation for Derivatives Partial derivatives of a function u = u(x, t): ux =

∂u ∂2u ∂nu ∂u ∂2u ∂2u (n) , u = , . . . , u = . , ut = , uxx = , u = xt tt x ∂x ∂t ∂x2 ∂x∂t ∂t2 ∂xn

Ordinary derivatives of a function f = f (x): fx′ =

df d3 f d4 f dn f d2 f ′′′ ′′′′ (n) ′′ , f = , f = , f = for n > 4. , fxx = xxx xxxx x dx dx2 dx3 dx4 dxn

Remarks 1. The book often uses the abbreviations ODE and PDE, which stand for ‘ordinary differential equation’ and ‘partial differential equation,’ respectively. xvii

xviii

S OME N OTATIONS AND R EMARKS

2. If a formula or a solution involves a derivatives of a function, it is assumed that the derivative exists. 3. If a formula or a solution involves an indefinite or definite integral, it is assumed that the integral exists. f (x) , appearing in formulas or solutions imply a−2 that a 6= 2; this is not usually stated explicitly. 4. Expressions of the form

5. The simple solutions that only involve a single independent variable appearing in the original equation are omitted throughout the book. 6. In general, the book adopts a simple and straightforward classification of the most common solutions by their appearance, which is unrelated to the type or appearance of the equations concerned (see the table below). Table Most common types of exact solutions for equations with two independent variables, x and t, and one unknown function, u No. Type of solution

Solution structure (x and t can be swapped)

1

Additive separable solution

u = ϕ(x) + ψ(t)

2

Multiplicative separable solution

u = ϕ(x)ψ(t)

∗

u = U (z), z = αx + βt, αβ 6= 0

3

Traveling wave solution

4

Self-similar solution

u = tα F (z), z = xtβ

5

Generalized self-similar solution

u = ϕ(t)F (z), z = ψ(t)x

6

Generalized traveling wave solution

u = U (z), z = ϕ(t)x + ψ(t)

7

Generalized separable solution

u = ϕ1 (x)ψ1 (t) + · · · + ϕn (x)ψn (t)

8

Functional separable solution (special case)

u = U (z), z = ϕ(x) + ψ(t)

9

Functional separable solution

u = U (z), z = ϕ1 (x)ψ1 (t) + · · · + ϕn (x)ψn (t)

∗

Both independent variables can play the role of space coordinates.

1. Methods of Generalized Separation of Variables 1.1. Simple Separable Solutions 1.1.1. Multiplicative and Additive Separable Solutions Linear equations of mathematical physics. The method of generalized separation of variables is the most common analytical method for solving linear equations of mathematical physics [54, 172, 261, 262, 344, 388]. For equations with two independent variables, x and t, and one unknown function, u = u(x, t), this method suggests searching for exact solutions as the product of functions with different arguments: u = ϕ(x)ψ(t).

(1.1.1.1)

The functions ϕ = ϕ(x) and ψ = ψ(t) are described by linear ordinary differential equations (ODEs) and determined in the course of a subsequent analysis. ◮ Example 1.1. Let us look at the linear heat equation

ut = uxx .

(1.1.1.2)

Its exact solutions are sought in the product form (1.1.1.1). Substituting (1.1.1.1) into (1.1.1.2) gives ϕψt′ = ψϕ′′xx . (1.1.1.3) Separating the variables by dividing both sides of this equations by ϕψ, we get ϕ′′ ψt′ = xx . ψ ϕ

(1.1.1.4)

The left-hand side of this equation only depends on the variable t, while the righthand side depends on x alone. This is possible only if both sides of equation (1.1.1.4) are individually equal to the same constant quantity, so that ψt′ = C, ψ

ϕ′′xx = C, ϕ

(1.1.1.5)

where C is the so-called constant of separation, which is a free parameter. For C = −λ2 < 0, the general solutions to ODEs (1.1.1.5) are given by ϕ = A1 cos(λx) + A2 sin(λx), DOI: 10.1201/9781003042297-1

ψ = A3 exp(−λ2 t),

(1.1.1.6) 1

2

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

where A1 , A2 , and A3 are arbitrary constants. Since ϕ and ψ appear in solution (1.1.1.1) as a product, the constant A3 can be set, without loss of generality, equal to unity. To the different values of λ in (1.1.1.6), e.g., λ = λ1 , . . . , λ = λn , there correspond different solutions. Since equation (1.1.1.2) is linear, these solutions can be added together by virtue of the linear superposition principle. As a result, an exact solution to equation (1.1.1.2) can be written as the sum u=

n X

ϕk (x)ψk (t),

(1.1.1.7)

k=1

where ϕk (x) = Ak1 cos(λk x) + Ak2 sin(λk x),

ψk (t) = Ak3 exp(−λ2k t),

(1.1.1.8)

with Ak1 , Ak2 , and Ak3 being arbitrary constants. The solution of initial-boundary value problems for equation (1.1.1.2) on a finite closed interval x1 ≤ x ≤ x2 is sought in the form of the infinite series (1.1.1.7) with n = ∞. The constants λk , Ak1 , and Ak2 are determined from the boundary conditions (as well as a normalization condition), while the constants Ak3 are determined ◭ from the initial condition [261, 262, 344]. Remark 1.1. Equation (1.1.1.3) is a simple functional differential equation. In general, functional equations are equations that involve functions with different arguments, while functional differential equations involve functions with different arguments and derivatives of these functions.

Nonlinear first-order partial differential equations. The integration of isolated classes of nonlinear first-order partial differential equations (PDEs) is based on seeking exact solutions in the form of the sum of functions with different arguments [161, 261, 277]: u = ϕ(x) + ψ(t). (1.1.1.9) Many nonlinear first-, second-, and higher-order PDEs also admit solutions of the from (1.1.1.9). ◮ Example 1.2. The free vertical drop of a point body near the Earth’s surface is described by the nonlinear first-order PDE (Hamilton–Jacobi equation) [131, 202]:

ut + au2x = bx,

(1.1.1.10)

where u is Hamilton’s principal function (often denoted S in analytical mechanics), 1 t is time, x is the vertical coordinate measured downward, m = 2a is the mass of the body, and g = 2ab is the gravitational acceleration. We seek an exact solution to equation (1.1.1.10) as the sum of functions with different arguments (1.1.1.9). Substituting (1.1.1.9) into (1.1.1.10) and moving the term a(ϕ′x )2 to the right-hand side, we find that ψt′ = −a(ϕ′x )2 + bx.

(1.1.1.11)

3

1.1. Simple Separable Solutions

The left-hand side of this equation only depends on t, while the right-hand side depends on x alone. Just as in Example 1.1, equating both sides of (1.1.1.11) with the same constant, C, we arrive at the ODEs ψt′ = C,

−a(ϕ′x )2 + bx = C.

(1.1.1.12)

The general solutions of these equations are expressed as ψ = Ct + C2 ,

2a ϕ=± 3b



bx − C a

3/2

+ C3 ,

(1.1.1.13)

where C2 and C3 are arbitrary constants. Substituting (1.1.1.13) into (1.1.1.9), renaming C = −C1 , and setting C3 = 0 (either function in (1.1.1.13) contains an arbitrary additive constant, so when the two functions are added together, one of the constants can be set equal to zero), we obtain the following exact solution to equation (1.1.1.10): 2a u = −C1 t ± 3b



bx + C1 a

3/2

+ C2 ,

where C1 and C2 are arbitrary constants. In the theory of nonlinear partial differential equations in two independent variables, such solutions with two arbitrary constants are called complete integrals. These solutions allow one to construct ◭ general solutions to the equations concerned in parametric form [161, 277]. Nonlinear second- and higher-order equations of mathematical physics can also have exact solutions of the form (1.1.1.1) or (1.1.1.9). Such solutions will be referred to as multiplicative separable or additive separable, respectively [274, 278].

1.1.2. Simple Cases of Separation of Variables in Nonlinear Partial Differential Equations Nonlinear equations in two independent variables. In simple cases, separation of variables in nonlinear partial differential equations in two independent variables is carried out following the same scheme as in linear equations. Exact solutions are sought as the product or sum of functions with different arguments. Substituting (1.1.1.1) or (1.1.1.9) into the equation concerned and performing a simple rearrangement, one arrives at a single equation (for equations in two independent variables) with either side dependent on a single variable. The equality is possible only if both sides equal the same constant quantity. As a result, one obtains two ordinary differential equations for the unknown functions, one for ϕ = ϕ(x) and the other one for ψ = ψ(t). Exact solutions obtained using this kind of separation of variables will be referred to as simple separable solutions. Remark 1.2. Exact solutions of the form (1.1.1.1) or (1.1.1.9) are not simple separable solutions if at least one of the unknown functions, ϕ or ψ , is described by an overdetermined system of ordinary differential equations.

4

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Let us illustrate the above with specific examples. ◮ Example 1.3. The heat equation with a power-law nonlinearity

ut = a(uk ux )x

(1.1.2.1)

admits an exact solution as the product of functions with different arguments. Indeed, substituting (1.1.1.1) into (1.1.2.1), we get ϕψt′ = aψ k+1 (ϕk ϕ′x )′x . Dividing both sides by ϕψ k+1 gives a(ϕk ϕ′x )′x ψt′ = . ψ k+1 ϕ The left-hand side of this equation only depends on t, while the right-hand side depends on x alone. The equality is possible only if ψt′ = C, ψ k+1

a(ϕk ϕ′x )′x = C, ϕ

(1.1.2.2)

where C is an arbitrary constant. The solution of the first ODE in (1.1.2.2) is expressed in terms of elementary functions.∗ The other ODE can be solved in implicit form. The procedure for finding separable solutions of the form (1.1.1.1) to the nonlinear equation (1.1.2.1) is very much the same as that for solving the linear heat equation (1.1.1.2) and other linear partial differential equations. The fundamental difference between linear and nonlinear differential equations is that the superposition principle does not apply to solutions of nonlinear equations. This means that solutions (1.1.1.1) of equations (1.1.2.1), obtained by integrating ◭ ODE (1.1.2.2) and taken at different C, cannot be added together. ◮ Example 1.4. The wave equation with an exponential nonlinearity

utt = a(eλu ux )x

(1.1.2.3)

has an additive separable solution. By substituting (1.1.1.9) into (1.1.2.3) and dividing the resulting equation by eλψ , we obtain ′′ e−λψ ψtt = a(eλϕ ϕ′x )′x .

(1.1.2.4)

The left-hand side of this equation only depends on t, while the right-hand side depends on x alone. Equating both sides of (1.1.2.4) with the same constant, C, we get ′′ = C, a(eλϕ ϕ′x )′x = C. (1.1.2.5) e−λψ ψtt ∗ This book is devoted to the methods for constructing exact solutions to nonlinear partial differential equations. Therefore, to not distract the reader from the main topic, we will often omit solutions to the much simpler, ordinary differential equations arising in the last stage of the analysis (such as those in Examples 1.3 and 1.4). Solution methods for ordinary differential equations are described in the handbooks [162, 219, 273, 276], where one can find a large number of solutions to similar and more complicated equations.

1.1. Simple Separable Solutions

5

Both ODEs (1.1.2.5) are autonomous (i.e., implicitly independent of the respective variables), and hence, their order can be reduced [273, 276] to obtain simpler equations. Moreover, the second equation in (1.1.2.5) directly reduces, by a single integration, to a first-order separable ODE. As a result, one can obtain a solution to ◭ equation (1.1.2.3) of the form (1.1.1.9) in terms of elementary functions. ◮ Example 1.5. The heat equation in an anisotropic medium with a logarithmic

source [f (x)ux ]x + [g(y)uy ]y = au ln u

(1.1.2.6)

admits a multiplicative separable solution u = ϕ(x)ψ(y).

(1.1.2.7)

Indeed, substituting (1.1.2.7) into (1.1.2.6) followed by dividing the resulting equation by ϕψ and moving some terms to the left- or right-hand side, we obtain 1 1 [f (x)ϕ′x ]′x − a ln ϕ = − [g(y)ψy′ ]′y + a ln ψ. ϕ ψ The left-hand side of this equation only depends on x, while the right-hand side depends on y alone. Equating these with the same constant, one obtains ordinary differential equations for ϕ(x) and ψ(y). ◭ Table 1.1 gives other examples of simple, additive or multiplicative, separable solutions for some nonlinear equations. Nonlinear PDEs with three or more independent variables. Nonlinear equations of mathematical physics with three or more independent variables can also have multiplicative or additive separable solutions. ◮ Example 1.6. The nonlinear heat equation of arbitrary dimension

ut = a

  n X ∂u ∂ uk ∂xi ∂xi i=1

(1.1.2.8)

admits a multiplicative separable solution of the form u = t−1/k ϕ(x1 , . . . , xn ),

(1.1.2.9)

with the function ϕ = ϕ(x1 , . . . , xn ) described by the stationary equation   n X 1 ∂ k ∂ϕ a ϕ + ϕ = 0. ∂x ∂x k i i i=1

◭

Systems of nonlinear equations of mathematical physics. Some systems of nonlinear equations of mathematical physics also have simple separable solutions. We will illustrate this with a specific example.

6

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Table 1.1. Some nonlinear equations of mathematical physics that admit simple separable solutions (C, C1 , and C2 are arbitrary constants) Equation

Equation name

Form of solutions

Determining equations

ut = auxx +bu ln u

Heat equation with source

u = ϕ(x)ψ(t)

aϕ′′ xx/ϕ−b ln ϕ = −ψt′ /ψ+b ln ψ = C

ut = a(uk ux)x +bu

Heat equation with source

u = ϕ(x)ψ(t)

ut = a(uk ux)x +buk+1

Heat equation with source

u = ϕ(x)ψ(t)

ut = a(eλuux)x +b

Heat equation with source

u = ϕ(x)+ψ(t)

e−λψ (ψt′ −b) = a(eλϕϕ′x)′x = C

ut = a(eu ux)x +beu

Heat equation with source

u = ϕ(x)+ψ(t)

e−ψ ψt′ = a(eϕϕ′x)′x +beϕ = C

ut = auxx +bu2x

Potential Burgers equation

u = ϕ(x)+ψ(t)

′ 2 ψt′ = aϕ′′ xx +b(ϕx) = C

ut = aukxuxx

Filtration equation

u = ϕ(x)+ψ(t), u = f (x)g(t)

ut = f (ux)uxx

Filtration equation

u = ϕ(x)+ψ(t)

ψt′ = f (ϕ′x )ϕ′′ xx = C

utt = a(uk ux)x

Wave equation

u = ϕ(x)ψ(t)

′′ /ψ k+1 = a(ϕk ϕ′ )′ /ϕ = C ψtt x x

utt = a(eλuux)x

Wave equation

u = ϕ(x)+ψ(t)

′′ = a(eλϕϕ′ )′ = C e−λψ ψtt x x

utt = auxx +bu ln u

Wave equation with source

u = ϕ(x)ψ(t)

′′ ψtt /ψ−b ln ψ = aϕ′′ xx/ϕ+b ln ϕ = C

uxx +a(uk uy )y = 0

Anisotropic steady heat equation

u = ϕ(x)ψ(y)

k+1 = −a(ψ k ψ ′ )′ /ψ = C ϕ′′ xx/ϕ y y

uxx +auy uyy = 0 u2xy = uxxuyy ut = auxxx +bu2x uy uxy −uxuyy = auyyy

Equation of steady u = ϕ(x)+ψ(y), transonic gas flow u = f (x)g(y) Monge–Amp`ere equation

= a(ϕk ϕ′x)′x/ϕ = C ψt′ /ψk+1 = a(ϕk ϕ′x)′x/ϕ+bϕk = C

ψt′ = a(ϕ′x )k ϕ′′ xx = C1 , ′′ gt′ /g k+1 = a(fx′ )k fxx /f = C2

′ ′′ ϕ′′ xx = −aψy ψyy = C1, ′′ ′′ fxx /f = −agy′ gyy /g = C2

′′ ′′ u = ϕ(x)+ψ(y), ϕxx = 0 or ψyy = 0, ′ 2 ′′ ′′ u = f (x)g(y) (fx) /(f fxx ) = ggyy /(gy′ )2 = C

Potential Korteweg– u = ϕ(x)+ψ(t) de Vries equation Boundary layer equation

(ψt′ −bψ)/ψk+1

′ 2 ψt′ = aϕ′′′ xxx +b(ϕx) = C

′ ′′′ ′′ u = ϕ(x)+ψ(y), −ϕx = aψyyy /ψyy = C1, ′ ′′′ ′ 2 ′′ −1 u = f (x)g(y) fx = agyyy[(gy ) −ggyy ] = C2

◮ Example 1.7. Let us look at the nonlinear system composed of two reaction– diffusion type equations ut = auxx + uf (u/v), (1.1.2.10) vt = bvxx + vg(u/v),

which involve two arbitrary functions, f and g, as well as two constants, a and b.

7

1.1. Simple Separable Solutions

We look for solutions to system (1.1.2.10) in the multiplicative form u = ϕ1 (x)ψ1 (t),

v = ϕ2 (x)ψ2 (t).

(1.1.2.11)

We divide equations (1.1.2.10) by u and v, respectively, and insert (1.1.2.11) to obtain the functional differential equations     ϕ1 ψ1 ϕ1 ψ1 (ϕ1 )′′xx (ϕ2 )′′xx (ψ1 )′t (ψ2 )′t =a +f =b +g , . ψ1 ϕ1 ϕ2 ψ2 ψ2 ϕ2 ϕ2 ψ2 (1.1.2.12) Their left-hand sides only depend on t, while the first terms on the right-hand sides depend on x alone. The variables are separated if both functions f (. . .) and g(. . .) are only dependent on x or t. Let us explore the two possibilities in order. 1◦ . By setting ψ1 (t) = ψ2 (t) = ψ(t) in (1.1.2.12) and separating the variables, we arrive at the following three equations: ψt′ = Cψ, a(ϕ1 )′′xx − Cϕ1 + ϕ1 f (ϕ1 /ϕ2 ) = 0, b(ϕ2 )′′xx − Cϕ2 + ϕ2 g(ϕ1 /ϕ2 ) = 0,

(1.1.2.13) (1.1.2.14) (1.1.2.15)

where C is an arbitrary constant. Integrating equation (1.1.2.13) gives a multiplicative separable solution to system (1.1.2.10): u = exp(Ct)ϕ1 (x),

v = exp(Ct)ϕ2 (x),

with the functions ϕ1 = ϕ1 (x) and ϕ2 = ϕ2 (x) satisfying the nonlinear systems of second-order ODEs (1.1.2.14)–(1.1.2.15). Below we specify two simple families of exact solutions to the systems of ordinary differential equations (1.1.2.14)–(1.1.2.15). (i) A family of solutions with trigonometric functions: ϕ1 (x) = k[A cos(λx) + B sin(λx)],

ϕ2 (x) = A cos(λx) + B sin(λx),

where A and B are arbitrary constants; the constants k and λ are determined from the algebraic (or transcendental) system of equations −aλ2 − C + f (k) = 0, −bλ2 − C + g(k) = 0.

(ii) A family of solutions with exponentials: ϕ1 (x) = k(Ae−λx + Beλx ),

ϕ2 (x) = Ae−λx + Beλx ,

where A and B are arbitrary constants; the constants k and λ are determined from the algebraic (or transcendental) system of equations aλ2 − C + f (k) = 0, bλ2 − C + g(k) = 0.

8

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

2◦ . By setting ϕ1 (x) = ϕ2 (x) = ϕ(x) in (1.1.2.12) and separating the variables, we arrive at another system of ODEs ϕ′′xx = Cϕ, (ψ1 )′t = aCψ1 + ψ1 f (ψ1 /ψ2 ), (ψ2 )′t = bCψ2 + ψ2 g(ψ1 /ψ2 ).

(1.1.2.16) (1.1.2.17) (1.1.2.18)

Integrating equation (1.1.2.16), we obtain, depending on the sign of C, two different nondegenerate multiplicative separable solutions to the system of PDEs (1.1.2.10): (i) if C = −λ2 < 0,

u = [A cos(λx) + B sin(λx)]ψ1 (t), v = [A cos(λx) + B sin(λx)]ψ2 (t), (1.1.2.19)

(ii) if C = λ2 > 0, u = (Ae−λx + Beλx )ψ1 (t), v = (Ae−λx + Beλx )ψ2 (t),

(1.1.2.20)

where A and B are arbitrary constants; the functions ψ1 = ψ1 (t) and ψ2 = ψ2 (t) satisfy the nonlinear system of second-order ODEs (1.1.2.17)–(1.1.2.18). Note that system (1.1.2.17)–(1.1.2.18) admits an exponential exact solution with ◭ ψ1 = A1 eβt and ψ2 = A2 eβt . Some generalizations. Below are two propositions [8] that allow one to generalize simple separation solutions of special forms. Proposition 1. Suppose that the equation F (u, ux , ut , uxx , uxt , utt , . . .) = 0

(1.1.2.21)

admits a simple separable solution of the special form u = tβ ϕ(x),

β 6= 0,

(1.1.2.22)

which is invariant (remains the same) under the dilation transformation t 7−→ λt,

u 7−→ λβ u (λ > 0 is an arbitrary constant).

(1.1.2.23)

Suppose equation (1.1.2.21) is also invariant under transformation (1.1.2.23). Then equation (1.1.2.21) admits a more complicated solution of the form u = (t + C1 )β θ(z),

z = x + C2 ln |t + C1 | + C3 ,

where C1 , C2 , and C3 are arbitrary constants. ◮ Example 1.8. The nonlinear wave equation

utt = a(uk ux )x

(1.1.2.24)

has a multiplicative separable solution of the form (1.1.2.22) with β = −2/k. Therefore, equation (1.1.2.24) also has a more complicated solution u = (t + C1 )−2/k θ(z),

z = x + C2 ln |t + C1 | + C3 .

The function θ = θ(z) is determined by the ordinary differential equation k+4 2(k + 2) ′′ θ− C2 θz′ + C22 θzz = a(θk θz′ )′z . k2 k

◭

9

1.1. Simple Separable Solutions

◮ Example 1.9. The nonlinear wave equation (1.1.2.24) also admits a solution

of the form u = x2/k ψ(t),

(1.1.2.25)

in which the function ψ = ψ(t) satisfies the autonomous ODE ′′ ψtt = 2a(k + 2)k −2 ψ k+1 .

Then, by swapping the independent variables, x ⇄ t, in Proposition 1, we conclude that equation (1.1.2.24) also has a more general solution of the form u = (x + C1 )2/k ω(ζ),

ζ = t + C2 ln |x + C1 | + C3 ,

in which the function ω = ω(ζ) is described by the ODE ′′ ωζζ = a(k + 2)k −2 ω k (2ω + C2 kωζ′ ) + aC2 k −1 [ω k (2ω + C2 kωζ′ )]′ζ .

◭

Proposition 1 can also be applied to equations in three or more independent variables. ◮ Example 1.10. The nonlinear heat equation in n coordinates (1.1.2.8) has a multiplicative separable solution of the form (1.1.2.9). The equation and its solutions are invariant under transformation (1.1.2.23) with β = −1/k. It follows that equation (1.1.2.8) also admits a more complex solution of the form

u = (t + C)−1/k θ(z1 , . . . , zn ),

zi = xi + Ai ln(t + C) + Bi ,

i = 1, . . . , n, (1.1.2.26) where Ai , Bi , and C are arbitrary constants. Substituting (1.1.2.26) into (1.1.2.8) yields the following stationary equation for θ = θ(z1 , . . . , zn ):   X n n X ∂ ∂θ ∂θ 1 a Ai θk − + θ = 0. ∂z ∂z ∂z k i i i i=1 i=1 ◭ Proposition 2. Suppose that equation (1.1.2.21) does not change under scaling of the unknown function u 7−→ λu, (1.1.2.27)

where λ > 0 is an arbitrary constant. Then equation (1.1.2.21) admits an exact solution of the form u = ekt θ(z), z = px + qt, (1.1.2.28) where k, p, and q are arbitrary constants (pq 6= 0). ◮ Example 1.11. The nonlinear heat-type equation

ut = auxx + uf ux /u




(1.1.2.29)

is invariant under the scaling transformation (1.1.2.27). Therefore, by virtue of Proposition 2, this equation has a solution of the form (1.1.2.28), where the function θ = θ(z) satisfies the nonlinear ordinary differential equation ′′ kθ + qθz′ = ap2 θzz + θf (pθz′ /θ).

◭

10

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

1.1.3. Examples of Nontrivial Separation of Variables in Nonlinear Equations In many cases, the variables in nonlinear partial differential equations get separated differently than in linear equations. To illustrate this fact, below we will give a few specific examples of nonlinear equations that admit solutions of the form (1.1.1.1) or (1.1.1.9), which, however, do not belong to simple separable solutions. ◮ Example 1.12. Consider the second-order equation with a cubic nonlinearity

ut = f (t)uxx + uu2x − au3 ,

(1.1.3.1)

where f (t) is an arbitrary function. We look for exact solutions as the product of functions with different arguments. Substituting (1.1.1.1) into (1.1.3.1) and dividing the resulting equation by f (t)ϕ(x)ψ(t), we obtain ψt′ ϕ′′ ψ2 = xx + [(ϕ′x )2 − aϕ2 ]. fψ ϕ f

(1.1.3.2)

The left-hand side of the functional differential equation (1.1.3.2) only depends on t. Its right-hand side contains the sum of two terms, one of which depends on x alone, while the other is the product of two functions with different arguments. There is no way to represent equation (1.1.3.2) as the equality of two functions with different arguments. This situation differs significantly from that in examples given in Subsections 1.1.1 and 1.1.2. However, equation (1.1.3.1) yet may have solutions of the form (1.1.1.1). 1◦ . It can be directly verified that the functions h Z i √
ϕ(x) = C exp ±x a , ψ(t) = exp a f (t) dt ,

(1.1.3.3)

where C is an arbitrary constant, define two solutions to the functional differential equation (1.1.3.2) with a > 0. Both solutions for ϕ nullify the expression in square brackets in (1.1.3.2), which allows us to separate the variables. 2◦ . For a > 0, there is a more general solution to the functional differential equation (1.1.3.2): √
√
ϕ(x) = C1 exp x a + C2 exp −x a ,  −1/2 Z Z , F = a f (t) dt, ψ(t) = eF C3 + 8aC1 C2 e2F dt

where C1 , C2 , and C3 are arbitrary constants. In this case, the function ϕ = ϕ(x) happens to have the property that both x-dependent terms in equation (1.1.3.2) are simultaneously constant: ϕ′′xx /ϕ = a,

(ϕ′x )2 − aϕ2 = 4aC1 C2 .

This is the circumstance that allows us to separate the variables. Note that the function ψ = ψ(t) satisfies the Bernoulli equation ψt′ = af (t)ψ − 4aC1 C2 ψ 3 .

11

1.1. Simple Separable Solutions

3◦ . There is another solution to the functional differential equation (1.1.3.2) with a > 0: √
√
ϕ(x) = C1 sin x −a + C2 cos x −a , h i−1/2 Z Z , F = a f (t) dt, ψ(t) = eF C3 + 2a(C12 + C22 ) e2F dt

where C1 , C2 , and C3 are arbitrary constants. The function ϕ = ϕ(x) has the property that both x-dependent terms in equation (1.1.3.2) are simultaneously constant, while ◭ ψ = ψ(t) satisfies the Bernoulli equation ψt′ = af (t)ψ − a(C12 + C22 )ψ 3 . ◮ Example 1.13. Consider the third-order partial differential equation with a

quadratic nonlinearity uy uxx + aux uyy = buxxx + cuyyy .

(1.1.3.4)

We will look for exact solutions to equation (1.1.3.4) using additive separation of variables: u = f (x) + g(y). (1.1.3.5) Substituting (1.1.3.5) into (1.1.3.4) gives the functional differential equation ′′ ′′ ′′′ ′′′ gy′ fxx + afx′ gyy = bfxxx + cgyyy ,

(1.1.3.6)

which cannot be rewritten as the equality of two functions with different arguments. However, it is easy to guess that the functional differential equation (1.1.3.6) holds if gy′ = C1 fx′

= C1

=⇒ g(y) = C1 y + C2 ,

f (x) = C3 eC1 x/b + C4 x

=⇒ f (x) = C1 x + C2 , g(y) = C3 e

aC1 y/c

(case 1),

+ C4 y (case 2),

where C1 , . . . , C4 are arbitrary constants. In these cases, two out of four terms in (1.1.3.6) vanish simultaneously, which allows the separation of variables. Equation (1.1.3.4) also admits a more complicated exact solution of the form (1.1.3.5): cλ x + C2 eλy − abλy + C3 , u = C1 e−aλx + a where C1 , C2 , C3 , and λ are arbitrary constants. The separation mechanism is different here: the two nonlinear terms on the left-hand side of (1.1.3.6) contain identical expressions that are only different in sign. Although inexpressible as the sum of functions with different arguments, these expressions cancel out when added together, which results in the separation of variables: +

′′ gy′ fxx ′′ afx′ gyy

= C1 C2 a2 λ3 eλy−aλx − C1 b(aλ)3 e−aλx = −C1 C2 a2 λ3 eλy−aλx + C2 cλ3 eλy

′′ ′′ ′′′ ′′′ gy′ fxx + afx′ gyy = −C1 b(aλ)3 e−aλx + C2 cλ3 eλy = bfxxx + cgyyy

◭

12

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

◮ Example 1.14. Let us look at the second-order equation with a cubic nonlin-

earity (1 + u2 )(uxx + uyy ) − 2uu2x − 2uu2y = au(1 − u2 ).

(1.1.3.7)

We seek an exact solution to equation (1.1.3.7) using multiplicative separation of variables: u = f (x)g(y). (1.1.3.8) Substituting this expression into (1.1.3.7) gives the functional differential equation ′′ ′′ (1 + f 2 g 2 )(gfxx + f gyy ) − 2f g[g 2(fx′ )2 + f 2 (gy′ )2 ] = af g(1 − f 2 g 2 ), (1.1.3.9)

which is inexpressible as the equality of functions with different arguments. Nevertheless, equation (1.1.3.7) admits a solution of the form (1.1.3.8). We will show that the functions f = f (x) and g = g(y) that satisfy the nonlinear first-order ordinary differential equations [337] (fx′ )2 = Af 4 + Bf 2 + C, (gy′ )2 = Cg 4 + (a − B)g 2 + A,

(1.1.3.10)

where A, B, and C are arbitrary constants, turn the functional differential equation (1.1.3.9) into an identity. Indeed, differentiating (1.1.3.10) gives two corollaries ′′ fxx = 2Af 3 + Bf, ′′ gyy = 2Cg 3 + (a − B)g.

(1.1.3.11)

Using (1.1.3.10) and (1.1.3.11) to eliminating the derivatives from (1.1.3.9), we arrive at an identity. It is noteworthy that the equations in (1.1.3.10) are integrable ◭ by quadrature. Remark 1.3. With the change of variable w = 4 arctan u, equation (1.1.3.7) can be reduced to a steady-state heat equation with a nonlinear sinusoidal source, ∆w = a sin w, where ∆ is the Laplace operator.

The above examples illustrate some characteristic features of multiplicative and additive separable solutions that are not simple separable solutions. Sections 1.3–1.5 will describe fairly general methods for constructing similar and more complicated exact solution to nonlinear partial differential equations.

1.2. Structure of Generalized Separable Solutions 1.2.1. General Form of Solutions. The Classes of Nonlinear Differential Equations of Interest For clarity and simplicity, we will restrict ourselves to the consideration of nonlinear equations of mathematical physics in two independent variables, x and y, and one dependent variable, u; one of the independent variables may be treated as time.

13

1.2. Structure of Generalized Separable Solutions

Linear partial differential equations. It is well known that linear equations of mathematical physics with constant coefficients and many linear equations with variable coefficients can have solutions as the sum of pairwise products of functions with different arguments as in Example 1.1 (see also [261, 262, 344]): u(x, y) = ϕ1 (x)ψ1 (y) + ϕ2 (x)ψ2 (y) + · · · + ϕn (x)ψn (y),

(1.2.1.1)

where ui = ϕi (x)ψi (y) are particular solutions. The functions ϕi (x), as well as the functions ψi (y), with the different subscripts i are unrelated to one another. It is noteworthy that linear partial differential equations often also admit exact solutions of the form (1.2.1.1) in which the pairwise products of functions with different arguments, ϕi (x)ψi (y), are not particular solutions of these equations. ◮ Example 1.15. The linear heat equation

ut = auxx, with y = t, admits the following solutions [262]: u = x2 + 2at, u = x3 + 6atx, u = x4 + 12atx2 + 12a2 t2 , n X (2n)(2n − 1) . . . (2n − 2k + 1) (at)k x2n−2k , u = x2n + k! k=1

u = x2n+1 +

n X (2n + 1)(2n) . . . (2n − 2k + 2) (at)k x2n−2k+1 , k! k=1

u = e−µx cos(µx) cos(2aµ2 t) + e−µx sin(µx) sin(2aµ2 t),

u = e−µx sin(µx) cos(2aµ2 t) − e−µx cos(µx) sin(2aµ2 t), where n is a positive integer and µ is an arbitrary constant. Individual terms in these ◭ solutions do not solve the heat equations. Nonlinear partial differential equations. Many nonlinear partial differential equations of mathematical physics with quadratic or power-law nonlinearities can be written as f1 (x)g1 (y)Π1 [u] + f2 (x)g2 (y)Π2 [u] + · · · + fm (x)gm (y)Πm [u] = 0, (1.2.1.2) where Πi [u] are differential forms that represent products of nonnegative integer powers of u and its partial derivatives ux , uy , uxx, uxy , uyy , uxxx , etc. Such equations also admit exact solutions of the form (1.2.1.1) (e.g., see [110–114, 275, 278, 346, 347]). These solutions will be referred to as generalized separable solutions. Unlike linear equations, the functions ϕi (x) with different subscripts i in nonlinear equations are often related to one another (and possibly to ψj (y)). In general, the functions ϕi (x) and ψj (y) are not known in advance and are subject to determination

14

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

in a subsequent analysis. Several examples of exact solutions to nonlinear equations of the form (1.2.1.1) are given in Subsections 1.1.2 and 1.1.3 for the simplest cases of n = 1 and n = 2 (with ψ1 = ϕ2 = 1). The methods for seeking exact solution of the form (1.2.1.1) to nonlinear PDEs will be referred to as methods of generalized separation of variables. It is noteworthy that, in practice, the most common structure of generalized separable solutions used to solve nonlinear equations of mathematical physics involves three unknown functions [110, 111, 275, 278]: u(x, y) = ϕ(x)θ(y) + ψ(x).

(1.2.1.3)

The variables x and y on the right-hand side can be swapped. In the special case ψ(x) = 0, this is a multiplicative separable solution, while if ϕ(x) = 1, this is a additive separable solution. Remark 1.4. The function θ = θ(y) in (1.2.1.3) is defined up to translation and dilation, ¯ 2 , where C1 and C2 are arbitrary constants, leads to since the linear transformation θ = C1 θ+C ¯ ¯ , in which ϕ a relation of the same form, u(x, y) = ϕ(x) ¯ θ(y)+ ψ(x) ¯ = C1 ϕ and ψ¯ = ψ +C2 ϕ. Remark 1.5. Expressions of the form (1.2.1.1) are frequently employed in applied and computational mathematics to construct analytical or numerical solutions of differential equations using Bubnov–Galerkin type projection methods [101, 102, 316]. Remark 1.6. Differential equations that have nonlinearities other than those in (1.2.1.2) can also admit solutions of the form (1.2.1.1) (see Example 1.38 in Section 1.5).

1.2.2. Functional Differential Equations Arising in Generalized Separation of Variables In general, after substituting solution (1.2.1.1) into the differential equation (1.2.1.2), one arrives at the following functional differential equation for determining ϕi (x) and ψi (y): Φ1 [X]Ψ1 [Y ] + Φ2 [X]Ψ2 [Y ] + · · · + Φk [X]Ψk [Y ] = 0,

(1.2.2.1)

where Φj [X] and Ψj [Y ] are functionals that depend on x and y, respectively:
Φj [X] ≡ Φj x, ϕ1 , ϕ′1 , ϕ′′1 , . . . , ϕn , ϕ′n , ϕ′′n ,
(1.2.2.2) Ψj [Y ] ≡ Ψj y, ψ1 , ψ1′ , ψ1′′ , . . . , ψn , ψn′ , ψn′′ .

For clarity, the formulas are written out for a second-order equation (1.2.1.2). For higher-order equations, the right-hand sides of formulas (1.2.2.2) will contain higherorder derivatives of ϕi and ψi . Sections 1.4 and 1.5 below will describe two fairly simple general methods for solving functional differential equations of the form (1.2.2.1)–(1.2.2.2). Section 1.3 will also present a simple but less general method that reduces the amount of computation. In addition, Section 1.6 will describe a fairly general special method that does not involve the analysis of functional differential equations (1.2.2.1)–(1.2.2.2). Remark 1.7. Unlike ordinary differential equations, equations of the form (1.2.2.1) involve several functions (and their derivatives) dependent on different arguments. These causes one to resort to special (nonstandard) solution methods.

15

1.3. Simplified Method for Constructing Generalized Separable Solutions

1.3. Simplified Method for Constructing Generalized Separable Solutions 1.3.1. Description of a Simplified Method Based on a Priori Setting of a System of Coordinate Functions For differential equations of the form (1.2.1.2) with a quadratic or power-law nonlinearity that are explicitly independent of y (all fi = const), a simplified approach can be used to find exact solutions. This approach suggests seeking solutions in the form of a finite sum (1.2.1.1) with preset coordinate functions ψi (y). The system of functions ψi (y) is assumed to be defined by a set of linear differential equations with constant coefficients. The most frequent solutions to these equations are ψi (y) = y i ,

ψi (y) = eλi y ,

ψi (y) = sin(αi y),

ψi (y) = cos(βi y). (1.3.1.1)

Finite sets of these functions (in various combinations) can be used to seek generalized separable solutions of the form (1.2.1.1), with the parameters λi , αi , and βi to be found. The other system of functions, ϕi (x), is determined by solving the nonlinear ODEs resulting from substituting solution (1.2.1.1) into the equation. This approach is not as general as the methods described below in Sections 1.4 and 1.5. However, the explicit specification of the coordinate functions {ψi (y)} makes the procedure of constructing exact solutions much simpler. It is noteworthy that some solutions of the form (1.2.1.1) may be lost with this approach. Importantly, the overwhelming majority of (generalized separable) solutions known so far for partial differential equations with quadratic nonlinearity are defined by coordinate functions of the form (1.3.1.1), most frequently with n = 2. Remark 1.8. The simplified method outlined here is similar to the method of undetermined coefficients, which is commonly used to find particular solutions to linear nonhomogeneous ordinary differential equations with a special right-hand side as well as to compute integrals, construct recurrence relations, and others [41, 69, 82]. Therefore, this approach may also be referred to as the method of undetermined functions.

1.3.2. Examples of Constructing Exact Solutions to Nonlinear Equations in Two Independent Variables We will give several specific examples demonstrating the application of the simplified method to constructing generalized separable solutions of nonlinear second- and third-order partial differential equations. ◮ Example 1.16. Consider the Guderley equation

uxx = auy uyy ,

(1.3.2.1)

which is employed to describe transonic gas flows [128], where γ = a − 1 is the adiabatic index.

16

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

1◦ . First group of solutions. We note right away that equation (1.3.2.1) has the obvious degenerate generalized separable solution u = (C1 x + C2 )y + C3 x + C4 ,

(1.3.2.2)

where C1 , . . . , C4 are arbitrary constants, which follows from the condition that uxx = uyy = 0. We will look for generalized separable solutions u(x, y) = ϕ(x)y k + ψ(x)

(1.3.2.3)

other than the degenerate solution (1.3.2.2). The functions ϕ(x) and ψ(x) and constant k 6= 0 are to be determined. It is noteworthy that similar two-term solutions to PDEs appear quite frequently in practice and are the simplest generalized separable solutions, along with solutions that have eλy instead of y k . On substituting (1.3.2.3) into (1.3.2.1) and on rearranging, we arrive at the equation ′′ ϕ′′xx y k − ak 2 (k − 1)ϕ2 y 2k−3 + ψxx = 0,

(1.3.2.4)

which must hold identically for any y. ′′ ′′ Let us look at the cases ψxx = 0 and ψxx 6= 0. ′′ (i) First case. For ψxx = 0, we get a two-term separable equation, which is satisfied if we set k = 3,

ϕ′′xx − 18aϕ2 = 0.

(1.3.2.5)

The general solutionR to the autonomous equation for ϕ(x) can be written in the implicit form x = ± (12aϕ3 +C1 )−1/2 dϕ+C2 . Furthermore, the equation admits a 1 (x+C1 )−2 , which generates a three-parameter power-law particular solution, ϕ = 3a exact solution to equation (1.3.2.1): u=

1 (x + C1 )−2 y 3 + C2 x + C3 . 3a

(1.3.2.6)

′′ (ii) Second case. The function ψxx 6= 0 can be balanced with the second term in (1.3.2.4) by setting k = 3/2. This results in a two-term equation, which can be satisfied with

ϕ′′xx = 0,

′′ ψxx =

9 8

aϕ2 .

(1.3.2.7)

These equations are easy to integrate resulting in a four-parameter exact solution: u = (C1 x + C2 )y 3/2 +

3a (C1 x + C2 )4 + C3 x + C4 . 32C12

(1.3.2.8)

2◦ . Titov’s solution (composite solution). It follows from (1.3.2.6) and (1.3.2.8) that equation (1.3.2.25) has two similar solutions, u1 = ϕ1 y 3/2 + ψ1 and u2 = ϕ2 y 3 + ψ2 , whose structures differ only in the exponent of y. This fact suggests that a more general solution to equation (1.3.2.25) may exist that would involve both

1.3. Simplified Method for Constructing Generalized Separable Solutions

17

terms with the different exponents. To check out this hypothesis, we substitute the proposed combined solution u(x, y) = ϕ1 (x)y 3 + ϕ2 (x)y 3/2 + ψ(x)

(1.3.2.9)

into Guderley’s equation (1.3.2.1). By collecting the coefficients of the different y 3n/2 (n = 0, 1, 2), we get (ϕ′′1 − 18aϕ21 )y 3 + (ϕ′′2 −

45 4

aϕ1 ϕ2 )y 3/2 + ψ ′′ −

2 9 8 aϕ2

= 0.

(1.3.2.10)

To ensure that this relation holds for any y, we have to set the coefficients of y 3n/2 to zero. This results in the system of ODEs ϕ′′1 − 18aϕ21 = 0,

ϕ′′2 −

45 4 aϕ1 ϕ2 ′′ ψ − 98 aϕ22

= 0,

(1.3.2.11)

= 0.

This proves that equation (1.3.2.1) admits solution (1.3.2.9), which was obtained in [346]. It can be shown that system (1.3.2.11) admits the exact solution 1 (x + C1 )−2 , 3a ϕ2 = C2 (x + C1 )5/2 + C3 (x + C1 )−3/2 , 3a 2 9 3 ψ= C (x + C1 )7 + aC2 C3 (x + C1 )3 + aC 2 (x + C1 )−1 + C4 x + C5 . 112 2 8 16 3 ϕ1 =

3◦ . Second group of solutions. Guderley’s equation also has polynomial solutions in y, which can be obtained from the following considerations. We will seek solutions as a polynomial of degree n in y with coefficients dependent on x: u = Pn ,

Pn =

n X

ψi (x)y i .

(1.3.2.12)

i=0

Differentiating (1.3.2.12) with respect to both variables and assuming that (ψn )′′xx 6= 0, we find that uxx = Qn ,

′ uy = Pn−1 ,

′′ uyy = Pn−2 ,

(1.3.2.13)

′ ′′ where Qn , Pn−1 , and Pn−2 are polynomials in y of degree n, n − 1, and n − 2, respectively. Substituting (1.3.2.13) into (1.3.2.1), we see that the left-hand side of the resulting equation is a polynomial of degree n, while the right-hand side is a polynomial of 2n − 3 (when polynomials are multiplied, their degrees are added up). For the polynomial solution to exist, both left- and right-hand sides must be polynomials of the same degree; it follows that n = 3. Consequently, solution (1.3.2.12) to Guderley’s equation can only be a cubic polynomial in y:

u = ψ1 (x) + ψ2 (x)y + ψ3 (x)y 2 + ψ4 (x)y 3 .

(1.3.2.14)

18

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

By direct verification, it is easy to see that expression (1.3.2.14) is indeed a solution to equation (1.3.2.1). The determining functions ψi = ψi (x) (i = 1, . . . , 4) are described by the following system of ordinary differential equations [114]: ψ1′′ = 2aψ2 ψ3 , ψ2′′ = 2a(3ψ2 ψ4 + 2ψ32 ), ψ3′′ = 18aψ3 ψ4 , ψ4′′ = 18aψ42 . This system is easy to integrate for ψ4 = 0 to obtain two simple solutions quadratic in y, u = C1 y 2 + 2aC12 x2 y + u

1 2 3 4 3 a C1 x , 2 4 1 = C1 xy + ( 3 aC1 x + C2 x + C3 )y 1 2 3 7 + 63 a C1 x + 16 aC1 C2 x4 + 31 aC1 C3 x3 2

(1.3.2.15) + C4 x + C5 ,

which are special cases of solution (1.3.2.14); C1 , . . . , C5 are arbitrary constants. Note that the first solution in (1.3.2.15) was obtained in [128]. A more complicated solution to (1.3.2.14) with ψ4 6= 0 can be found in [275]. In conclusion, we give two additive separable solutions to equation (1.3.2.1): u=

1 1 aC1 x2 + C2 x + C3 ± (2C1 y + C4 )3/2 . 2 3C1

◭

◮ Example 1.17. Consider the nonhomogeneous Monge–Amp`ere equation

u2xy − uxx uyy = f (x),

(1.3.2.16)

where f (x) is an arbitrary function. Note that Monge–Amp`ere type equations arise in differential geometry, gas dynamics, and meteorology [168, 169, 203, 320]. We look for generalized separable solutions in the form u(x, y) = ϕ(x)y k + ψ(x),

(1.3.2.17)

with the functions ϕ(x) and ψ(x) and constant k to be determined. Substituting (1.3.2.17) into (1.3.2.16) and rearranging, we get ′′ k−2 y − f (x) = 0. (1.3.2.18) [k 2 (ϕ′x )2 − k(k − 1)ϕϕ′′xx ]y 2k−2 − k(k − 1)ϕψxx

This is a polynomial equation in y; it involves y 2k−2 and y k−2 and must hold identically for any y. Therefore, to compensate for the function f (x) at y = 0, we must set one of the two exponents of y to zero. This gives two possible values of k: k = 1 or k = 2. First case. If k = 1, (1.3.2.18) reduces to the equation (ϕ′x )2 − f (x) = 0.

1.3. Simplified Method for Constructing Generalized Separable Solutions

19

Rp It has two solutions: ϕ(x) = ± f (x) dx. They generate two solutions of equation (1.3.2.16) in the form (1.3.2.17): Z p u(x, y) = ±y f (x) dx + ψ(x), where ψ(x) is an arbitrary function. Second case. If k = 2, equating the functional coefficients of the different powers of y with zero, we obtain two equations: 2(ϕ′x )2 − ϕϕ′′xx = 0, ′′ 2ϕψxx + f (x) = 0. Their general solutions are given by 1 ϕ(x) = , C1 x + C2

1 ψ(x) = − 2

Z

x

0

(x − t)(C1 t + C2 )f (t) dt + C3 x + C4 ,

where C1 , . . . , C4 are arbitrary constants. The nonlinear equation u2xy + g(x)uxx uyy = f (x) is a simple generalization of the Monge–Amp`ere equation (1.3.2.16); f (x) and g(x) are arbitrary functions. It also admits exact solutions of the form (1.3.2.17). For an extensive list of exact solutions to the Monge–Amp`ere type equation (1.3.2.16) as well as more general related equations, see the handbook [275] (see ◭ also [169]). ◮ Example 1.18. Let us demonstrate a possible line of reasoning for constructing generalized separable solutions to a porous medium equation with a quadratic nonlinearity (Boussinesq’s equation [40]):

ut = a(uux )x ,

(1.3.2.19)

which describes an unsteady flow of groundwater in the presence of a free surface, where u is the groundwater pressure. Remark 1.9. The heat equation with thermal diffusivity linearly dependent on temperature, vt = [(av + b)vx ]x , can be reduced with the change of variable u = v + (b/a) to equation (1.3.2.19); for many metals, a linear approximation of the thermal diffusivity is applicable in a wide range of temperatures.

Equation (1.3.2.19) is a special case of equation (1.1.2.1) with k = 1; it has a simple separable solution quadratic in x: u = ϕ(t)x2 ,

ϕ(t) = −1/(6at).

(1.3.2.20)

A natural question arises: Can we find, using the simple solution (1.3.2.20), a more complicated generalized separable solution to equation (1.3.2.19)?

20

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Let us try the solution u(x, t) = ϕ(t)x2 + ψ(t)xk ,

k 6= 2,

(1.3.2.21)

whose first term coincides with solution (1.3.2.20). The function ψ(t) and exponent k in the second term are to be determined. Substituting (1.3.2.21) into (1.3.2.19) and rearranging, we obtain (ϕ′t − 6aϕ2 )x2 + [ψt′ − a(k + 1)(k + 2)ϕψ]xk − ak(2k − 1)ψ 2 x2k−2 = 0. (1.3.2.22) Since this equation must hold identically for any x, the functional coefficients of the different powers of x in (1.3.2.22) must be zero. As we look for solutions with ψ(t) 6= 0, we get two cases, k = 0 and k = 1/2, both nullifying the coefficient of x2k−2 . We will consider these cases in order. First case. For k = 0, we get the following system of equations for ϕ = ϕ(t) and ψ = ψ(t): ϕ′t − 6aϕ2 = 0, ψt′ − 2aϕψ = 0. Its general solution is given by ϕ(t) = −

1 , 6a(t + C1 )

ψ(t) =

C2 , |t + C1 |1/3

(1.3.2.23)

where C1 and C2 are arbitrary constants. Second case. (Barenblatt–Zel’dovich dipole solution [23].) For k = 1/2, the system for ϕ = ϕ(t) and ψ = ψ(t) is expressed as ϕ′t − 6aϕ2 = 0,

ψt′ −

15 4

aϕψ = 0.

Its general solution is ϕ(t) = −

1 , 6a(t + C1 )

ψ(t) =

C2 . |t + C1 |5/8

(1.3.2.24)

As a result, we arrive at the following two generalized separable solutions of equation (1.3.2.19) [388]: 1 C2 (x + C3 )2 + , 6a(t + C1 ) |t + C1 |1/3 1 C2 u=− (x + C3 )2 + (x + C3 )1/2 . 6a(t + C1 ) |t + C1 |5/8

u=−

For more generality, we have added an arbitrary translation to the space variable x.◭

21

1.3. Simplified Method for Constructing Generalized Separable Solutions

Remark 1.10. Equation (1.3.2.19) also admits a simple additive separable solution

u = C1 x + aC12 t + C2 ,

where C1 and C2 are arbitrary constants. For invariant solutions to (1.3.2.19), see [229] (see also [85, 150, 231]). More complicated solutions to equation (1.3.2.19) can be found in [272, 275]. Remark 1.11. The wave equation with a quadratic nonlinearity

utt = a(uux )x ,

also admits solutions of the form (1.3.2.21) with k = 0 and k = 1/2. ◮ Example 1.19. In certain cases, fairly complex solutions with generalized separation of variables can be found using a different approach, other than that used in Example 1.18. This approach starts with looking for simple solutions and then proceeds to constructing more complicated ones. We will demonstrate this with an example of a nonlinear diffusion equation with a volume chemical reaction of the second order:

ut = a(uux )x − bu2 .

(1.3.2.25)

1◦ . Exponential solutions in x. We seek a generalized separable solution to equation (1.3.2.25) in the form u(x, t) = ϕ(t)eλx + ψ(t).

(1.3.2.26)

The functions ϕ = ϕ(t) and ψ = ψ(t) and constant λ are to be determined in the subsequent analysis. Substituting (1.3.2.26) into (1.3.2.25) and collecting the coefficients of the different exponentials enλx (n = 0, 1, 2), we obtain (b − 2aλ2 )ϕ2 e2λx + [ϕ′t + (2b − aλ2 )ϕψ]eλx + ψt′ + bψ 2 = 0.

(1.3.2.27)

Since this equation must hold identically for any x, the functional coefficients of enλx must be set equal to zero. As a result, we arrive at the algebraic differential system b − 2aλ2 = 0,

ϕ′t + (2b − aλ2 )ϕψ = 0, ψt′

(1.3.2.28)

2

+ bψ = 0,

which admits two solutions λ=±



b 2a

1/2

,

ϕ=

C1 , |t + C2 |3/2

where C1 and C2 are arbitrary constants.

ψ=

1 , b(t + C2 )

(1.3.2.29)

22

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

2◦ . Compound exponential solution in x. It follows from expressions (1.3.2.26) and (1.3.2.29) that equation (1.3.2.25) has two solutions, u1,2 = ϕe±λx + ψ, which only differ in the sign of λ. This leads us to believe that equation (1.3.2.25) may have a more complex solution that involves both exponential terms. To check out this hypothesis, let us substitute the supposed solution u(x, t) = ϕ1 (t)e

−λx

+ ϕ2 (t)e

λx

+ ψ(t),



λ=

b 2a

1/2

.

(1.3.2.30)

into (1.3.2.25). On rearranging, we obtain [(ϕ1 )′t +

−λx 3 2 bϕ1 ψ]e

+ [(ϕ2 )′t +

λx 3 2 bϕ2 ψ]e

+ ψt′ + b(2ϕ1 ϕ2 + ψ 2 ) = 0.

Equating the functional coefficients of enλx (n = 0, ±1) with zero, we arrive at the system of first-order ODEs (ϕ1 )′t +

3 2 bϕ1 ψ ′ (ϕ2 )t + 32 bϕ2 ψ ′ ψt + b(2ϕ1 ϕ2 + ψ 2 )

= 0, = 0,

(1.3.2.31)

= 0.

This proves that equation (1.3.2.25) admits a solution of the form (1.3.2.30). Eliminating ψ from the first two equations in (1.3.2.31), we obtain (ϕ1 )′t /ϕ1 = (ϕ2 )′t /ϕ2 . It follows that ϕ1 = Aϕ(t) and ϕ2 = Bϕ(t), where A and B are arbitrary constants. Consequently, the generalized separable solution (1.3.2.30) becomes u(x, t) = ϕ(t)(Ae

−λx

λx

+ Be ) + ψ(t),

λ=



b 2a

1/2

,

(1.3.2.32)

with the functions ϕ = ϕ(t) and ψ = ψ(t) satisfying the nonlinear system of ODEs ϕ′t + ψt′

3 2

bϕψ = 0,

2

+ b(2ABϕ + ψ 2 ) = 0.

(1.3.2.33)

This system is autonomous; hence, it reduces, through the elimination of t, to a single homogeneous ODE, which can be integrated [273]. For AB > 0, system (1.3.2.33) admits two simple solutions ϕ=±

1 √ , 3b AB (t + C)

ψ=

2 , 3b(t + C)

(1.3.2.34)

which define two multiplicative separable solutions (1.3.2.32). 3◦ . Trigonometric solution in x. In formulas (1.3.2.30) and (1.3.2.32), it was implicitly assumed that ab > 0. If ab < 0, we have λ = iβ,

β=

 1/2 b − , 2a

i2 = −1.

23

1.3. Simplified Method for Constructing Generalized Separable Solutions

This means that solution (1.3.2.32) can be rewritten in terms of trigonometric functions:  1/2 b u(x, t) = ϕ(t)[A1 cos(βx) + B1 sin(βx)] + ψ(t), β = − , (1.3.2.35) 2a where A1 and B1 are arbitrary constants. Substituting (1.3.2.35) into (1.3.2.25) and performing calculations similar to those in Item 2◦ , we arrive at the following nonlinear system of ODEs for ϕ = ϕ(t) and ψ = ψ(t): ψt′

+

b[ 12 (A21

+

ϕ′t +

= 0,

B12 )ϕ

+ ψ ] = 0.

3 2 bϕψ 2 2

(1.3.2.36)

This system admits two simple solutions 2 p , 2 3b A1 + B12 (t + C)

2 , 3b(t + C)

(1.3.2.37)

which define two multiplicative separable solutions (1.3.2.35).

◭

ϕ=±

ψ=

Remark 1.12. Solution (1.3.2.35) and system (1.3.2.36) can be obtained directly from solution (1.3.2.32) and system (1.3.2.33) by formally setting

eλx = eiβx = cos(βx) + i sin(βx), e−λx = e−iβx = cos(βx) − i sin(βx), A = 21 (A1 + iB1 ), B = 21 (A1 − iB), A1 = A + B, B1 = i(B − A). ◮ Example 1.20. Let us look at the von Mises equation with a cubic nonlinearity

utt + 2ux uxt + auxx + but uxx + cu2x uxx = 0,

(1.3.2.38)

For certain values of the constants a, b, and c, it describes a one-dimensional flow of a compressible gas [345]. We will seek solutions of equation (1.3.2.38) as a polynomial of degree n in x with coefficients dependent on t. Let Pn denote the polynomial. Reasoning as in Example 1.16 (see Item 3◦ ), we substitute the proposed solution u = Pn into the equation. With each term of equation (1.3.2.38), we associate the degree of the polynomial resulting from substituting u = Pn into this term. This can be represented as the table term of equation degree of polynomial

utt n

ux uxt 2n − 2

uxx n−2

ut uxx 2n − 2

u2x uxx 3n − 4

At least two out of five resulting polynomials must have the same maximum degree for these polynomials to balance each other. By matching up the degrees n, 2n − 2, and 3n − 4 pairwise, we get n = 2, in which case four polynomials will have the same degree. Considering the above, we look for an exact solution to equation (1.3.2.38) as a quadratic polynomial in x (specified by von Mises [114, 345]): u = ψ1 (t) + ψ2 (t)x + ψ3 (t)x2 .

(1.3.2.39)

24

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Substituting (1.3.2.39) into (1.3.2.38) yields the following system of ODEs for ψi = ψi (t): ψ1′′ + 2ψ2 ψ2′ + 2aψ3 + 2bψ3 ψ1′ + 2cψ22 ψ3 = 0, ψ2′′ + 2(2 + b)ψ3 ψ2′ + 4(ψ3′ + 2cψ32 )ψ2 = 0, ψ3′′

+ 2(4 +

b)ψ3 ψ3′

+

8cψ33

(1.3.2.40)

= 0.

The first equation of system (1.3.2.40) is linear in ψ1 and so admits order reduction with the substitution z = ψ1′ . Hence, its solution can be expressed in terms of the solutions to the second and third equations. The second equations of system (1.3.2.40) is linear in ψ2 and has the particular solution ψ2 = ψ3 (since the second equation coincides with the third one at ψ2 = ψ3 ). Considering the above, we find that system (1.3.2.40) admits two four-parameter exact solutions C22 a (t + C1 )−1 − (t + C1 ) + C3 (t + C1 )1−2bβ + C4 , 4β b −1 ψ2 = C2 (t + C1 ) , ψ3 = β(t + C1 )−1 , ψ1 =

(1.3.2.41)

where C1 , . . . , C4 are arbitrary constants and β = β1,2 are roots of the quadratic equation 4cβ 2 − (4 + b)β + 1 = 0. ◭ Remark 1.13. The more general, than (1.3.2.38), nonlinear PDE

utt + f1 (t, uxx ) + uf2 (t, uxx ) + ut f3 (t, uxx ) + u2x f4 (t, uxx ) + ux uxt f5 (t, uxx ) = 0

where f1 (t, w), . . . , f5 (t, w) are arbitrary functions with two arguments, also admits an exact solution of the form (1.3.2.39). ◮ Example 1.21. The steady-state longitudinal flow of a viscous incompressible fluid in the laminar boundary layer around a horizontal thin flat plate is described by the following nonlinear third-order equation for the stream function:

uy uxy − ux uyy = νuyyy ,

(1.3.2.42)

where x and y are a longitudinal and a transverse coordinate, while ν is the kinematic viscosity. For the derivation of this equation, see [192, 329] and Subsection 3.2.3. We look for a generalized separable solution to equation (1.3.2.42) in the form u(x, y) = xk ϕ(y) + ψ(y).

(1.3.2.43)

The functions ϕ(y) and ψ(y) and constant k are to be determined. Substituting (1.3.2.43) into (1.3.2.42) and collecting the coefficients of the different powers of x, we obtain ′′′ ′′ )−νxk ϕ′′′ kx2k−1 [(ϕ′y )2 −ϕϕ′′yy ]+kxk−1 (ϕ′y ψy′ −ϕψyy yyy −νψyyy = 0. (1.3.2.44)

This equation involves x2k−1 , xk−1 , and xk and must hold identically for any x. ′′′ To compensate for the last term in (1.3.2.44) for ψyyy 6≡ 0, we must set one of the

25

1.3. Simplified Method for Constructing Generalized Separable Solutions

exponents of x equal to zero. Discarding the degenerate case k = 0, we get two allowed values, k = 1 and k = 1/2. Consider the two cases in order. First case. Substituting k = 1 into (1.3.2.44) and rearranging the terms, we obtain ′ ′ ′′ ′′′ x[(ϕ′y )2 − ϕϕ′′yy − νϕ′′′ yyy ] + [ϕy ψy − ϕψyy − νψyyy ] = 0.

For this equality to hold for any x, we must set the two expressions in square brackets to zero. As a result, we arrive at the following system of ordinary differential equations for ϕ = ϕ(y) and ψ = ψ(y): (ϕ′y )2 − ϕϕ′′yy − νϕ′′′ yyy = 0, ′′ ′′′ ϕ′y ψy′ − ϕψyy − νψyyy = 0.

For example, this system admits the solution ϕ=

6ν , y + C1

ψ=

C2 C3 + + C4 , y + C1 (y + C1 )2

where C1 , . . . , C4 are arbitrary constants. Second case. For k = 1/2, we arrive at a degenerate solution, which is of little ′′′ interest. The case of arbitrary k and ψyyy ≡ 0, also results in a degenerate solution. Some other exact solutions of nonlinear PDE (1.3.2.42) can be found in the ◭ handbook [275] (see also Example 1.28). Remark 1.14. The boundary layer equation has a remarkable property [231, 235], which can be formulated as follows: if u(x, y) is a solution to equation (1.3.2.42), then

u1 = u(x, y + θ(x)),

where θ(x) is an arbitrary differentiable function, is also a solution to this equation. The wider class of hydrodynamic type equations of any order uy uxy − ux uyy = F (x, u, uy , uyy , . . . , u(n) y )

possesses the same property [272, 274]. Remark 1.15. The nonlinear nth-order partial differential equation

f (y)uy uxy + g(y)ux uyy = h(y)u(n) + p(y)u + q(y), y

where f (y), g(y), h(y), p(y), and q(y) are arbitrary functions, also admits a generalized separable solution of the form (1.3.2.43) with k = 1. ◮ Example 1.22. Consider the third-order nonlinear partial differential equation with a mixed derivative

uxt + u2x − uuxx = νuxxx ,

(1.3.2.45)

which arises in fluid dynamics [15, 243, 245]. We look for exact solutions of the form u = ϕ(t)eλx + ψ(t),

λ 6= 0.

(1.3.2.46)

26

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

On substituting (1.3.2.46) into (1.3.2.45), we get ϕ′t − λϕψ = νλ2 ϕ. Solving this equation for ψ and substituting the resulting expression into (1.3.2.46), we obtain a solution to equation (1.3.2.45) in the form u = ϕ(t)eλx +

1 ϕ′t (t) − νλ, λ ϕ(t)

(1.3.2.47)

where ϕ(t) is an arbitrary function and λ is an arbitrary constant. The nth-order nonlinear equation uxt + u2x − uuxx = f (t)u(n) x , where f (t) is an arbitrary function, generalizes the hydrodynamic type equation (1.3.2.45) and admits an exact solution of the form (1.3.2.46): u = ϕ(t)eλx +

1 ϕ′t (t) − λn−2 f (t), λ ϕ(t)

where ϕ(t) is an arbitrary function and λ is an arbitrary constant.

◭

Remark 1.16. The hydrodynamic type equation (1.3.2.45) has a remarkable property [244, 272], which can be stated as follows: if u(x, t) is a solution to equation (1.3.2.45), then

u1 = u(x + θ(t), t) + θt′ (t),

where θ(t) is an arbitrary differentiable function, is also a solution to this equation. Remark 1.17. The wider class of hydrodynamic type equations of any order

uxt = a(t)uuxx + F (t, ux , uxx , . . . , u(n) x )

(1.3.2.48)

possesses a similar property [274]. Specifically, if u(x, t) is a solution to equation (1.3.2.48), then θ′ (t) u1 = u(x + θ(t), t) + t , a(t) where θ(t) is an arbitrary differentiable function, is also a solution to this equation.

1.3.3. Equations in Three or More Independent Variables. Exact Solutions to the Navier–Stokes Equations Generalized separable solutions to PDEs in three or more independent variables. The methods of generalized separation of variables can also be employed to construct exact solutions for nonlinear equations of mathematical physics with three or more independent variables. (Increasing the number of independent variables is not essential; it just makes it technically more challenging to use these methods in practice.) In particular, for equations with three independent variables x, y, and t, the unknown u can be sought as a finite series u=

n X

k=1

ϕk (x, t)ψk (y).

(1.3.3.1)

1.3. Simplified Method for Constructing Generalized Separable Solutions

27

With the simplified approach, the functions ψk (y) are preset from a priori considerations, while the functions ϕk (x, t) are determined in the analysis. The variables x, y, and t in (1.3.3.1) are all interchangeable. Navier–Stokes equations. Exact solutions. To illustrate the above, we look at three unsteady two-dimensional equations of motion of an incompressible viscous fluid that include two Navier–Stokes equations and one continuity equation [192, 329]: 1 Ut + U Ux + V Uy = − px + ν∆U, ρ 1 Vt + U Vx + V Vy = − py + ν∆V, ρ Ux + Vy = 0.

(1.3.3.2)

Here, t is time, x and y are Cartesian coordinates, U and V are the fluid velocity components, p is pressure, ρ is density, ν is the kinematic viscosity, and ∆ is the Laplace operator. By introducing a stream function u such that U = uy ,

V = −ux

(1.3.3.3)

followed by eliminating pressure p, through cross-differentiation of the first and second equations, one can reduce system (1.3.3.2) to a single fourth-order nonlinear equation [87, 192, 329]: (∆u)t + uy (∆u)x − ux (∆u)y = ν∆∆u,

∆u = uxx + uyy .

(1.3.3.4)

Omitting intermediate calculations, below we list a few exact solutions to equation (1.3.3.4) (see [245] for details). 1◦ . Generalized separable solution:   u = e−λy A(t)eβx + B(t)e−βx + ϕ(t)x + ψ(t)y,   Z Z A(t) = C1 exp ν(λ2 + β 2 )t − β ψ(t) dt − λ ϕ(t) dt ,   Z Z B(t) = C2 exp ν(λ2 + β 2 )t + β ψ(t) dt − λ ϕ(t) dt , where ϕ(t) and ψ(t) are arbitrary functions, while C1 , C2 , λ, and β are arbitrary constants. 2◦ . Generalized separable solution:   u = e−λy A(t) sin(βx) + B(t) cos(βx) + ϕ(t)x + ψ(t)y.

Here, ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions, λ and β are arbitrary constants, while A = A(t) and B = B(t) are functions satisfying the nonautonomous linear

28

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

system of ODEs   A′t = ν(λ2 − β 2 ) − λϕ(t) A + βψ(t)B,   Bt′ = ν(λ2 − β 2 ) − λϕ(t) B − βψ(t)A,

whose general solution is expressed as    Z   Z  Z 2 2 A(t) = exp ν(λ − β )t − λ ϕ dt C1 sin β ψ dt + C2 cos β ψ dt ,    Z   Z  Z 2 2 B(t) = exp ν(λ − β )t − λ ϕ dt C1 cos β ψ dt − C2 sin β ψ dt . In particular, for ϕ =

ν 2 λ (λ

− β 2 ) and ψ = a, we get a family of periodic solutions

A(t) = C1 sin(aβt) + C2 cos(aβt), B(t) = C1 cos(aβt) − C2 sin(aβt). 3◦ . Generalized separable solution: u = A(t) exp(k1 x + λ1 y) + B(t) exp(k2 x + λ2 y) + ϕ(t)x + ψ(t)y, where ϕ(t) and ψ(t) are arbitrary functions. The four constants k1 , λ1 , k2 , and λ2 are connected by either of the two relations k12 + λ21 = k22 + λ22

(first family of solutions),

k1 λ2 = k2 λ1

(second family of solutions).

The functions A = A(t) and B = B(t) satisfy the independent linear ODEs   A′t = ν(k12 + λ21 ) + λ1 ϕ(t) − k1 ψ(t) A,   Bt′ = ν(k22 + λ22 ) + λ2 ϕ(t) − k2 ψ(t) B.

Integrating these equations yields   Z Z 2 2 A(t) = C1 exp ν(k1 + λ1 )t + λ1 ϕ(t) dt − k1 ψ(t) dt ,   Z Z B(t) = C2 exp ν(k22 + λ22 )t + λ2 ϕ(t) dt − k2 ψ(t) dt .

Using transformations of the independent variables to construct exact solutions. For nonlinear PDEs in three or more independent variables, the methods of generalized separation of variables can be applied several times successively as the dimensionality of the arising reduced equations decreases. In addition, transformations of the independent variables can sometimes be used in the initial or intermediate stage to obtain exact solutions.

1.3. Simplified Method for Constructing Generalized Separable Solutions

29

To illustrate the above, we revisit the nonlinear hydrodynamic type equation with three independent variables (1.3.3.4). It is easy to show that this equation admits generalized separable solutions of the form u = xf (t, y) + g(t, y),

(1.3.3.5)

where the functions f = f (t, y) and g = g(t, y) satisfy the system of two fourthorder partial differential equations ftyy + fy fyy − f fyyy = νfyyyy , gtyy + gy fyy − f gyyy = νgyyyy .

(1.3.3.6) (1.3.3.7)

Equation (1.3.3.6) contains one unknown function f and is independent of equation (1.3.3.7). The most comprehensive survey of exact solution to equation (1.3.3.6) and system (1.3.3.6)–(1.3.3.7) can be found in [275] (see also [15, 87, 245, 313]). Below, we briefly outline a few more complicated solutions obtained in [294]. We will need two useful propositions stated below. These allow one to generalize existing solutions and construct new, more complex solutions to equation (1.3.3.6) and system (1.3.3.6)–(1.3.3.7) using an already known simple solution to equation (1.3.3.6). Proposition 1. Suppose f = f (t, y) is a solution to equation (1.3.3.6). Then equation (1.3.3.7) admits the exact solution g = Cfy + a(t)f − a′t (t)y,

(1.3.3.8)

where C is an arbitrary constant and a = a(t) is an arbitrary function. This proposition can be proved by substituting expression (1.3.3.8) into (1.3.3.7) and taking into account equation (1.3.3.6) and its corollary 2 ftyyy + fyy − f fyyyy = νfyyyyy

obtained by differentiating (1.3.3.6) with respect to y. Proposition 2. Suppose f = f (t, y) is a solution to equation (1.3.3.6). Then system (1.3.3.6)–(1.3.3.7) admits the exact solution f1 = f (t, y + b) + b′t , g1 = Cfy (t, y + b) + af (t, y + b) − a′t y,

(1.3.3.9)

where a = a(t) and b = b(t) are arbitrary functions. ◮ Example 1.23. Solution of system (1.3.3.6)–(1.3.3.7), rational in y. It is easy to verify that equation (1.3.3.6) has a stationary solution f = 6ν/y. Using the formulas (1.3.3.9), we obtain the following exact solution to system (1.3.3.6)–(1.3.3.7):

f=

6ν + b′t , y+b

g=

C1 6νa + − a′t y. (y + b)2 y+b

This solution involves two arbitrary functions, a = a(t) and b = b(t), and one arbitrary ◭ constant, C1 = −6νC.

30

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

To construct more complicated exact solutions, we introduce the new independent variable z = λ(t)y, where λ = λ(t) is an arbitrary function, instead of y. As a result, system (1.3.3.6)–(1.3.3.7) reduces to λftzz + λ′t zfzzz + 2λ′t fzz + λ2 (fz fzz − f fzzz ) = νλ3 fzzzz , λgtzz +

λ′t zgzzz

+

2λ′t gzz

2

3

+ λ (gz fzz − f gzzz ) = νλ gzzzz .

(1.3.3.10) (1.3.3.11)

For g = f , equation (1.3.3.11) coincides with equation (1.3.3.10). Therefore, in searching for generalized separable solutions, we take f and g to be similar (in z). Below we summarize the results [294] without including intermediate calculations. 1. Solution involving exponentials. System (1.3.3.6)–(1.3.3.7) admits exact solutions of the form f = a(t)e−λ(t)y + b(t)y + c(t), g = α(t)e−λ(t)y + β(t)y.

(1.3.3.12)

The six functional coefficients a = a(t), b = b(t), c = c(t), α = α(t), β = β(t), and λ = λ(t) satisfy three equations λ′t − bλ = 0,

a′t + 3ab + acλ − νaλ2 = 0,

α′t

(1.3.3.13) 2

+ 2bα + aβ + cαλ − ναλ = 0.

The second and third equations have been rearranged using the first one. The functions a, α, and λ in (1.3.3.13) will be treated as arbitrary. Then the other three functions can be found without integration by the formulas 1 ′ λ′t , c=− (a + 3ab) + νλ, λ aλ t 1 β = (−α′t − 2bα − cαλ + ναλ2 ). a b=

(1.3.3.14)

Below are two examples that illustrate the usage of formulas (1.3.3.12) and (1.3.3.14) to construct solutions for model hydrodynamic problems. ◮ Example 1.24. Let us look at the special case of solution (1.3.3.12) with

λ = const,

b = 0,

c = νλ −

a′t , aλ

α = aσ,

β = −σt′ ,

(1.3.3.15)

where a = a(t) and σ = σ(t) are arbitrary functions. The expressions (1.3.3.15) satisfy system (1.3.3.13) and are obtained from (1.3.3.14). Substituting (1.3.3.15) into (1.3.3.12) and taking into account (1.3.3.5), we obtain the stream function  a′  u = x ae−λy + νλ − t + aσe−λy − σt′ y. (1.3.3.16) aλ

1.3. Simplified Method for Constructing Generalized Separable Solutions

31

The fluid velocity components are given by formulas (1.3.3.3): U = −aλxe−λy − aσλe−λy − σt′ ,

V = −ae−λy − νλ +

a′t . aλ

(1.3.3.17)

By setting y = 0 in (1.3.3.17), we get U |y=0 = −aλx − aσλ − σt′ ,

V |y=0 = −a − νλ +

a′t . aλ

(1.3.3.18)

We choose the free functions a and σ in the form a=−

νλ , C1 exp(νλ2 t) + 1

σ=

C2 exp(νλ2 t) , C1 exp(νλ2 t) + 1

(1.3.3.19)

where C2 is an arbitrary constant and C1 is an arbitrary constant such that C1 ≥ 0 or C1 < −1. Then the boundary conditions (1.3.3.18) simplify significantly to become U |y=0 = A(t) x, where A(t) =

V |y=0 = 0,

(1.3.3.20)

νλ2 . C1 exp(νλ2 t) + 1

For λ > 0, formulas (1.3.3.17) with the functions (1.3.3.19) describe unsteady flows of a viscous fluid in the half-plane 0 ≤ y < ∞, caused by the extension (for A > 0) or compression (for A < 0) of the surface y = 0 by the law (1.3.3.20) under special initial conditions (that correspond to t = 0 in (1.3.3.19)). In the special case C1 = C2 = 0, the above formulas give a steady-state solution [79] that models extrusion. For C2 = 0 and C1 6= 0, formulas (1.3.3.17) and (1.3.3.19) define the ◭ solution obtained in [15]. ◮ Example 1.25. Substituting (1.3.3.12) and (1.3.3.14) with α = β = 0 into (1.3.3.5), we get the steam function  λ′ λ′ a′  u = x ae−λy + t y + νλ − 3 2t − t . (1.3.3.21) λ λ aλ

The respective velocity components are given by  λ′  U = x −aλe−λy + t , λ λ′t λ′ a′ −λy V = −ae − y − νλ + 3 2t + t , λ λ aλ

(1.3.3.22)

where a = a(t) and λ = λ(t) are arbitrary functions. Solution (1.3.3.22) satisfies the following boundary conditions as y → ∞: U → Λ(t)x,

V → −Λ(t)y,

where Λ = λ′t /λ.

These conditions are used to model viscous flows near a stagnation point (e.g., see [15, 87, 138, 358, 359]).

32

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Let us look at the special case a= √

a0 , t+C

λ0 λ= √ , t+C

a0 = −

1 (νλ0 + 2), λ0

(1.3.3.23)

where C > 0 and λ0 > 0 are arbitrary constants, in more detail. On the surface y = 0, solution (1.3.3.22) with (1.3.3.23) satisfies the conditions U |y=0 = −A(t)x,

V |y=0 = 0,

(1.3.3.24)

where

2νλ20 + 3 . t+C Therefore, solution (1.3.3.22) with (1.3.3.23) describes an unsteady motion of a ◭ viscous fluid near a shrinking plane. A(t) = −

2. Solution involving trigonometric functions. 2.1. The system of equations (1.3.3.6)–(1.3.3.7) admits exact solutions of the form f = a(t) cos[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) cos[λ(t)y + σ(t)] + s(t) sin[λ(t)y + σ(t)] + β(t)y.

(1.3.3.25)

The eight functional coefficients a = a(t), b = b(t), c = c(t), s = s(t), α = α(t), β = β(t), λ = λ(t), and σ = σ(t) satisfy five equations λ′t − bλ = 0, σt′ − cλ = 0,

a′t + 3ab + νaλ2 = 0,

α′t + 2bα + aβ + ναλ2 s′t + 2bs + νsλ2 = 0.

(1.3.3.26) = 0,

The last three equations have been rearranged using the first two. The functions λ, σ, and α will be treated as arbitrary. Then the other five functions are expressed as  Z  λ′ σ′ a0 2 a = 3 exp −ν λ dt , b = t , c = t , λ λ λ  Z  (1.3.3.27) 1 ′ s 0 2 2 β = − (αt + 2bα + ναλ ), s = 2 exp −ν λ dt , a λ where a0 and s0 are arbitrary constants. ◮ Example 1.26. Let us look at the special case of equation (1.3.3.25) with

λ = const, a = a1 E(t),

σ = π/2,

b = c = 0,

s = s1 E(t),

α = β = ω = 0,

E(t) = exp(−νλ2 t),

(1.3.3.28)

33

1.3. Simplified Method for Constructing Generalized Separable Solutions

where a1 = a0 λ−3 and s1 = s0 λ−2 are arbitrary constants. The expressions (1.3.3.28) satisfy system (1.3.3.26) and agree with formulas (1.3.3.27). Substituting (1.3.3.28) into (1.3.3.25) and taking into account (1.3.3.5), we obtain the stream function u = −a1 E(t)x sin(λy) + s1 E(t) cos(λy).

(1.3.3.29)

Using formulas (1.3.3.3), we get the fluid velocity components U = −a1 λE(t)x cos(λy) − s1 λE(t) sin(λy),

V = a1 E(t) sin(λy),

E(t) = exp(−νλ2 t).

(1.3.3.30)

By setting y = 0 in (1.3.3.30), we find U |y=0 = −a1 λE(t)x,

V |y=0 = 0.

It follows from these conditions that the boundary y = 0 extends if a1 λ < 0 or shrinks if a1 λ > 0. Solution (1.3.3.30) is periodic in y. Thus, formulas (1.3.3.30) describe a fluid flow in the strip 0 ≤ y ≤ 2π/λ, whose boundaries deform in a consistent ◭ manner (e.g., in extrusion). 2.2. System (1.3.3.6)–(1.3.3.7) also admits exact solutions of the form f = a(t) tan[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) tan[λ(t)y + σ(t)] + β(t)y.

(1.3.3.31)

The seven functional coefficients λ = λ(t), σ = σ(t), α = α(t), β = β(t), a = a(t), b = b(t), and c = c(t) must satisfy five equations λ′t − bλ = 0,

σt′ − cλ = 0,

α′t − 4νλ2 α − 6νλβ = 0,

a = −6νλ,

b = 2νλ2 .

Assuming c(t) and β(t) to be arbitrary and solving these equations, we obtain Z c dt 1 6ν λ=±√ , σ=± √ + C2 , a = ∓ √ , C1 − 4νt C1 − 4νt C1 − 4νt  Z p 2ν 6ν α=± , β C1 − 4νt dt + C3 , b = C1 − 4νt C1 − 4νt (1.3.3.32) where C1 , C2 , and C3 are arbitrary constants. Here, the upper or lower signs must be taken simultaneously. 3. Solutions involving hyperbolic functions. 3.1. System (1.3.3.6)–(1.3.3.7) admits exact solutions in the form f = a(t) cosh[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) cosh[λ(t)y + σ(t)] + s(t) sinh[λ(t)y + σ(t)] + β(t)y.

(1.3.3.33)

Here, λ = λ(t), σ = σ(t), and α = α(t) are arbitrary functions, while the other functional coefficients are expressed as  Z  λ′ σ′ a0 a = 3 exp ν λ2 dt , b = t , c = t , λ λ λ  Z  (1.3.3.34) s0 1 β = − (α′t + 2bα − ναλ2 ), s = 2 exp ν λ2 dt , a λ

34

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

where a0 and s0 are arbitrary constants. 3.2. System (1.3.3.6)–(1.3.3.7) admits another solution f = a(t) sinh[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) sinh[λ(t)y + σ(t)] + s(t) cosh[λ(t)y + σ(t)] + β(t)y, where λ = λ(t), σ = σ(t), and α = α(t) are arbitrary functions, while the functional coefficients a = a(t), b = b(t), c = c(t), s = s(t), and β = β(t) are also defined by formulas (1.3.3.34). 3.3. System (1.3.3.6)–(1.3.3.7) admits exact solutions of the form f = a(t) tanh[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) tanh[λ(t)y + σ(t)] + β(t)y.

(1.3.3.35)

The seven functional coefficients λ = λ(t), σ = σ(t), α = α(t), β = β(t), a = a(t), b = b(t), and c = c(t) must satisfy five equations λ′t − bλ = 0,

σt′ − cλ = 0,

α′t + 4νλ2 α + 6νλβ = 0,

a = 6νλ,

b = −2νλ2 .

Assuming c(t) and β(t) to be arbitrary and solving these equations, we obtain Z c dt 6ν 1 , σ=± √ + C2 , a = ± √ , λ=±√ 4νt + C1 4νt + C1 4νt + C1 Z p  6ν 2ν α=∓ β 4νt + C1 dt + C3 , b = − , 4νt + C1 4νt + C1 (1.3.3.36) where C1 , C2 , and C3 are arbitrary constants. Here, the upper or lower signs must be taken simultaneously. 3.4. System (1.3.3.6)–(1.3.3.7) also admits exact solutions of the form f = a(t) coth[λ(t)y + σ(t)] + b(t)y + c(t), g = α(t) coth[λ(t)y + σ(t)] + β(t)y,

(1.3.3.37)

where c(t) and β(t) are arbitrary functions, while the other functional coefficients are defined by formulas (1.3.3.36). Remark 1.18. For other exact solutions to two- and three-dimensional steady-state and unsteady Navier–Stokes equations, see, for example, [12, 14–16, 18, 19, 28, 37, 52, 78, 79, 83, 87, 123, 126, 148, 189, 192, 194, 195, 205, 208, 209, 218, 231, 245, 275, 281, 294, 306, 312, 313, 318, 319, 329, 358, 359, 373].

Properties of equation (1.3.3.4). Below we indicate some important properties of equation (1.3.3.4) that allow one to multiply and generalize its exact solutions. Specifically, the following statement holds true (see [52, 189, 231, 312] for details).

35

1.4. Solution of Functional Differential Equations by the Method of Differentiation

Suppose u(x, y, t) is a solution to equation (1.3.3.4). Then the functions u1 = −u(y, x, t),

u2 = u(C1 x + C2 , C1 y + C3 , C12 t + C4 ) + C5 , u3 = u(x cos α + y sin α, −x sin α + y cos α, t),

u4 = u(x cos βt + y sin βt, −x sin βt + y cos βt, t) −

2 1 2 β(x

+ y 2 ),

u5 = u(x + ϕ(t), y + ψ(t), t) + ψt′ (t)x − ϕ′t (t)y + χ(t),

with C1 , . . . , C5 , α, β being arbitrary constants and ϕ(t), ψ(t), χ(t) being arbitrary functions, are also solutions to this equation.

1.4. Solution of Functional Differential Equations by the Method of Differentiation 1.4.1. Description of the Method of Differentiation In looking for exact solutions, one often has to deal with functional differential equations of the form (1.2.2.1)–(1.2.2.2). It is essential to be able to solve such equations. This section outlines a simple and quite general procedure for solving such equations, which involves three consecutive steps described below. 1◦ . Suppose that Ψk 6≡ 0. Dividing equation (1.2.2.1) by Ψk and differentiating with respect to y, we obtain an equation of the same form but with fewer terms: e 1 [X]Ψ e 1 [Y ] + Φ e 2 [X]Ψ e 2 [Y ] + · · · + Φ e k−1 [X]Ψ e k−1 [Y ] = 0, Φ e j [X] = Φj [X], Ψ e j [Y ] = (Ψj [Y ]/Ψk [Y ])′ . Φ

(1.4.1.1)

y

If we repeat the same procedure (k − 3) times more, we will arrive at a two-term separable equation: b 1 [X]Ψ b 1 [Y ] + Φ b 2 [X]Ψ b 2 [Y ] = 0. Φ

(1.4.1.2)

Now we will look at two possible situations. b 2 [X] 6≡ 0 and Ψ b 1 [Y ] 6≡ 0. Let us move Nondegenerate case. Suppose that Φ the second term of equation (1.4.1.2) to the right and then divide both sides by b 2 [X]Ψ b 1 [Y ]. This results in an equation with either side dependent on its own Φ variable. Equating, as usual, both sides with the same constant quantity, C, we obtain two ordinary differential equations b 1 [X] + C Φ b 2 [X] = 0, Φ

b 1 [Y ] − Ψ b 2 [Y ] = 0. CΨ

(1.4.1.3)

Degenerate cases. There are four cases in which both terms in the sum (1.4.1.2)

36

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

are zero simultaneously: b 1 [X] ≡ 0, Φ b 2 [X] ≡ 0 Φ b 1 [Y ] ≡ 0, Ψ b 2 [Y ] ≡ 0 Ψ b 1 [X] ≡ 0, Ψ b 2 [Y ] ≡ 0 Φ

b 2 [X] ≡ 0, Ψ b 1 [Y ] ≡ 0 Φ

b 1 [Y ] and Ψ b 2 [Y ] are any; =⇒ Ψ b 1 [X] and Φ b 2 [X] are any; =⇒ Φ

b 2 [X] and Ψ b 1 [Y ] are any; =⇒ Φ b 1 [X] and Ψ b 2 [Y ] are any. =⇒ Φ

(1.4.1.4)

b1 = Ψ b 2 = 0 and Φ b2 = Ψ b 1 = 0, are corollaries of Although the last two cases, Φ equation (1.4.1.3) in the limit cases C → 0 and C → ∞, they should be treated separately, since the associated limit solutions to equations (1.4.1.3) may not exist. 2◦ . The above solutions of the two-term equations (1.4.1.2) must be substituted into the original functional differential equation (1.2.2.1)–(1.2.2.2) to get rid of the redundant constants of integration, which arise because equation (1.4.1.2) is obtained from (1.2.2.1) by differentiation. 3◦ . The case Ψk ≡ 0 must be treated separately, since the equation was first divided by Ψk . Likewise, all other cases of vanishing functionals by which all intermediate functional differential equations were divided must be studied separately. Remark 1.19. The functional differential equation (1.2.2.1)–(1.2.2.2) can have one or more solutions or no solutions at all. Remark 1.20. In each step, the number of terms in the original functional differential equation can be reduced by differentiating by either y or x. For example, in the first step, one can suppose that Φi 6≡ 0 (for any i such that 1 ≤ i ≤ k), divide equation (1.2.2.1) by Φi , and differentiate with respect to x to obtain a similar equation with fewer terms.

In a practical application of the presented method, one should first remove the terms that involve either the highest derivatives or the most complex nonlinearities. One will eventually arrive at the simplest possible two-term equation of the form (1.4.1.2).

1.4.2. Examples of Constructing Generalized Separable Solutions by the Differentiation Method Below we give a few specific examples of applying the differentiation method to construct generalized separable solutions of nonlinear PDEs. ◮ Example 1.27. Consider the second-order parabolic equation with a quadratic

nonlinearity ut = auuxx + bu2x + c.

(1.4.2.1)

We look for exact separable solutions in the form u = ϕ(t)θ(x) + ψ(t). Substituting (1.4.2.2) into (1.4.2.1) and collecting similar terms, we get   ′′ ′′ + b(θx′ )2 . ψt′ − c + ϕ′t θ = aϕψθxx + ϕ2 aθθxx

(1.4.2.2)

(1.4.2.3)

1.4. Solution of Functional Differential Equations by the Method of Differentiation

37

First of all, we remove the most complex nonlinearity, which is the one in square brackets. On dividing relation (1.4.2.3) by ϕ2 and differentiating with respect to t and x, we obtain the two-term functional differential equation ′′′ (ϕ′t /ϕ2 )′t θx′ = a(ψ/ϕ)′t θxxx .

(1.4.2.4)

Separating the variables, we arrive at the ordinary differential equations ′′′ θxxx = Kθx′ ,

(1.4.2.5)

(ϕ′t /ϕ2 )′t

(1.4.2.6)

=

aK(ψ/ϕ)′t ,

where K is an arbitrary constant. The general solution of equation (1.4.2.5) is given by  2  if K = 0, A1 x + A2 x + A3 λx −λx θ = A1 e + A2 e (1.4.2.7) + A3 if K = λ2 > 0,   2 A1 sin(λx) + A2 cos(λx) + A3 if K = −λ < 0,

where A1 , A2 , and A3 are arbitrary constants. Integrating (1.4.2.6) yields B ϕ= , ψ(t) is any if K = 0, t + C1 (1.4.2.8) 1 ϕ′t , ϕ(t) is any if K 6= 0, ψ = Bϕ + aK ϕ where B is an arbitrary constant. On substituting solutions (1.4.2.7) and (1.4.2.8) into (1.4.2.3), one can remove the redundant constants and define the functions ψ and ϕ. Below we summarize the results. 1◦ . Solution for b 6= −a and b 6= − 12 a (corresponds to K = 0): a (x + C3 )2 c(a + 2b) − (t + C1 ) + C2 (t + C1 ) a+2b − , 2(a + b) 2(a + 2b)(t + C1 ) where C1 , C2 , and C3 are arbitrary constants. 2◦ . Solution for b = −a: 1 ϕ′t + ϕ(A1 eλx + A2 e−λx ) (corresponds to K = λ2 > 0). u= aλ2 ϕ The function ϕ = ϕ(t) is determined from the autonomous ordinary differential equation ′′ Ztt = acλ2 + 4a2 λ4 A1 A2 e2Z , ϕ = eZ ,

u=

whose solution can be found in implicit form. In the
special case A1 = 0 or A2 = 0, one easily finds that ϕ = C1 exp 12 acλ2 t2 + C2 t . 3◦ . Solution for b = −a: 1 ϕ′ u = − 2 t + ϕ[A1 sin(λx) + A2 cos(λx)] (corresponds to K = −λ2 < 0), aλ ϕ where the function ϕ = ϕ(t) is determined from the autonomous ordinary differential equation ′′ Ztt = −acλ2 + a2 λ4 (A21 + A22 )e2Z , ϕ = eZ , whose solution can be found in implicit form.

38

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

4◦ . For b = − 12 a, there is a degenerate solution u = ct + A1 . The degenerate solution ϕ = const, ψ = const of the two-term functional differential equation (1.4.2.4) leads to a stationary solution of the original equation. Here ◭ and henceforth, such solutions will not be considered. Remark 1.21. The generalized separable solutions to equation (1.4.2.1) were obtained in [108] in a different way.

◮ Example 1.28. Consider once again the third-order nonlinear PDE

uy uxy − ux uyy = νuyyy ,

(1.4.2.9)

which describes a laminar boundary layer on a flat plane (see equation (1.3.2.42)). We look for generalized separable solutions to the equation in the form u = ϕ(x)θ(y) + ψ(x).

(1.4.2.10)

On substituting (1.4.2.10) into (1.4.2.9) and on canceling by ϕ, we arrive at the functional differential equation ′′ ′′ ′′′ ϕ′x [(θy′ )2 − θθyy ] − ψx′ θyy = νθyyy .

(1.4.2.11)

First, we remove the highest derivative. To this end, we differentiate (1.4.2.11) with respect to x to obtain ′′ ′′ ′′ (1.4.2.12) θyy . ] = ψxx ϕ′′xx [(θy′ )2 − θθyy Nondegenerate case. On separating the variables in (1.4.2.12), we get ′′ ψxx = C1 ϕ′′xx , ′′ ′′ − C1 θyy = 0. (θy′ )2 − θθyy

Integrating yields ϕ(x) is any,

ψ(x) = C1 ϕ(x) + C2 x + C3 ,

θ(y) = C4 eλy − C1 , (1.4.2.13)

where C1 , . . . , C4 , and λ are constants of integration. On substituting (1.4.2.13) into (1.4.2.11), we establish the relationship between the constants to obtain C2 = −νλ. Ultimately, taking into account the above and formulas (1.4.2.10) and (1.4.2.13), we arrive at a solution to equation (1.4.2.9) of the form (1.4.2.10): u = ϕ(x)eλy − νλx + C, where ϕ(x) is an arbitrary function, while C and λ are arbitrary constants (C = C3 and C4 = 1). Degenerate case. It follows from (1.4.2.12) that ϕ′′xx = 0,

′′ ψxx = 0,

θ(y) is any.

(1.4.2.14)

Integrating the first two equations in (1.4.2.14) twice yields ϕ(x) = C1 x + C2 , where C1 , . . . , C4 are arbitrary constants.

ψ(x) = C3 x + C4 ,

(1.4.2.15)

1.4. Solution of Functional Differential Equations by the Method of Differentiation

39

Substituting (1.4.2.15) into (1.4.2.11), we arrive at an ordinary differential equation for θ = θ(y): ′′ ′′′ C1 (θy′ )2 − (C1 θ + C3 )θyy = νθyyy . (1.4.2.16) Formulas (1.4.2.10) and (1.4.2.15) together with equation (1.4.2.16) determine an exact solution of equation (1.4.2.9). An extensive list of exact solutions to the boundary layer equation (1.4.2.9) as ◭ well as related hydrodynamic equations can be found in the handbook [275]. ◮ Example 1.29. Two-dimensional steady-state motions of a viscous incompressible fluid are described by the Navier–Stokes equations and continuity equation (1.3.3.2) with Ut = Vt = 0. By introducing a stream function u by formulas (1.3.3.3), once can reduce these equations to a single fourth-order steady-state nonlinear equation:

uy (∆u)x − ux (∆u)y = ν∆∆u,

∆u = uxx + uyy .

(1.4.2.17)

The following property of equation (1.4.2.17) is worth noting: if u(x, y) is a solution to the equation, then −u(y, x) is also its solution. In this example, we will omit solutions that can be obtained using this property. For other properties of equation (1.4.2.17), see [275]. We will seek separable solutions to equation (1.4.2.17) in the form u = ϕ(x) + ψ(y).

(1.4.2.18)

Substituting (1.4.2.18) into (1.4.2.17) gives ′ ′′′ ′′′′ ′′′′ ψy′ ϕ′′′ xxx − ϕx ψyyy = νϕxxxx + νψyyyy .

(1.4.2.19)

By differentiating both sides of (1.4.2.19) with respect to x and y, we eliminate the terms with the highest derivatives to obtain ′′ ′′ ′′′′ ψyy ϕ′′′′ xxxx − ϕxx ψyyyy = 0.

(1.4.2.20)

′′ Nondegenerate case. For ϕ′′xx 6≡ 0 and ψyy 6≡ 0, by separating the variables in (1.4.2.20), we arrive at the linear ordinary differential equations with constant coefficients ′′ ϕ′′′′ xxxx = Cϕxx , ′′′′ ′′ ψyyyy = Cψyy ,

(1.4.2.21) (1.4.2.22)

which have different solutions depending on the sign of the integration constant C. 1◦ . For C = 0, the general solutions to equations (1.4.2.21) and (1.4.2.22) are expressed as ϕ(x) = A1 + A2 x + A3 x2 + A4 x3 , (1.4.2.23) ψ(y) = B1 + B2 y + B3 y 2 + B4 y 3 ,

40

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

where Ak and Bk are arbitrary constants (k = 1, 2, 3, 4). Substituting (1.4.2.23) into (1.4.2.19) gives three sets of values of the constants: A4 = B4 = 0, An , Bn are any (n = 1, 2, 3); Ak = 0, Bk are any (k = 1, 2, 3, 4); Bk = 0, Ak are any (k = 1, 2, 3, 4). The first two sets determine two well-known polynomial solutions [192], of the second and third degree in the independent variables, to equations (1.4.2.17): u = C1 x2 + C2 x + C3 y 2 + C4 y + C5 , u = C1 y 3 + C2 y 2 + C3 y + C4 ,

(1.4.2.24)

where C1 , . . . , C5 are arbitrary constants. 2◦ . For C = λ2 > 0, the general solutions to equations (1.4.2.21) and (1.4.2.22) are given by ϕ(x) = A1 + A2 x + A3 eλx + A4 e−λx , (1.4.2.25) ψ(y) = B1 + B2 y + B3 eλy + B4 e−λy . Substituting (1.4.2.25) into (1.4.2.19), cancelling by λ3 , and combining like terms, we obtain A3 (νλ − B2 )eλx + A4 (νλ + B2 )e−λx + B3 (νλ + A2 )eλy + B4 (νλ − A2 )e−λy = 0. Equating the coefficients of the exponentials with zero, we get three sets of values of the constants: A3 = A4 = B3 = 0, A2 = νλ (case 1), A3 = B3 = 0, A2 = νλ, B2 = −νλ (case 2), A3 = B4 = 0, A2 = −νλ, B2 = −νλ (case 3). The other constants can assume any values. These sets of values determine three solutions to equation (1.4.2.17) of the form (1.4.2.18) [245]: u = C1 e−λy + C2 y + C3 + νλx, u = C1 e−λx + νλx + C2 e−λy − νλy + C3 ,

u = C1 e−λx − νλx + C2 eλy − νλy + C3 , where C1 , C2 , C3 , and λ are arbitrary constants.

3◦ . For C = λ2 < 0, the general solutions to equations (1.4.2.21) and (1.4.2.22) are

ϕ(x) = A1 + A2 x + A3 cos(λx) + A4 sin(λx), ψ(y) = B1 + B2 y + B3 cos(λy) + B4 sin(λy).

(1.4.2.26)

The substitution of (1.4.2.26) into (1.4.2.19) does not lead to any new real-valued solutions.

1.4. Solution of Functional Differential Equations by the Method of Differentiation

41

′′ Degenerate cases. If ϕ′′xx ≡ 0 or ψyy ≡ 0, equation (1.4.2.20) becomes an identity for any ψ = ψ(y) and any ϕ = ϕ(x), respectively. These cases must be treated separately. For example, if ϕ′′xx ≡ 0, we get ϕ(x) = Ax + B, where A and B are arbitrary constants. Substituting this function into (1.4.2.19), we arrive at the ′′′ ′′′′ equation −Aψyyy = νψyyyy , whose general solution is ψ(y) = C1 exp(−Ay/ν) + C2 y 2 + C3 y + C4 . Thus, we find another solution to equation (1.4.2.17) of the form (1.4.2.18):

u = C1 e−λy + C2 y 2 + C3 y + C4 + νλx

(A = νλ, B = 0).

It was obtained in [312] with the group analysis. An extensive list of exact solutions to equation (1.4.2.17) as well as related ◭ hydrodynamic equations can be found in [87, 275]. It is noteworthy that the results obtained for nonlinear equations of mathematical physics using generalized separation of variables are often easy to generalize substantially and extend to entire classes of nonlinear equations of higher order (sometimes even arbitrary order) with more complex coefficients dependent on the independent variables. We will illustrate this with a specific example. ◮ Example 1.30. Let us look at the nth-order nonlinear equation with a varying

coefficient uy uxy − ux uyy = f (x)u(n) y ,

(1.4.2.27)

where f (x) is an arbitrary function and n is an arbitrary positive integer. In the special case of n = 3 and f (x) = ν = const, it coincides with the boundary layer equation (1.3.2.42). Just as in Example 1.28, we seek a generalized separable solution to equation (1.4.2.27) in the form (1.4.2.10). On substituting (1.4.2.10) in (1.4.2.27) and on cancelling by ϕ, we get the functional differential equation ′′ ′′ ϕ′x [(θy′ )2 − θθyy ] − ψx′ θyy = f (x)θy(n) .

(1.4.2.28)

To remove the highest derivative, we divide equation (1.4.2.28) by f = f (x) and then differentiate with respect to x to obtain ′′ ′′ ] − (ψx′ /f )′x θyy = 0. (ϕ′x /f )′x [(θy′ )2 − θθyy

(1.4.2.29)

Nondegenerate case. By separating the variables in (1.4.2.29), we obtain (ψx′ /f )′x = C1 (ϕ′x /f )′x , ′′ ′′ (θy′ )2 − θθyy − C1 θyy = 0.

Integrating yields ϕ(x) is any, ψ(x) = C1 ϕ(x)+C2

Z

f (x) dx+C3 , θ(y) = C4 eλy −C1 , (1.4.2.30)

where C1 , . . . , C4 , and λ are constants of integration. Substituting (1.4.2.30) into (1.4.2.28), we find the relationship between the constants: C2 = −λn−2 . Considering

42

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

the above as well as formulas (1.4.2.10) and (1.4.2.30), we ultimately arrive at a solution to equation (1.4.2.27) of the form (1.4.2.10): Z

u = ϕ(x)eλy − λn−2 f (x) dx + C, where ϕ(x) is an arbitrary function, while C and λ are arbitrary constants (C = C3 and C4 = 1). Degenerate case. From equation (1.4.2.29) we get (ϕ′x /f )′x = 0,

(ψx′ /f )′x = 0,

θ(y) is any.

(1.4.2.31)

Integrating the first two equations of (1.4.2.31) twice, we find that ϕ(x) = C1

Z

f (x) dx + C2 ,

ψ(x) = C3

Z

f (x) dx + C4 ,

(1.4.2.32)

where C1 , . . . , C4 are arbitrary constants. Substituting (1.4.2.32) into (1.4.2.28) yields an ordinary differential equation for θ = θ(y): ′′ C1 (θy′ )2 − (C1 θ + C3 )θyy = θy(n) . (1.4.2.33) Hence, we obtain an exact solution for (1.4.2.27) in the form   Z Z u = C1 f (x) dx + C2 θ(y) + C3 f (x) dx + C4 , with the function θ = θ(y) satisfying the autonomous ODE (1.4.2.33). Note that equation (1.4.2.33) has exact solutions in the power-law and exponential forms 1 [Kn (y + A)2−n − C3 ], C1 1 θ(y) = Aeλy − (λn−2 + C3 ), C1

θ(y) =

Kn = (−1)n−1

(2n − 3)! (n − 2)!

(n = 2, 3, . . .);

where A, C1 , and C3 are arbitrary constants (C1 6= 0). For n = 1, one should set K1 = 1 in the first solution. Clearly, the procedures for constructing generalized separable solutions to the boundary layer equation (1.3.2.42) and the significantly more complicated equation ◭ (1.4.2.27) are not only very much the same but also similar in complexity. For functional differential equations with many terms, the differentiation methods can be applied several times to refine the solution structures successively. We will illustrate this with a specific example. ◮ Example 1.31. Consider the heat type equation with a quadratic nonlinearity

ut = auuxx + bu2x , which is a special case of equation (1.4.2.1) with c = 0.

(1.4.2.34)

1.4. Solution of Functional Differential Equations by the Method of Differentiation

43

We look for solutions to equation (1.4.2.34) in the form u = ψ(t)ξ(x) + ϕ(t)η(x),

(1.4.2.35)

with all four functions on the right to be determined. Substituting (1.4.2.35) into (1.4.2.34) gives ′′ ψt′ ξ + ϕ′t η = ψ 2 [aξξxx + b(ξx′ )2 ] ′′ + ψϕ[a(ξη)′′xx + 2(b − a)ξx′ ηx′ ] + ϕ2 [aηηxx + b(ηx′ )2 ].

(1.4.2.36)

Nondegenerate case. Multiplying equation (1.4.2.36) by suitable functions dependent on t and differentiating with respect to t (three times), we get rid of the second term on the left as well as the second and third terms on the right to obtain the two-term equation ′′ p1 (t)ξ = p2 (t)[aξξxx + b(ξx′ )2 ].

(1.4.2.37)

The functions p1 (t) and p2 (t) are dependent on ψ and ϕ; their specific forms are inessential for the subsequent analysis. It is easy to verify that the function ξ(x) = x2

(1.4.2.38)

satisfies equation (1.4.2.37). The scaling factor in the function has been set equal to one, which can always be done by renormalizing ψ in (1.4.2.35). Multiplying equation (1.4.2.36) by other suitable functions dependent on t and then differentiating with respect to t, we get rid of the first term on the left as well as the first and third terms on the right to obtain the equation q1 (t)η = q2 (t)[a(ξη)′′xx + 2(b − a)ξx′ ηx′ ].

(1.4.2.39)

Inserting the function (1.4.2.38) leads to a linear homogeneous equation in x, q1 (t)η = q2 (t)[a(x2 η)′′xx + 4(b − a)xηx′ ],

(1.4.2.40)

which is satisfied by any power-law function η(x) = xk .

(1.4.2.41)

Now substituting (1.4.2.38) and (1.4.2.41) into (1.4.2.36) and collecting the coefficients of like powers of x, we find that [ψt′ − 2(a + 2b)ψ 2 ]x2 + {ϕ′t − [a(k 2 − k + 2) + 4bk]ψϕ}xk

− k[a(k − 1) + bk]ϕ2 x2k−2 = 0, (1.4.2.42)

This relation must hold identically for any x. The last term in (1.4.2.42) can be balanced in the exponent with the first two terms only if k = 2; this leads to a simple separable solution, which is of little interest. The only remaining cases are a k = 0, k = , (1.4.2.43) a+b both ensuring that the coefficient of x2k−2 is zero.

44

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

For k = 0, equation (1.4.2.34) has a two-term quadratic solution in x, which was considered in Example 1.27 (see the formula in Item 1◦ with c = 0). The other value of k in (1.4.2.43) leads to a power-law exact solution in x: a

u = ψ(t)x2 + ϕ(t)x a+b ,

(1.4.2.44)

where b 6= −a; the functions ψ = ψ(t) and ϕ = ϕ(t) satisfy the ODEs ψt′ − Aψ 2 = 0,

A = 2(a + 2b),

ϕ′t − Bψϕ = 0,

B = a(a + 2b)(2a + 3b)(a + b)−2 ,

(1.4.2.45)

which follow from setting the functional coefficients of x2 and xk in (1.4.2.42) to zero. The general solution to (1.4.2.45) is expressed as ψ=−

1 , A(t + C1 )

ϕ = C2 (t + C1 )−B/A .

where C1 and C2 are arbitrary constants. Degenerate cases. Now let us look as the degenerate case where the left- and right-hand sides of the two-term equation (1.4.2.37) vanish simultaneously. 1◦ . Let p1 (t) and the expression in square brackets in (1.4.2.37) be zero. Then we arrive at the simple nonlinear equation ′′ aξξxx + b(ξx′ )2 = 0,

(1.4.2.46)

which admits several solutions analyzed below. (i) For any a and b, equation (1.4.2.46) has a degenerate solution ξ = 1, which generates the exact solutions of equation (1.4.2.34) described in Example 1.27; equation (1.4.2.34) coincides with (1.4.2.1) at c = 0, while solution (1.4.2.35) with ξ = 1 coincides, up to renaming, with solution (1.4.2.2). a

(ii) Equation (1.4.2.46) with b 6= −a has a power-law solution ξ = x a+b , which ultimately leads to the previously considered solution of the form (1.4.2.44) to the original equation (1.4.2.34); this solution corresponds to swapping the functions ξ and η in (1.4.2.35). (iii) Equation (1.4.2.46) with b = −a has an exponential solution ξ = eλx , which leads to the solutions considered in Example 1.27 (see Item 2◦ with c = 0 and A2 = 0). 2◦ . Let the functions p1 (t) and p2 (t) in (1.4.2.37) vanish simultaneously. Then we ultimately arrive at the exponential and trigonometric solutions considered previ◭ ously in Example 1.27 (see Items 2◦ and 3◦ with c = 0). ◮ Example 1.32. Consider the unnormalized Boussinesq equation

utt + a(uux )x + buxxxx = 0.

(1.4.2.47)

This equation arises in several physical applications: propagation of long waves in shallow water, one-dimensional nonlinear lattice waves, vibration of a nonlinear string, and ion acoustic waves in a plasma [39, 330, 348].

1.4. Solution of Functional Differential Equations by the Method of Differentiation

45

As in Example 1.31, we look for solutions to equation (1.4.2.47) in the form (1.4.2.35). Substituting (1.4.2.35) into (1.4.2.47) yields ′′ ψtt ξ + ϕ′′tt η + aψ 2 (ξξx′ )′x + aψϕ(ξη)′′xx ′′′′ ′′′′ + aϕ2 (ηηx′ )′x + bψξxxxx + bϕηxxxx = 0.

(1.4.2.48)

Nondegenerate case. Multiplying relation (1.4.2.48) by suitable functions dependent on t followed by differentiating with respect to t, we get rid of all terms except for the first and third ones to obtain the two-term equation p1 (t)ξ = p2 (t)(ξξx′ )′x .

(1.4.2.49)

The functions p1 (t) and p2 (t) depend on ψ and ϕ; their specific forms are inessential for the subsequent analysis. One can check by direct verification that the quadratic function ξ = x2 satisfies equation (1.4.2.49). Multiplying equation (1.4.2.48) by other suitable functions dependent on t followed by differentiating with respect to t, we get rid of all terms except for the second and fourth ones to obtain the equation q1 (t)η = q2 (t)(ξη)′′xx .

(1.4.2.50)

Inserting ξ = x2 gives a linear equation homogeneous in x, q1 (t)η = q2 (t)(x2 η)′′xx ,

(1.4.2.51)

which is satisfied by any power-law function η = xk . Substituting ξ = x2 and η = xk into (1.4.2.48) and collecting the coefficients of like powers of x, we find that ′′ (ψtt + 6aψ 2 )x2 + [ϕ′′tt + a(k + 1)(k + 2)ψϕ]xk

+ ak(2k − 1)ϕ2 x2k−2 + bk(k − 1)(k − 2)(k − 3)ϕxk−4 = 0. (1.4.2.52) This equation must hold identically for any x. The last two terms in (1.4.2.52) can be balanced in the exponent only if k = −2. In this case, ϕ = −12 b/a = const and the functional coefficient of xk in (1.4.2.52) vanishes. As a result, we arrive at the following exact solution to the Boussinesq equation (1.4.2.47) [72]: u = ψ(t)(x + C1 )2 −

12 b . a(x + C1 )2

(1.4.2.53)

For generality, a translation by C1 has been added to x. The function ψ = ψ(t) is determined from the autonomous ordinary differential equation ′′ + 6aψ 2 = 0, ψtt

(1.4.2.54)

whose general solution can be written in explicit form. Equation (1.4.2.54) has a simple particular solution ψ = −a−1 (t + C2 )−2 . To k = 0 in (1.4.2.52) there corresponds a degenerate solution, which is not considered here. The articles [72, 73] (see also [274, 275]) present a number of other exact solu◭ tions to equation (1.4.2.47).

46

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

1.5. Solution of Functional Differential Equations by the Splitting Method 1.5.1. Preliminary Remarks. Description of the Method. The Splitting Principle Although the number of terms in the functional differential equation (1.2.2.1) decreases when one employs the differentiation method, some redundant constants of integrations arise. These constants must be removed in the final step (see Section 1.4). Moreover, the order of the resulting equation can be higher than that of the original one. To avoid these difficulties, instead of solving the functional differential equation, it is often easier to solve a standard bilinear functional equation followed by solving a system of ordinary differential equations. This way, the original problem splits into two simpler problems. Below we outline the main steps of this method. 1◦ . First, we treat equation (1.2.2.1) as a bilinear functional equation: k X

Φn Ψn = 0,

(1.5.1.1)

n=1

where Φn = Φn [X] and Ψn = Ψn [Y ] are the unknown variables (n = 1, . . . , k), while X and Y are independent variables. The splitting principle. All solutions to the bilinear functional equation (1.5.1.1) can be represented as a number of linear combinations of Φ1 , . . . , Φk together with linear combinations of Ψ1 , . . . , Φk : k X

αni Φn = 0,

i = 1, . . . , l;

n=1 k X

(1.5.1.2) βnj Ψn = 0,

j = 1, . . . , m,

n=1

where 1 ≤ l ≤ k − 1 and 1 ≤ m ≤ k − 1. The constants αni and βnj in (1.5.1.2) are chosen so that the bilinear relation (1.5.1.1) holds identically (this can always be done as shown later in Subsection 1.5.2). Importantly, relations (1.5.1.2) are purely algebraic in nature and are independent of any particular expressions of the differential forms (1.2.2.2). 2◦ . In the second stage, we successively replace the variables Φi and Ψj in solutions (1.5.1.2) with the differential forms Φi [X] and Ψj [Y ] from (1.2.2.2). As a result, we obtain systems of ordinary differential equations, which are often overdetermined, to find the functions ϕp (x) and ψq (y). By solving these systems, one obtains generalized separable solutions of the form (1.2.1.1). 3◦ . Finally, in addition to the linear relations (1.5.1.2), one must treat separately the degenerate cases where one or more of the differential forms Φn and/or Ψn vanish.

1.5. Solution of Functional Differential Equations by the Splitting Method

47

Remark 1.22. Birkhoff [29] was the first to apply the splitting principle to seek generalized self-similar solutions to the Navier–Stokes equations. Remark 1.23. Importantly, the bilinear functional equation (1.5.1.1) (or (1.2.2.1)) used in the splitting principle as well as its solutions for fixed k remain the same for different classes of the original nonlinear equations of mathematical physics. Remark 1.24. The splitting principle can be proved by mathematical induction [104, 261] as follows. 1. Indeed, this principle holds for k = 2 (see formulas (1.4.1.2) and (1.4.1.3)). 2. Suppose that the principle holds for arbitrary k > 2, that is, the solutions are described by linear relations of the form (1.5.1.2). 3. Let us look at equation (1.5.1.1) with k + 1 rather than k terms. Dividing it by Ψk+1 and differentiating with respect to Y , we arrive at an equation of the same form but with fewer terms (see equation (1.4.1.1) in which k − 1 should be replaced with k). Its solutions are ˜ n = Φn given by formulas of the form (1.5.1.2) in which Φn and Ψn must be substituted by Φ ˜ n = (Ψn /Ψk+1 )′y , respectively. The first set of linear relations for Φ ˜ n = Φn remains and Ψ ˜ n , with respect to Y followed by unchanged. Integrating the second set of relations, for Ψ multiplying by Ψk+1 , we again arrive at linear relations: k X

βnj Ψn + Cj Ψk+1 = 0,

j = 1, . . . , m,

n=1

where Cj are arbitrary constants. Thus, from the assumption that the solutions to the functional equation (1.5.1.1) for any k > 2 are determined by the linear relations (1.5.1.2), it follows that the solutions to the functional equation with k + 1 terms also satisfy linear relations. This is what was to be proved. Remark 1.25. The bilinear functional equation (1.5.1.1) and its solutions (1.5.1.2) play a key role in the method of functional separation of variables (see Subsections 2.5.2 and 2.7.1).

For clarity, Fig. 1.1 displays the main steps of finding generalized separable solutions by the splitting method.

1.5.2. Solutions of Bilinear Functional Equations 1◦ . In practice, to obtain solutions to the bilinear functional equation (1.5.1.1), one should proceed as follows. First, one chooses a few first elements Φ1 , . . . , Φp (p < k) out of the entire set Φ1 , . . . , Φk and represents them as linear combinations of the remaining elements Φp+1 , . . . , Φk . This defines the first set of relations (1.5.1.2). On replacing Φ1 , . . . , Φp in (1.5.1.1) with their linear combinations in terms of Φp+1 , . . . , Φk , one arrives at a relation of the form k X

q=p+1

Ωq Φq = 0,

Ωq =

k X

aqs Ψs ,

s=1

where aqs are some constants. Setting the functional coefficients Ωq (q = p+1, . . . , k) to zero gives the second set of relations (1.5.1.2). Due to the symmetry of equations (1.5.1.1) with respect to the Φ’s and Ψ’s, one can start with choosing elements from the set Ψ1 , . . . , Ψk rather than Φ1 , . . . , Φk .

48

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Figure 1.1. The schematic of constructing generalized separable solutions by the splitting method.

◮ Example 1.33. Consider the bilinear functional equation

Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 + Φ5 Ψ5 = 0.

(1.5.2.1)

Suppose that the first three functions, Φ1 , Φ2 , and Φ3 , are linear combinations of the last two: Φ1 = A1 Φ4 + B1 Φ5 ,

Φ2 = A2 Φ4 + B2 Φ5 ,

Φ3 = A3 Φ4 + B3 Φ5 , (1.5.2.2)

where A1 , A2 , A3 and B1 , B2 , B3 are arbitrary constants. Substituting (1.5.2.2) into (1.5.2.1) and collecting the terms as the coefficients of Φ4 and Φ5 , we get (A1 Ψ1 + A2 Ψ2 + A3 Ψ3 + Ψ4 )Φ4 + (B1 Ψ1 + B2 Ψ2 + B3 Ψ3 + Ψ5 )Φ5 = 0. By setting the expressions in parentheses to zero, we obtain Ψ4 = −A1 Ψ1 − A2 Ψ2 − A3 Ψ3 , Ψ5 = −B1 Ψ1 − B2 Ψ2 − B3 Ψ3 .

(1.5.2.3)

Formulas (1.5.2.2) and (1.5.2.3) give one of the solutions to equation (1.5.2.1). The ◭ other solutions can be found likewise.

1.5. Solution of Functional Differential Equations by the Splitting Method

49

2◦ . Below we give the nondegenerate solutions to two simple functional equations of the form (1.5.1.1) (or (1.2.2.1)), which will be used subsequently to construct exact solutions of specific partial differential equations. The three-term bilinear functional equation Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 = 0,

(1.5.2.4)

where all Φi have one argument and all Ψi are functions of another argument, has two solutions Φ1 = A1 Φ3 , Φ2 = A2 Φ3 , Ψ3 = −A1 Ψ1 − A2 Ψ2 ; Ψ1 = A1 Ψ3 , Ψ2 = A2 Ψ3 , Φ3 = −A1 Φ1 − A2 Φ2 ,

(1.5.2.5)

where A1 and A2 are arbitrary constants. All functions on the right-hand sides of (1.5.2.5) are assumed to be arbitrary. In solutions (1.5.2.5), all functions Φi (first solution) or all functions Ψi (second solution) are proportional to each other. The functional equation with four terms Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 = 0,

(1.5.2.6)

has a solution Φ1 = A1 Φ3 + A2 Φ4 , Φ2 = A3 Φ3 + A4 Φ4 , Ψ3 = −A1 Ψ1 − A3 Ψ2 , Ψ4 = −A2 Ψ1 − A4 Ψ2 ,

(1.5.2.7)

which involves four arbitrary constants A1 , . . . , A4 . The functions on the right-hand sides in (1.5.2.7) are all assumed to be arbitrary. Equation (1.5.2.6) also has two simpler solutions involving three arbitrary constants: Φ1 = A1 Φ4 , Φ2 = A2 Φ4 , Φ3 = A3 Φ4 , Ψ4 = −A1 Ψ1 − A2 Ψ2 − A3 Ψ3 ; Ψ1 = A1 Ψ4 , Ψ2 = A2 Ψ4 , Ψ3 = A3 Ψ4 , Φ4 = −A1 Φ1 − A2 Φ2 − A3 Φ3 . (1.5.2.8) In solutions (1.5.2.8), the functions Φi or the functions Ψi are all proportional to each other. In practice, solution (1.5.2.7) to the bilinear functional equation (1.5.2.6) are used more frequently than solutions (1.5.2.8); this allows one to find generalized separable solutions to nonlinear partial differential equations. 3◦ . It can be shown that the bilinear functional equation (1.5.1.1) (or (1.2.2.1)) has (k − 1) different nondegenerate solutions: Φi = Ci,1 Φm+1 + Ci,2 Φm+2 + · · · + Ci,k−m Φk , Ψm+j = −C1,j Ψ1 − C2,j Ψ2 − · · · − Cm,j Ψm , i = 1, . . . , m;

j = 1, . . . , k − m;

(1.5.2.9)

m = 1, . . . , k − 1;

where Ci,j are arbitrary constants. The functions Φm+1 , . . . , Φk and Ψ1 , . . . , Ψm on the right-hand sides of relations (1.5.2.9) are assumed to be arbitrary. The total

50

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

number of linear relations in the first and second rows of (1.5.2.9) equals k, which is the number of terms in the bilinear functional equation (1.5.1.1). It is apparent that for a fixed m, solution (1.5.2.9) involves m(k − m) arbitrary constants. Remark 1.26. For a fixed m, there are m(k − m) arbitrary constants Ci,j in solution (1.5.2.9). For a fixed k, the following solutions have the maximum number of arbitrary constants:

Solution number m= m=

1 2 1 2

Number of arbitrary constants 1 4 1 4

k (k ± 1)

k

2

Condition for k k is an even integer,

2

(k − 1)

k is an odd integer.

In practice, it is these solutions of bilinear functional equations that most frequently lead to nontrivial generalized separable solutions of nonlinear partial differential equations.

By virtue of discrete symmetries of bilinear functional equations with respect to permutations of the unknowns, the symbols Φ and Ψ can be swapped in solutions (1.5.2.5), (1.5.2.7), (1.5.2.8), and (1.5.2.9). Specifically, the permutations Φp ⇄ Ψp are allowed as well as the simultaneous pairwise permutations Φp ⇄ Φq and Ψp ⇄ Ψq (this can be repeated as many times as required to ensure that all possible permutations are done). Remark 1.27. Subsection 2.7.1 will describe some special solutions of the bilinear functional equation (1.5.1.2) for arbitrary k.

1.5.3. Examples of Constructing Generalized Separable Solutions by the Splitting Method Below we give a few specific examples of utilizing the splitting method to construct generalized separable solutions to nonlinear PDEs. ◮ Example 1.34. Let us look at the nonlinear hyperbolic type equation

utt = a(uux )x + f (t)u + g(t),

(1.5.3.1)

where f (t) and g(t) are arbitrary functions. We seek generalized separable solutions of the form u = ϕ(t)θ(x) + ψ(t).

(1.5.3.2)

Substituting (1.5.3.2) into (1.5.3.1) and rearranging, we get ′′ ′′ = 0. + (f ϕ − ϕ′′tt )θ + f ψ + g − ψtt aϕ2 (θθx′ )′x + aϕψθxx

This equation can be represented as the four-term bilinear functional equation (1.5.2.6) with ′′ , Φ1 = (θθx′ )′x , Φ2 = θxx 2

Ψ1 = aϕ ,

Φ3 = θ;

Ψ2 = aϕψ, Ψ3 = f ϕ −

Φ4 = 1, ϕ′′tt ,

′′ Ψ4 = f ψ + g − ψtt .

(1.5.3.3)

51

1.5. Solution of Functional Differential Equations by the Splitting Method

Substituting (1.5.3.3) into solution (1.5.2.7), we arrive at a system of ordinary differential equations for the unknown functions θ = θ(x), ϕ = ϕ(t), and ψ = ψ(t): (θθx′ )′x = A1 θ + A2 ,

′′ θxx = A3 θ + A4 ;

′′ f ϕ − ϕ′′tt = −A1 aϕ2 − A3 aϕψ, f ψ + g − ψtt = −A2 aϕ2 − A4 aϕψ,

(1.5.3.4)

where A1 , . . . , A4 are arbitrary constants. The first two equations in (1.5.3.4) make up an overdetermined system for the single function θ. The second equation is linear and so is easy to integrate. Depending on the value of A3 , its solution is expressed in terms of trigonometric functions (for A3 < 0), hyperbolic functions (for A3 > 0), or a quadratic polynomial (for A3 = 0). Substituting one by one these solutions into the first equation, we conclude that the only compatible solution to both these equations is the quadratic polynomial θ(x) = B2 x2 + B1 x + B0 ,

(1.5.3.5)

in which the constants B0 , B1 , and B2 are related to the constants A1 , . . . , A4 as follows: A1 = 6B2 ,

A2 = B12 − 4B0 B2 ,

A3 = 0,

A4 = 2B2 ,

(1.5.3.6)

Inserting the expressions (1.5.3.6) into the last two equations in (1.5.3.4), we obtain the following system for ϕ(t) and ψ(t): ϕ′′tt = 6aB2 ϕ2 + f (t)ϕ, ′′ ψtt = [2aB2 ϕ + f (t)]ψ + a(B12 − 4B0 B2 )ϕ2 + g(t).

(1.5.3.7)

Formulas (1.5.3.2) and (1.5.3.5) together with system (1.5.3.7) define a generalized separable solution to equation (1.5.3.1). The first equation in (1.5.3.7) is solved independently; it is linear if B2 = 0 and integrable by quadrature if f (t) = const. The second equation in (1.5.3.7) is linear in ψ, once ϕ is known. For θ 6≡ 0, ϕ 6≡ 0, and ψ 6≡ 0 as well as arbitrary f and g, equation (1.5.3.1) has no other solutions of the form (1.5.3.2). Note that the constant a in equations (1.5.3.1), (1.5.3.4), and (1.5.3.7) can be ◭ replaced with a(t). Remark 1.28. Equation (1.5.3.1) has a more general, polynomial solution in x [114]:

u = x2 ψ1 (t) + xψ2 (t) + ψ3 (t).

(1.5.3.8)

The functions ψi = ψi (t) are determined from the ordinary differential equations (a prime stands for a derivative with respect to t) ψ1′′ = 6aψ12 + f (t)ψ1 , ψ2′′ = [6aψ1 + f (t)]ψ2 ,

(1.5.3.9)

ψ3′′ = [2aψ1 + f (t)]ψ3 + aψ22 + g(t).

The second equation in 1.5.3.9 has a particular solution ψ2 = ψ1 . Hence, its general solution can be written in the form [273]: Z dt ψ2 = C1 ψ1 + C2 ψ1 . ψ12

52

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

◮ Example 1.35. Consider the nonlinear Monge–Amp`ere type equation

u2xy + kuxx uyy = f (x)g(y).

(1.5.3.10)

For k = −1, equations of this form are encountered in differential geometry, gas dynamics, and meteorology. We look for generalized separable solutions with the form u = ϕ(x)θ(y) + ψ(x).

(1.5.3.11)

On substituting (1.5.3.11) into (1.5.3.10) and on collecting like terms, we obtain ′′ ′′ ′′ kϕϕ′′xx θθyy + kϕψxx θyy + (ϕ′x )2 (θy′ )2 − f (x)g(y) = 0.

(1.5.3.12)

This functional differential equation can be represented as the four-term bilinear functional equation (1.5.2.6) by setting ′′ Φ1 = kϕϕ′′xx , Φ2 = kϕψxx , Φ3 = (ϕ′x )2 , Φ4 = f (x); ′′ Ψ1 = θθyy ,

′′ Ψ2 = θyy ,

Ψ3 = (θy′ )2 ,

Ψ4 = −g(y).

(1.5.3.13)

On substituting (1.5.3.13) into (1.5.3.10), we get kϕϕ′′xx = A1 (ϕ′x )2 + A2 f (x), ′′ ′′ (θy′ )2 = −A1 θθyy − A3 θyy ,

′′ kϕψxx = A3 (ϕ′x )2 + A4 f (x); ′′ ′′ g(y) = A2 θθyy + A4 θyy .

(1.5.3.14)

The function ϕ(x) can be determined from the first equation. Then, by double simple integration of the second equation, we find ψ(x): Z A3 [ϕ′t (t)]2 + A4 f (t) 1 x (x − t) dt + B1 x + B2 , ψ(x) = k x0 ϕ(t) where B1 and B2 are arbitrary constants. The third equation serves to determine θ(y) and the last equation serves to determine the admissible function g(y). For A1 = −k, the first equation in (1.5.3.14) is integrable by quadrature for any f (x): Z 2A2 x 2 ϕ = (x − t)f (t) dt + C1 x + C2 , k x0 where C1 and C2 are arbitrary constants and x0 is any number for which the integral exists (if the integrand does not have singularities, we can set x0 = 0). The third equation in (1.5.3.14) is easy to integrate; for A1 6= 0, without loss of generality, it can be assumed that A3 = 0 (this is achieved by translating θ by a constant, which leads to redefining ψ in (1.5.3.11); see Remark 1.4). For A1 6= 0 and A3 = 0, power-law functions and exponentials are solutions to the equation. In the special case A1 = 0, the equation is solved by a logarithmic function. The last relation in (1.5.3.14) serves to identify the admissible functions g(y). These results are summarized in the first three rows of Table 1.2. The table also lists three degenerate solutions, which correspond to vanishing second derivatives of the determining functions in solution (1.5.3.11).

53

1.5. Solution of Functional Differential Equations by the Splitting Method

Table 1.2. Exact solutions of the form (1.5.3.11) for the Monge–Amp`ere type equation (1.5.3.10); a, b, n, and λ are free parameters (the sum C1 x + C2 y + C3 can be added to all solutions; C1 , C2 , and C3 are arbitrary constants). No. Function f (x) Function g(y)

Any

1

ay n + by 2n+2 (n 6= −1, −2)

2

Any

a ln y + b y2

3

Any

aeλy + be2λy

4

Any

Any

5

Any

1

Generalized separable solution u(x, y) Z x a f (t) u = ϕ(x)y n+2 + dt; (x − t) k(n + 1)(n + 2) x0 ϕ(t) k(n + 1)(n + 2)ϕϕ′′xx + (n + 2)2 (ϕ′x )2 = bf (x) Z 1 x [ϕ′ (t)]2 − bf (t) u = ϕ(x) ln y + dt; (x − t) t k x0 ϕ(t) kϕϕ′′xx + af (x) = 0 Z x a f (t) u = ϕ(x)eλy + 2 dt; (x − t) kλ x0 ϕ(t)

u=

C1 k

kλ2 ϕϕ′′xx + λ2 (ϕ′x )2 = bf (x) Z y x 1 (y − ξ)g(ξ) dξ (x − t)f (t) dt + C1 y 0 x0 Z xp u = ±y f (x) dx + ψ(x);

Z

x0

ψ(x) is an arbitrary function

6

u = (ax + b)θ(y) + c(ax + b)[ln(ax + b) − 1]; ′′ a2 ckθyy + a2 (θy′ )2 − g(y) = 0

Any

1

It is noteworthy that the determining equations (1.5.3.14) and the first five solutions remain valid if k = k(x) is an arbitrary function; in these solutions, the fraction 1/k must first be carried over into the integrand, and then replaced with the function 1/k(t). ◭ ◮ Example 1.36. Let us look at the third-order nonlinear hydrodynamic type

equation uxt + u2x − uuxx = νuxxx .

(1.5.3.15)

We seek exact solutions in the form u = ϕ(t)θ(x) + ψ(t).

(1.5.3.16)

Substituting (1.5.3.16) into (1.5.3.15) gives   ′′′ ′′ ′′ − νϕθxxx = 0. ϕ′t θx′ − ϕψθxx + ϕ2 (θx′ )2 − θθxx

(1.5.3.17)

This functional differential equation can be reduced to the four-term bilinear functional equation (1.5.2.6) by setting Φ1 = ϕ′t , Φ2 = ϕψ, Ψ1 =

θx′ ,

Ψ2 =

′′ −θxx ,

Φ3 = ϕ2 , Ψ3 =

(θx′ )2

Φ4 = νϕ; −

′′ θθxx ,

′′′ Ψ4 = −θxxx .

(1.5.3.18)

54

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

First group of solutions. Substituting expressions (1.5.3.18) into (1.5.2.7) yields the system of ordinary differential equations ϕ′t = A1 ϕ2 + A2 νϕ, (θx′ )2

−

′′ θθxx

=

−A1 θx′

ϕψ = A3 ϕ2 + A4 νϕ; +

′′ A3 θxx ,

′′′ ′′ θxxx = A2 θx′ − A4 θxx .

(1.5.3.19)

The last two equations in (1.5.3.19) make up an overdetermined system of ODEs for the single function θ; the last one is linear and so is easy to integrate. It can be shown that the two equations have compatible solutions only if θ is linearly related to its derivative: θx′ = B1 θ + B2 . (1.5.3.20) We eliminate the derivatives from the last two equations in (1.5.3.19) using relation (1.5.3.20) and its corollaries obtained by differentiation. As a result, we find that the six constants B1 , B2 , A1 , A2 , A3 , and A4 must satisfy three conditions B1 (A1 + B2 − A3 B1 ) = 0, B2 (A1 + B2 − A3 B1 ) = 0, B12

Integrating (1.5.3.20) gives ( θ=

(1.5.3.21)

+ A4 B1 − A2 = 0.

B3 exp(B1 x) − B2 x + B3

B2 B1

for B1 6= 0,

(1.5.3.22)

for B1 = 0,

where B3 is an arbitrary constant. From the first two equations in (1.5.3.19), we find the functions ϕ and ψ:  A2 ν   if A2 6= 0,  C exp(−A2 νt) − A1 ψ = A3 ϕ + A4 ν, (1.5.3.23) ϕ= 1   − if A2 = 0, A1 t + C

where C is an arbitrary constant. Formulas (1.5.3.22) and (1.5.3.23) as well as relations (1.5.3.21) allow us to find the following generalized separable solutions to equation (1.5.3.15): C1 e−λx + 1 + νλ λt + C2 u = C1 e−λ(x+βνt) + ν(λ + β) u=

−λx

if

A2 = 0, B1 = −A4 , B2 = −A1 − A3 A4 ;

if

A1 = A3 = B2 = 0, A2 = B12 + A4 B1 ;

νβ + C1 e + ν(λ − β) if 1 + C2 e−νλβt x + C1 u= + C3 if t + C2

u=

A1 = A3 B1 − B2 , A2 = B12 + A4 B1 ; A2 = B1 = 0, B2 = −A1 ,

where C1 , C2 , C3 , β, and λ are arbitrary constants, which can be expressed in terms of Ak and Bk .

55

1.5. Solution of Functional Differential Equations by the Splitting Method

Second group of solutions. Let all terms dependent on x in equation (1.5.3.17) be proportional to θx′ . Then we obtain the system of ordinary differential equations ′′′ ′′ ′′ θxxx = A1 θx′ , θxx = A2 θx′ , (θx′ )2 − θθxx = A3 θx′ ;

ϕ′t − A2 ϕψ + A3 ϕ2 − A1 νϕ = 0.

(1.5.3.24)

The overdetermined subsystem consisting of the first three equations in (1.5.3.24) admits two compatible solutions: solution 1: solution 2:

θ = e−λx θ=x

if if

A1 = λ2 , A2 = −λ, A3 = 0; A1 = A2 = 0, A3 = 1.

(1.5.3.25)

Solutions (1.5.3.25) together with the solutions of the last equation in (1.5.3.24), which are easy to obtain, ultimately lead to the following two solutions of the original nonlinear differential equation (1.5.3.15): u = ϕ(t)e−λx − u=

ϕ′t (t) + νλ, λϕ(t)

x + ψ(t), t+C

where ϕ(t) and ψ(t) are arbitrary functions; C and λ are arbitrary constants.

◭

◮ Example 1.37. Let us return to the fourth-order nonlinear hydrodynamic type equation (1.4.2.17). Just as in Example 1.29, we look for its additive separable solutions in the form (1.4.2.18). We will arrive at the functional differential equation (1.4.2.19) and rewrite in the bilinear form

Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 = 0,

(1.5.3.26)

where ′′′ ′ Φ1 = ϕ′′′′ Φ4 = ν; xxxx , Φ2 = ϕxxx , Φ3 = ϕx , ′ ′′′ ′′′′ Ψ1 = ν, Ψ2 = −ψy , Ψ3 = ψyyy , Ψ4 = ψyyyy .

(1.5.3.27)

For convenience, the functions Φi are numbered in order from the highest to lowest derivative of ϕ. Now we take advantage of the solutions to the functional equation (1.5.3.26) presented in Subsection 1.5.2. 1◦ . Substituting (1.5.3.27) into formulas (1.5.2.7), which involve four free parameters, A1 to A4 , and satisfy identically the functional equation (1.5.3.26), we arrive at the following overdetermined system of linear ordinary differential equations with constant coefficients: ′ ′ ϕ′′′ ϕ′′′′ xxx = A3 ϕx + A4 ν; xxxx = A1 ϕx + A2 ν, ′′′ ′ ′′′′ ψyyy = −A1 ν + A3 ψy , ψyyyy = −A2 ν + A4 ψy′ .

(1.5.3.28)

These equations are easy to integrate: first, we look for solutions to the third-order ODEs and then determine the constants of integration and equation parameters by

56

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

inserting the resulting solutions into the fourth-order equations. Now we will consider the cases A3 > 0, A3 = 0, and A3 < 0 in order. (i) For A3 = λ2 > 0, the first two equations in (1.5.3.28) have the compatible solution ϕ = C1 e−λx + C2 νx + C3 , (1.5.3.29) where C1 , C2 , C3 , and λ are arbitrary constants, provided that the equation coefficients are defined as A1 = −λ3 ,

A2 = C2 λ3 ,

A3 = λ2 ,

A4 = −C2 λ2 .

(1.5.3.30)

The last two equations in (1.5.3.28) with coefficients (1.5.3.30) have two different compatible solutions: ψ = C4 e−λy − νλy ψ = C4 e

λy

− νλy

if C2 = λ;

(1.5.3.31)

if C2 = −λ,

where C4 is an arbitrary constant. Adding up solutions (1.5.3.29) and (1.5.3.31), we obtain two solutions of the original hydrodynamic type equation (1.4.2.17) in the form (1.4.2.18): u = C1 e−λx + νλx + C3 + C4 e−λy − νλy;

(1.5.3.32)

u = C1 e−λx − νλx + C3 + C4 eλy − νλy.

In addition, if

A1 = C5 λ2/ν,

A2 = −C5 λ3/ν,

A3 = λ2 ,

A4 = −λ3 ,

the overdetermined system (1.5.3.28) admits a degenerate solution in one variable, ϕ = νλx+C3 , and a nondegenerate solution in the other variable, ψ = C4 e−λy +C5 y. This results in the following solution to the original equation (1.4.2.17): u = νλx + C3 + C4 e−λy + C5 y.

(1.5.3.33)

It differs from the first solution (1.5.3.32) with C1 = 0 in the arbitrary coefficient, C5 , of y. (ii) For A3 = 0, the overdetermined system (1.5.3.28) has a compatible solution polynomial of degrees two and three in the independent variables. This leads to solutions (1.4.2.24). (iii) For A3 = λ2 < 0, the general solution of the second equation in (1.5.3.28) involves trigonometric functions, ϕ(x) = C1 cos(λx)+C2 sin(λx)+A4 νλ−2 x+C3 , where Ci are arbitrary constants. Inserting it into the first equation in (1.5.3.28), we find that C1 = C2 = 0. The last two equations in (1.5.3.28) are treated likewise. Thus, in this case, the solution is linear in both x and y and, hence, is a special case of the above solution from Item (ii). 2◦ . Now let us substitute expressions (1.5.3.27) into the upper row of formulas (1.5.2.8), which involve three arbitrary parameters, A1 , A2 , and A3 , and satisfy identically the functional equation (1.5.3.26). We arrive at the overdetermined system of linear ordinary differential equations with constant coefficients ϕ′′′′ ϕ′′′ ϕ′x = A3 ν; xxxx = A1 ν, xxx = A2 ν, ′′′′ ′′′ ψyyyy = −A1 ν + A2 ψy′ − A3 ψyyy .

(1.5.3.34)

1.5. Solution of Functional Differential Equations by the Splitting Method

57

The first three equations in (1.5.3.34) admit a simple compatible solution ϕ(x) = A3 νx,

A1 = A2 = 0,

A3 is any.

(1.5.3.35)

With the same values of Ai , the general solution of the last equation in (1.5.3.34) is given by ψ(y) = C1 exp(−A2 y) + C2 y 2 + C3 y + C4 , (1.5.3.36) where C1 , . . . , C4 are arbitrary constants. By adding up solutions (1.5.3.35) and (1.5.3.36), we get a solution to the original hydrodynamic type equation (1.4.2.17) in the form (1.4.2.18): u = νλx + C1 e−λy + C2 y 2 + C3 y + C4

(λ = A2 ).

It generalizes solution (1.5.3.33), since it contains an additional quadratic term, ◭ C2 y 2 . ◮ Example 1.38. Consider the equation with an exponential nonlinearity in the

highest derivative ut = f (x) exp(auxx ).

(1.5.3.37)

We look for exact solutions of the form u = ϕ(x)θ(t) + ψ(x).

(1.5.3.38)

Substituting (1.5.3.38) into (1.5.3.37), dividing the resulting equation by f (x), taking the logarithm, assuming that ϕ/f > 0, and performing simple rearrangements, we obtain ′′ aψxx − ln(ϕ/f ) + aθϕ′′xx − ln θt′ = 0. This functional differential equation can be rewritten in the bilinear form (1.5.2.4) with ′′ Φ1 = aψxx − ln(ϕ/f ), Φ2 = ϕ′′xx , Φ3 = 1; Ψ1 = 1, Ψ2 = aθ, Ψ3 = − ln θt′ .

Substituting these expression into the first solution in (1.5.2.5), we arrive at the ordinary differential equations ′′ aψxx − ln(ϕ/f ) = A1 ,

ϕ′′xx = A2 ,

ln θt′ = A1 + A2 aθ.

Integrating yields 1 A2 x2 + C1 x + C2 , 2 Z 1 1 x ϕ(ξ) ψ(x) = A1 x2 + C3 x + C4 + dξ, (x − ξ) ln 2a a x0 f (ξ)
1 θ(t) = − ln C5 − A2 aeA1 t . A2 a ϕ(x) =

(1.5.3.39)

Formulas (1.5.3.38) and (1.5.3.39) define an exact generalized separable solution ◭ to equation (1.5.3.37).

58

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method) 1.6.1. Subspaces Invariant under a Nonlinear Differential Operator. Description of the Method This section outlines the method of invariant subspaces∗ [108, 112, 114]. It does not involve the analysis of functional differential equations and differs significantly from the methods discussed previously in Sections 1.4 and 1.5. Consider the evolution equation ut = F [u],

(1.6.1.1)

where F [u] is a nonlinear differential operator of the form F [u] ≡ F (x, u, ux , . . . , u(n) x ).

(1.6.1.2)

Definition [114]. A finite-dimensional linear subspace  Lk = ϕ1 (x), . . . , ϕk (x) ,

(1.6.1.3)

whose elements are all possible linear combinations of linearly independent functions ϕ1 (x), . . . , ϕk (x), is said to be invariant under a differential operator F if F [Lk ] ⊆ Lk . This means that there exist some functions f1 , . . . , fk such that F

X k i=1

 X k Ci ϕi (x) = fi (C1 , . . . , Ck )ϕi (x)

(1.6.1.4)

i=1

for arbitrary constants C1 , . . . , Ck . Note that the functions ϕi (x) appearing in (1.6.1.4) must be independent of the constants C1 , . . . , Ck . Proposition 1. Suppose that a linear subspace (1.6.1.3) is invariant under a differential operator F . Then equation (1.6.1.1) has generalized separable solutions of the form [114]: k X u= ψi (t)ϕi (x), (1.6.1.5) i=1

in which the functions ψ1 (t), . . . , ψk (t) are described by an autonomous system of ordinary differential equations: ψi′ = fi (ψ1 , . . . , ψk ),

i = 1, . . . , k.

The prime stands for a derivative with respect to t. ∗ In

[274, 275, 278], it is referred to as the Titov–Galaktionov method.

(1.6.1.6)

59

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

This can be proved as follows. First, we substitute expression (1.6.1.5) into equation (1.6.1.1) and use relation (1.6.1.4) with the constants Ci replaced with the functions ψi = ψi (t). Then, collecting the coefficients of ϕi = ϕi (x), we obtain the relation k X [ψi′ − fi (ψ1 , . . . , ψk )]ϕi (x) = 0. i=1

Since the functions ϕi are linearly independent, all expressions in square brackets must be set equal to zero. This leads to the system of ordinary differential equations (1.6.1.6). The examples below illustrate the outlined method for constructing generalized separable solutions. ◮ Example 1.39. Consider the nonlinear heat equation with a linear source

ut = (uux )x + bu.

(1.6.1.7)

1◦ . Let us prove the three-dimensional linear subspace with power-law basis  that elements L3 = 1, x, x2 is invariant under the nonlinear differential operator F [u] = (uux )x + bu

(1.6.1.8)

that defines the right-hand side of equation (1.6.1.7). Indeed, for arbitrary C1 , C2 , and C3 , the following relation holds:   F C1 + C2 x + C3 x2 = 2C1 C3 + C22 + bC1 + (6C2 C3 + bC2 )x + (6C32 + bC3 )x2 .

It shows that under the action of operator (1.6.1.8), any quadratic polynomial becomes a quadratic polynomial. It follows from Proposition 1 that the nonlinear heat equation (1.6.1.7) has a generalized separable solution u = ψ1 (t) + ψ2 (t)x + ψ3 (t)x2 ,

(1.6.1.9)

with the functions ψi = ψi (t) (i = 1, 2, 3) described by the autonomous system of first-order ordinary differential equations ψ1′ = 2ψ1 ψ3 + ψ22 + bψ1 , ψ2′ = 6ψ2 ψ3 + bψ2 , ψ3′ = 6ψ32 + bψ3 . This system can be integrated successively starting from the last equation, which is a Bernoulli equation [219, 273]. As a result, we obtain the following exact solutions to equation (1.6.1.7): bebt A1 ebt − (x + A3 )2 6ebt + A2 (6ebt + A2 )1/3 A1 1 u= − (x + A3 )2 6(t + A2 ) (t + A2 )1/3 u=

where A1 , A2 , and A3 are arbitrary constants.

for b 6= 0, for b = 0,

60

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

2◦ . We will show √ that the two-dimensional linear subspace with power-law basis elements L2 = x, x2 is also invariant under the differential operator (1.6.1.8). Indeed, for arbitrary C1 and C2 , we have  √  √ 2 2 F C1 x + C2 x2 = ( 15 4 C1 C2 + bC1 ) x + (6C2 + bC2 )x .

Then, using Proposition 1, we conclude that the nonlinear equation (1.6.1.7) has a generalized separable solution, √ u = ψ1 (t) x + ψ2 (t)x2 ,

other than (1.6.1.9). The functions ψi = ψi (t) (i = 1, 2) are described by the system of first-order ordinary differential equations ψ1′ = ψ2′

=

15 4 ψ1 ψ2 + bψ1 , 6ψ22 + bψ2 .

Integrating this system gives the following solutions to equation (1.6.1.7): p A1 ebt bebt (x + A3 )2 x + A − 3 6ebt + A2 (6ebt + A2 )5/8 p A1 1 u= x + A3 − (x + A3 )2 6(t + A2 ) (t + A2 )5/8

u=

for b 6= 0, for b = 0.

For generality, a translation in x has been added.

◭

◮ Example 1.40. Now let us look at the nonlinear parabolic equation

ut = auxx + u2x + ku2 + bu + c.

(1.6.1.10)

We will show that for k > 0, the differential operator F [u] = auxx + u2x + ku2 + bu + c

(1.6.1.11)

defining the right-hand side √ of equation (1.6.1.10) has a two-dimensional linear subspace L2 = {1, cos(x k )}. Indeed, for arbitrary C1 and C2 , the relation √ √   F C1 + C2 cos(x k ) = k(C12 + C22 ) + bC1 + c + C2 (2kC1 − ak + b) cos(x k )

holds. Therefore, equation (1.6.1.10) admits a generalized separable solution of the form √ u = ψ1 (t) + ψ2 (t) cos(x k ), (1.6.1.12)

where ψ1 (t) and ψ2 (t) are functions satisfying the autonomous system of ordinary differential equations ψ1′ = k(ψ12 + ψ22 ) + bψ1 + c, (1.6.1.13) ψ2′ = ψ2 (2kψ1 − ak + b). ◭

61

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

Remark 1.29. For k > 0, the nonlinear differential F [u] in (1.6.1.11) admits a √ operator √  three-dimensional invariant subspace, L3 = 1, sin(x k ), cos(x k ) , that involves trigonometric functions. For k < 0, the nonlinear operator F [u] in (1.6.1.11) admits √ a three-dimensional √ invariant subspace with hyperbolic functions, L3 = 1, sinh(−x −k ), cosh(−x −k ) , or an  √ √ equivalent subspace with exponentials, L3 = 1, exp(−x −k ), exp(x −k ) . For k = 0, the operator F [u] in (1.6.1.11) admits a three-dimensional invariant subspace  involving power-law functions, L3 = 1, x, x2 .

Remark 1.30. The more general equation (1.6.1.10) with arbitrary a = a(t), b = b(t), and c = c(t) and k = const > 0, also has a generalized separable solution of the form (1.6.1.12), with ψ1 (t) and ψ2 (t) described by the system of ordinary differential equations (1.6.1.13).

◮ Example 1.41. Let us look at the fourth-order nonlinear PDE

ut = −a(uuxxx)x .

(1.6.1.14)

It describes a fluid flow in a porous medium or in a Hele-Shaw cell, which is formed by two immiscible fluids that are separated by a thin interface of thickness 2u [124, 334]. We will show that the five-dimensional linear subspace with power-law elements L5 = {1, x, x2 , x3 , x4 } is invariant under the nonlinear differential operator F [u] = (uuxxx)x ,

(1.6.1.15)

which defines the right-hand side of equation (1.6.1.14). Indeed, for arbitrary Ci , the relation   F C1 + C2 x + C3 x2 + C4 x3 + C5 x4 = 6(4C1 C5 + C2 C4 ) + 12 (4C2 C5 + C3 C4 )x + 18 (4C3 C5 + C42 )x2 + 120 C4 C5 x3 + 120 C52x4

holds; it shows that any polynomial of degree 4 becomes, under the action of the operator (1.6.1.15), a polynomial of degree 4 again. It follows from Proposition 1 that the fourth-order nonlinear equation (1.6.1.14) has a generalized separable solution u = ϕ1 (t) + ϕ2 (t)x + ϕ3 (t)x2 + ϕ4 (t)x3 + ϕ5 (t)x4 , where the functions ϕn = ϕn (t) are described by the system of ordinary differential equations [114]: ϕ′1 = −6 a(4ϕ1 ϕ5 + ϕ2 ϕ4 ), ϕ′2 = −12 a(4ϕ2ϕ5 + ϕ3 ϕ4 ), ϕ′3 = −18 a(4ϕ3ϕ5 + ϕ24 ), ϕ′4 = −120 aϕ4ϕ5 , ϕ′5 = −120 aϕ25.

This system is easy to integrate in reverse order, starting from the last equation.

◭

◮ Example 1.42. Consider the nonlinear differential operator (n) F [u] = u(m) x ux .

(1.6.1.16)

62

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

1◦ . It is easy to see that the (m + n + 1)-dimensional invariant subspace Lm+n = {1, x, x2 , . . . , xm+n }

(1.6.1.17)

is invariant under the nonlinear operator (1.6.1.16). In particular, for m = 0 and n = 2, we have F [u] = uuxx and L3 = {1, x, x2 }. 2◦ . Let us show that the three-dimensional subspace  (1.6.1.18) L3 = 1, x(m+n)/2 , xn+m is also invariant under the nonlinear operator (1.6.1.16). Indeed, we find successively that F [C1 + C2 x(m+n)/2 + C3 xn+m ] (m+n)/2 = (C1 + C2 x(m+n)/2 + C3 xn+m )(m) + C3 xn+m )(n) x (C1 + C2 x x

= (C2 a1 x(n−m)/2 + C3 b1 xn )(C2 a2 x(m−n)/2 + C3 b2 xm ) = C22 a1 a2 + C2 C3 (a1 b2 + a2 b1 )x(m+n)/2 + C32 b1 b2 xn+m , where a1 , b1 , a2 , and b2 are numeric coefficients independent of C1 , C2 , and C3 . This is what was to be proved. In particular, for m = 1 and n = 2, F [u] = ux uxx ◭ and L3 = {1, x3/2 , x3 }. Remark 1.31. The nonlinear differential operator

F [u] =

k X

ai u(m+i) u(n−i) x x

(ai are arbitrary constants),

i=0

which is more general than (1.6.1.16), also admits the (m + n + 1)-dimensional invariant subspace (1.6.1.17).

Table 1.3 lists some nonlinear differential operators and linear subspaces invariant under these operators [114, 275]. Adding the linear operator L[u] = αuxx + βux + γu + δ to the first seven nonlinear operators does not change the invariant subspaces (except for L2 for the third operator).

1.6.2. Some Modifications and Generalizations The Nonlinear Operator is Parametrically Dependent on t. We will look at more general equations of the form L1 [u] = L2 [w],

w = F [u],

(1.6.2.1)

where L1 [u] and L2 [w] are linear differential operators in t, L1 [u] ≡

m1 X

(i)

ai (t)ut ,

i=0

L2 [w] ≡

m2 X

(j)

bj (t)wt ,

(1.6.2.2)

j=0

while F [u] is a nonlinear differential operator in x, F [u] ≡ F (t, x, u, ux , . . . , u(n) x ), which can depend on t as a parameter.

(1.6.2.3)

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

63

Table 1.3. Some nonlinear differential operators and linear subspaces invariant under these operators (a, b, and c are constants). No. Nonlinear operator F [u] 1

auxx + bu2x

2

auxx + u2x + bu2

3

auuxx + bu2x + cu2

4

uuxx − u2x (special case of the 3rd operator)

5

uuxx − 32 u2x (special case of the 3rd operator)

6

uuxx − 34 u2x + au2 (special case of the 3rd operator)

7

[(au2 + bu + c)ux ]x

8

u2 uxx −

1 2

uu2x + au3

9

ux uxx

10

(u2 )xxxx

11

(n) (u2 )x

12

(m) (n)

ux ux

Subspaces invariant under the operator F [u]  L3 = 1, x, x2 } √ √  L3 = 1, sin(x b ), cos(x b ) if b > 0, p p  L3 = 1, sinh(x |b| ), cosh(x |b| ) if b < 0  2 L3 = 1, sin(λx ), cos(λx ) if c/(a + b) = λ > 0, L3 = 1, sinh(λx ), cosh(λx ) if c/(a + b) = −λ2 < 0, L3 = 1, x, x2 } if c = 0,  L2 = x2 , xβ }, β = a/(a + b) if c = 0, a 6= −b  L3 = 1, sin(λx ), cos(λx ) , λ is an arbitrary constant, L3 = 1, sinh(λx ), cosh(λx ) , λ is an arbitrary constant, L3 = 1, x, x2 }  L4 = 1, x, x2 , x3  L5 = 1, cos(kx), sin(kx), cos(2kx), sin(2kx) 2 if a = k > 0,  L5 = 1, cosh(kx), sinh(kx), cosh(2kx), sinh(2kx) 2  if a = −k < 0, 2 3 4 L5 = 1, x, x , x , x if a = 0  L2 = 1, x √ √  L3 = 1, cos( 2a x), sin( 2a x) if a > 0, p p  L3 = 1, cosh( 2|a| x), sinh( 2|a| x) if a < 0, L3 = 1, x, x2 if a = 0  L4 = 1, x, x2 , x3 ,  L3 = 1, x3/2 , x3 , L2 = 1, ϕ(x) , ϕ′x ϕ′′xx = p1 + p2 ϕ, p1 and p2 constants  L5 = 1, x, x2 , x3 , x4 ,  1/2 3/2 4 L3 = x , x , x Ln+1 = {1, x, x2 , . . . , xn }, L3 = xk/2 , xm/2 , xn , where k < n and m < n, with k, m, and 21 (k + m) being nonnegative integers Lm+n+1 {1, x, x2 , . . . , x m+n },  =(m+n)/2 L3 = 1, x , xn+m

Proposition 2. Suppose that the linear subspace (1.6.1.3) is invariant under the nonlinear differential operator F in the sense that for arbitrary constants C1 , . . . , Ck ,

64

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

the relation F

X k i=1

 X k Ci ϕi (x) = fi (t, C1 , . . . , Ck )ϕi (x)

(1.6.2.4)

i=1

holds; the functions ϕi (x) are independent of t and C1 , . . . , Ck . Then, equation (1.6.2.1) has generalized separable solutions of the form (1.6.1.5) with the functions ψ1 = ψ1 (t), . . . , ψk = ψk (t) satisfying the system of ordinary differential equations   L1 [ψi ] = L2 fi (t, ψ1 , . . . , ψk ) , i = 1, . . . , k. (1.6.2.5) ◮ Example 1.43. Consider the generalized Guderley equation

utt + a1 (t)ut = a2 (t)ux uxx ,

(1.6.2.6)

which is used, for a1 (t) = 0 and a2 (t) = a as well as a1 (t) = 1/t and a2 (t) = a, to describe transonic gas flows, where t plays the role of a space variable [128] and γ = a − 1 is the adiabatic index. Equation (1.6.2.6) is a special case of equation (1.6.2.1), where L1 [u] = utt + a1 (t)ut ,

L2 [w] = a2 (t)w,

F [u] = ux uxx .

1◦ . One can easily see, by direct verification, that the nonlinear differential operator F [u] = uxuxx admits a three-dimensional invariant subspace L3 = {1, x3/2 , x3 } [346]. It follows for Proposition 2 above that equation (1.6.2.6) has a generalized separable solution of the form u = ψ1 (t) + ψ2 (t)x3/2 + ψ3 (t)x3 ,

(1.6.2.7)

in which the functions ψi = ψi (t) (i = 1, 2, 3) satisfy the system of ordinary differential equations ψ1′′ + a1 (t)ψ1′ = ψ2′′ + a1 (t)ψ2′ = ψ3′′ + a1 (t)ψ3′ =

2 9 8 a2 (t)ψ2 , 45 4 a2 (t)ψ2 ψ3 , 18a2 (t)ψ32 .

2◦ . The operator F [u] = ux uxx also admits a four-dimensional invariant subspace L4 = {1, x, x2 , x3 } [114]. Hence, equation (1.6.2.6) also has a generalized separable solution of the form u = ψ1 (t) + ψ2 (t)x + ψ3 (t)x2 + ψ4 (t)x3 ,

(1.6.2.8)

with the functions ψi = ψi (t) (i = 1, 2, 3, 4) satisfying the system of ordinary differential equations ψ1′′ + a1 (t)ψ1′ = 2a2 (t)ψ2 ψ3 , ψ2′′ + a1 (t)ψ2′ = 2a2 (t)(3ψ2 ψ4 + 2ψ32 ), ψ3′′ + a1 (t)ψ3′ = 18a2 (t)ψ3 ψ4 , ψ4′′ + a1 (t)ψ4′ = 18a2 (t)ψ42 .

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

65

For arbitrary a1 (t) and a2 (t), this system admits exact solutions in explicit form if ψ3 = const and ψ4 = 0 and, in addition, reduces to a single second-order linear ODE for ψ1 = ψ1 (t) if ψ3 6= const and ψ4 = 0. In conclusion, we note that for a1 (t) = 1/t and a2 (t) = a, equation (1.6.2.6) has a quadratic polynomial solution in x [128]: u = Cx2 + aC 2 t2 x +

1 2 3 4 8a C t ,

where C is an arbitrary constant (special case of solution (1.6.2.8)). 3◦ . The operator F [u] = ux uxx can be shown to admit also a two-dimensional invariant subspace L2 = {1, ϕ(x)} with ϕ = ϕ(x) satisfying the following autonomous ordinary differential equation (derived using the method described in Subsection 1.6.3): ϕ′x ϕ′′xx = p1 + p2 ϕ,

(1.6.2.9)

where p1 and p2 are arbitrary constants. The general solution to equation (1.6.2.9) can be written in implicit form: Z
−1/3 2 3 dϕ + p3 , (1.6.2.10) x= 2 p2 ϕ + 3p1 ϕ + p0

where p0 and p3 are arbitrary constants. Consequently, equation (1.6.2.6) has a generalized separable solution of the form u = ψ1 (t) + ψ2 (t)ϕ(x), where ϕ(x) is defined implicitly by (1.6.2.10), while ψi = ψi (t) (i = 1, 2) are described by the system of ordinary differential equations ψ1′′ + a1 (t)ψ1′ = p1 a2 (t)ψ22 , ψ2′′ + a1 (t)ψ2′ = p2 a2 (t)ψ22 .

◭

The equation involves several nonlinear operators. Now consider more general, than (1.6.2.1), nonlinear equations of the form m X

Lj [wj ] = 0,

wj = Fj [u],

j = 1, . . . , m,

(1.6.2.11)

j=1

where Lj [w] are linear differential operators in t of the form (1.6.2.2), while Fj [u] are nonlinear differential operators in x of the form (1.6.1.2) (some of the operators Fj may be linear). Proposition 3. Suppose the finite-dimensional linear subspace (1.6.1.3) is invariant under all differential operators Fj [u]; that is, there exist functions fj1 , . . . , fjk such that X  X k k Fj Ci ϕi (x) = fji (C1 , . . . , Ck )ϕi (x). (1.6.2.12) i=1

i=1

66

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Then, equation (1.6.2.11) has a generalized separable solution of the form (1.6.1.5) with the functions ψ1 = ψ1 (t), . . . , ψk = ψk (t) satisfying system of ordinary differential equations m X j=1

  Lj fji (ψ1 , . . . , ψk ) = 0,

i = 1, . . . , k.

(1.6.2.13)

◮ Example 1.44. Let us rewrite the hydrodynamic type equation (1.3.2.45) as

the sum L1 [w1 ] + L2 [w2 ] = 0,

(1.6.2.14)

where L1 [w1 ] = (w1 )t ,

F1 [u] = ux ,

L2 [w2 ] = w2 ,

F2 [u] = u2x − uuxx − νuxxx.

The two-dimensional linear subspace L2 = {1, eλx}, where λ is an arbitrary constant, is invariant under both operators F1 [u] and F2 [u], since the relations F1 [C1 + C2 eλx ] = C2 λeλx ,

F2 [C1 + C2 eλx ] = −(C1 C2 λ2 + C2 νλ3 )eλx

hold. Therefore, equation (1.6.2.14) and the equivalent equation (1.3.2.45) both admit a generalized separable solution of the form u = ψ1 (t) + ψ2 (t)eλx .

(1.6.2.15)

The functions ψ1 (t) and ψ2 (t) satisfy the single equation ψ2′ − λψ1 ψ2 − νλ2 ψ2 = 0.

(1.6.2.16)

Now assuming ψ2 to be an arbitrary given function, we eliminate ψ1 from (1.6.2.15) ◭ using (1.6.2.16) to arrive at solution (1.3.2.47) in which ψ2 is renamed ϕ. Many other nonlinear equations with generalized separable solutions as well as some details and generalizations of the method can be found in [108, 111, 112, 114, 275, 340, 341, 346]. The nonlinear operator involves derivatives in both variables. Let us look at the general partial differential equation F [u] = 0,

(1.6.2.17)

where F is a nonlinear differential operator explicitly dependent on x and t and involving partial derivatives in these variables. Proposition 4. Let the finite-dimensional linear subspace (1.6.1.3) be invariant under the nonlinear operator F in the sense that for any set of functions C1 (t), . . . , Ck (t), the relation F

X k i=1

 X k Ci (t)ϕi (x) = fi [C(t)]ϕi (x) i=1

(1.6.2.18)

67

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

holds, where C(t) = {C1 (t), . . . , Ck (t)},


fi [C(t)] = fi t, C(t), C′t (t), C′′tt (t), . . . .

Then, equation (1.6.2.17) admits an exact solution of the form (1.6.1.5) with the functions ψi (t) satisfying the system of ordinary differential equations [114] fi [ψ(t)] = 0,

ψ(t) = {ψ1 (t), . . . , ψk (t)}.

(1.6.2.19)

Some examples of utilizing Proposition 4 for the construction of exact solutions to nonlinear PDEs can be found in [114].

1.6.3. Finding Linear Subspaces Invariant under a Given Nonlinear Operator Preliminary remarks. The main difficulty with the method of invariant subspaces for constructing exact solutions to specific equations is to find linear subspaces invariant under a given nonlinear operator. So far, we have not discussed this problem in the book, presuming that the reader knows in advance or has guessed from some considerations (e.g., intuitive) which set of linearly independent functions is suited for seeking exact solutions. Therefore, Propositions 1–4 stated in Subsections 1.6.1 and 1.6.2 are essentially equivalent to looking for exact solutions in the form of a bilinear sum (1.6.1.5) in which the functions ϕi (x) are set a priori; this corresponds to employing the method for constructing generalized separable solutions described above in Section 1.3. Below we describe a method to find, theoretically, the functions ϕi (x). The simple situation. In order to determine the independent basis functions P ϕi = ϕi (x), let us substitute the linear combination ki=1 Ci ϕi (x) into the nonlinear differential operator (1.6.1.2). This results in an expression like F

X k i=1

 Ci ϕi (x) = A1 (C)Φ1 [X] + A2 (C)Φ2 [X] + · · · + Am (C)Φm [X]

+ B1 (C)ϕ1 (x) + B2 (C)ϕ2 (x) + · · · + Bk (C)ϕk (x), (1.6.3.1)

where Aj (C) and Bi (C) only depend on C1 , . . . , Ck , while the functionals Φj [X] depend on x and are independent of C1 , . . . , Ck : Aj (C) ≡ Aj (C1 , . . . , Ck ), Bi (C) ≡ Bi (C1 , . . . , Ck ),

Φj [X] ≡

j = 1, . . . , m; i = 1, . . . , k;

Φj x, ϕ1 , ϕ′1 , ϕ′′1 , . . . , ϕk , ϕ′k , ϕ′′k


.

(1.6.3.2)

For simplicity, we have written out the formulas for the case of a second-order differential operator. For higher-order operators, the right-hand sides of relations (1.6.3.2) will contain higher-order derivatives of ϕi . The functionals Φ1 [X], . . . , Φm [X] are all assumed to be linearly independent, and the Aj (C) are linearly independent functions of C1 , . . . , Ck .

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

The finite-dimensional linear subspace (1.6.1.3) is invariant under the nonlinear differential operator F (1.6.1.2) if the functionals Φj [X] (j = 1, . . . , m) in (1.6.3.1) are all linear combinations of the basis functions ϕi (x) (i = 1, . . . , k). Consequently, we arrive at the following system (usually overdetermined) of ordinary differential equations [114, 275] for the basis functions:
Φj x, ϕ1 , ϕ′1 , ϕ′′1 , . . . , ϕk , ϕ′k , ϕ′′k = pj,1 ϕ1 +pj,2 ϕ2 +· · ·+pj,k ϕk , j = 1, . . . , m, (1.6.3.3) where pj,i are some constants independent of the parameters C1 , . . . , Ck . If for some collection of constants pi,j , system (1.6.3.3) is solvable (in practice, it suffices to find a particular solution), then the functions ϕi = ϕi (x) define a linear subspace invariant under the nonlinear differential operator (1.6.1.2). In this case, the functions appearing on the right-hand side of (1.6.1.4) are given by fi (C) = p1,i A1 (C) + p2,i A2 (C) + · · · + pm,i Am (C) + Bi (C), i = 1, . . . , k. (1.6.3.4) Remark 1.32. In general, for the nonlinear equation (1.6.2.17), the coefficients Aj and ′′ Bi in (1.6.3.1) and the functions fi in (1.6.3.4) will depend on C, Ct′ , Ctt , ... . Remark 1.33. The analysis of nonlinear differential operators is worth to begin with searching for two-dimensional invariant subspaces of the form L2 = {1, ϕ(x)}.

Proposition 1. Suppose a nonlinear differential operator F [u] admits a twodimensional invariant subspace L2 = {1, ϕ(x)}, where ϕ(x) = pϕ1 (x) + qϕ2 (x), p and q are arbitrary constants, and the functions 1, ϕ1 (x), and ϕ2 (x) are linearly independent. Then the operator F [u] also admits the three-dimensional invariant subspace L3 = {1, ϕ1 (x), ϕ2 (x)}. Proposition 2. Let two nonlinear differential operators F1 [u] and F2 [u] admit an invariant subspace Ln = {ϕ1 (x), . . . , ϕn (x)}. Then the nonlinear operator pF1 [u]+ qF2 [u], where p and q are arbitrary constants, also admits the same invariant subspace. In particular, for any constants a and b, the operators F [u] and aF [u] + bu admit identical invariant subspaces. ◮ Example 1.45. Consider the nonlinear differential operator (1.6.1.11). We look for its invariant subspaces in the form L2 = {1, ϕ(x)}. We have

F [C1 +C2 ϕ(x)] = C22 [(ϕ′x )2 +kϕ2 ]+C2 aϕ′′xx +kC12 +bC1 +c+(bC2 +2kC1 C2 )ϕ, where Φ1 [X] = (ϕ′x )2 + kϕ2 and Φ2 [X] = aϕ′′xx . Hence, the basis function ϕ(x) must satisfy the overdetermined system of ordinary differential equations (ϕ′x )2 + kϕ2 = p1 + p2 ϕ, ϕ′′xx = p3 + p4 ϕ,

(1.6.3.5)

where p1 = p1,1 , p2 = p1,2 , p3 = p2,1 /a, and p4 = p2,2 /a. Let us investigate system (1.6.3.5) for consistency. To this end, we differentiate the first equation with respect

69

1.6. Method of Invariant Subspaces (Titov–Galaktionov Method)

to x and then divide by ϕ′x to obtain ϕ′′xx = −kϕ + p2 /2. Using this relation to eliminate the second derivative from the second equation in (1.6.3.5), we get (p4 + k)ϕ + p3 − 12 p2 = 0. For this equation to hold, we must set p4 = −k,

p3 =

1 2

p2 .

(1.6.3.6)

The simultaneous solution of system (1.6.3.5) under condition (1.6.3.6) is given by ϕ(x) = px2 +qx if k = 0 √
√
ϕ(x) = p sin x k +q cos x k if k > 0 √
√
ϕ(x) = p sinh x −k +q cosh x −k if k < 0

(p1 = q 2, p2 = 4p), (p1 = kp2 +kq 2, p2 = 0), (p1 = −kp2 +kq 2, p2 = 0), (1.6.3.7)

where p and q are arbitrary constants. Since formulas (1.6.3.7) involve two arbitrary parameters, p and q, it follows from Proposition 1 that the nonlinear differential operator (1.6.1.11) admits the following invariant subspaces:  if k = 0, L3 = 1, x, x2 √ √  if k > 0, L3 = 1, sin(x k ), cos(x k ) √ √  if k < 0. L3 = 1, sinh(x −k ), cosh(x −k ) ◭ ◮ Example 1.46. Consider the heat equation with a quadratic nonlinearity

ut = f (x)(uux )x ,

(1.6.3.8)

where the nonlinear differential operator is explicitly dependent on the space variable x. We have F [u] = f (x)(uux )x . We look for its invariant subspaces of the form L2 = {1, ϕ(x)}. We get F [C1 + C2 ϕ(x)] = C22 f (x)[ϕϕ′′xx + (ϕ′x )2 ] + C1 C2 f (x)ϕ′′xx . Hence, Φ1 [X] = f (x)[ϕϕ′′xx + (ϕ′x )2 ] and Φ2 [X] = f (x)ϕ′′xx and the basis function ϕ(x) must satisfy the overdetermined system of ordinary differential equations f (x)[ϕϕ′′xx + (ϕ′x )2 ] = p1 + p2 ϕ, f (x)ϕ′′xx = p3 + p4 ϕ.

(1.6.3.9)

Let us find the form of admissible functions f (x) for which system (1.6.3.9) is consistent. In the nondegenerate case with ϕ′′xx 6= 0, eliminating f (x) gives an equation for ϕ: (p3 + p4 ϕ)[ϕϕ′′xx + (ϕ′x )2 ] = (p1 + p2 ϕ)ϕ′′xx . (1.6.3.10) Any solution to this equation generates a function f (x) such that f (x) =

p3 + p4 ϕ . ϕ′′xx

(1.6.3.11)

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

The substitution (ϕ′x )2 = 2w(ϕ) reduces (1.6.3.10) to a separable linear firstorder equation ′ [p4 ϕ2 + (p3 − p2 )ϕ − p1 ]wϕ = −2(p3 + p4 ϕ)w,

which is easy to integrate. Some cases where ϕ(x) can be expressed explicitly in terms of elementary functions are gathered in Table 1.4. Table 1.4. The determining functions f (x) appearing in equation (1.6.3.8) and the associated basis functions ϕ(x); a, b, n, and λ are arbitrary constants (n 6= 2, λ 6= 0). No.

Function f (x)

Function ϕ(x)

Parameters p1 , p2 , p3 , and p4

1

a

1 2 x + bx 2a

p1 = ab2 , p2 = 3, p3 = 1, p4 = 0

2

axn

x2−n

p1 = p4 = 0, p2 = a(2 − n)(3 − 2n), p3 = a(1 − n)(2 − n)

3

ax2

ln x

p1 = a, p2 = p3 = −a, p4 = 0

4

aeλx

e−λx

p1 = p4 = 0, p2 = 2aλ2 , p3 = aλ2

Exact solutions to equation (1.6.3.8) are expressed as u = ψ1 (t) + ψ2 (t)ϕ(x), where the functions ψ1 = ψ1 (t) and ψ2 = ψ2 (t) are determined by solving the autonomous system of two ordinary differential equations ψ1′ = p1 ψ22 + p3 ψ1 ψ2 , ψ2′ = p2 ψ22 + p4 ψ1 ψ2 . The prime denotes a derivative with respect to t. By dividing the former equation by the latter, one can reduce the system to a single homogeneous first-order equation, ◭ which is easy to integrate. Remark 1.34. The equation of a diffusion boundary layer near a solid surface with the diffusion coefficient linearly dependent on concentration is reduced to equation (1.6.3.8) with f (x) = a/x [274, 376]. Also reducible to equations of the form (1.6.3.8) is the equation

ut = a(z)[b(z)uuz ]z .

(1.6.3.12)

dz , where f (x) = a(z). Equation b(z) (1.6.3.12) with a(z) = z −n and b(z) = z n describes nonlinear heat and mass transfer in the radial symmetric case (n = 1 corresponds to a plane problem and n = 2 to a spatial one).

The reduction can be done with the substitution x =

Z

The complex situation. For a better understanding of the following material, we consider separately the last two terms in the upper row on the right-hand side of (1.6.3.1):∗ Am−1 (C)Φm−1 [X] + Am (C)Φm [X].

(1.6.3.13)

∗ In general, the coefficients A in (1.6.3.1), including A m−1 and Am in (1.6.3.13), as well as the j functions fi in (1.6.3.4) can depend on both C and its derivatives C′t , C′′ tt , . . . ; see Remark 1.32.

71

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

These correspond to the last two equations in system (1.6.3.3). There is a possibility that the coefficients in (1.6.3.13) are linearly related, for example, Am (C) = βAm−1 (C),

(1.6.3.14)

where β is a constant independent of C. In this case, replacing Am (C) with Am−1 (C) in (1.6.3.13), we get Am−1 (C)(Φm−1 [X] + βΦm [X]),

(1.6.3.15)

which corresponds to a new composite functional Φ∗m−1 [X] = Φm−1 [X]+βΦm [X]. By reasoning in the same manner as in Item 1◦ above, we arrive at a system of m − 1 equations for ϕi , which is obtained from (1.6.3.3) by discarding the last equation and replacing formally Φm−1 (. . .) with Φ∗m−1 (. . .). Then, in formulas (1.6.3.4) for the functional coefficients fi (C), we must set Am (C) = 0. Solutions to equation (1.6.1.1) with operator (1.6.1.2) are sought as the bilinear sum (1.6.1.5) with the functions ψ1 (t), . . . , ψk (t) satisfying the system of ODEs (1.6.1.6) augmented by equation (1.6.3.14); the functions fi on the right-hand sides of the equations are defined by formulas (1.6.3.4), in which C must be replaced with ψ. It follows that the system of equations for ψi will, in this case, be overdetermined. The above fully applies to the general nonlinear equation (1.6.2.17) as well. Instead of one simple relation (1.6.3.14) that links Am (C) to Am−1 (C), one can use more complicated linear relations that would link more coefficients. Furthermore, one can even introduce several linear relations like that. This would lead to fewer ODEs for ϕi (x) and overdetermined systems for ψi (t). The above situation corresponds to different solutions of the bilinear functional equation (1.5.1.1), which serves to obtain generalized separable solutions with the splitting method (see Section 1.5). Notably, in the splitting method, both independent variables, x and t, are peers; by contrast, in the method of invariant subspaces, x is chosen to be the primary variable, and t plays the role of a parameter. This is why finding exact solutions to nonlinear PDEs with the method of invariant subspaces is more difficult if there are several solutions, and all of them are significantly different.

1.7. Other Nonlinear Equations Having Generalized Separable Solutions 1.7.1. Nonlinear Partial Differential Equations with Delay Preliminary remarks. Reaction–diffusion equations with delay. Differential equations with delay are indispensable in mathematical modeling of complex phenomena and processes whose state depends not only on a given point in time, but also on one or more times in the past [371]. Apart from the unknown function u = u(x, t), partial differential equations with delay also involve the function u¯ = u(x, t − τ ), where τ is the delay time. Usually, τ is a positive constant; however, there are more complex models where τ = τ (t) is a given function of time.

72

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Differential equations with delay have qualitative features that are absent from differential equations without delay [270, 371]. Reaction–diffusion equations with delay are among the most common types of nonlinear delay PDEs that appear in the literature (e.g., see [207, 270, 285, 290, 291, 371]). These have the general form ut = [f (u)ux ]x + g(u, u ¯),

u¯ = u(x, t − τ ),

(1.7.1.1)

where the delay only appears in the kinetic function g(u, u ¯). The term ‘exact solution’ with regard to nonlinear delay partial differential equations is used in the following cases: (i) the solution is expressible in terms of elementary functions or in closed form using quadratures; (ii) the solution is expressible in terms of solutions to ordinary differential or delay ordinary differential equations (or systems of such equations). Combinations of cases (i) and (ii) are also allowed. This definition generalizes the concept of an exact solution with regard to nonlinear partial differential equations without delay (see the preface). Simple separable solutions. Some nonlinear delay PDEs admit simple separable solutions. We will illustrate this with examples of specific equations discussed in [282, 285, 287]. ◮ Example 1.47. Consider the nonlinear reaction–diffusion type PDE with delay

ut = auxx + uf (¯ u/u),

u ¯ = u(x, t − τ ),

(1.7.1.2)

where f (. . .) is an arbitrary function. 1◦ . Equation (1.7.1.2) admits a multiplicative separable solution periodic in the space coordinate x: u = [C1 cos(λx) + C2 sin(λx)]ψ(t),

(1.7.1.3)

where C1 , C2 , and λ are arbitrary constants and ψ = ψ(t) is a function satisfying the delay ordinary differential equation ¯ ψt′ = −aλ2 ψ + ψf (ψ/ψ).

(1.7.1.4)

Here and henceforth, ψ¯ = ψ(t − τ ). The delay ODE (1.7.1.4) has an exponential particular solution ψ(t) = C3 exp(−βτ ),

(1.7.1.5)

where C3 is an arbitrary constant and β is a root of the algebraic (or transcendental) equation aλ2 − β = f (eβτ ). 2◦ . Equation (1.7.1.2) admits another multiplicative separable solution: u = [C1 exp(−λx) + C2 exp(λx)]ψ(t),

(1.7.1.6)

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

73

where C1 , C2 , and λ are arbitrary constants and ψ = ψ(t) is a function satisfying the delay ordinary differential equation ¯ ψt′ = aλ2 ψ + ψf (ψ/ψ).

(1.7.1.7)

The delay ODE (1.7.1.7) has an exponential particular solution of the form (1.7.1.5), where β is a root of the algebraic (transcendental) equation aλ2 + β + f (eβτ ) = 0.

◭

Remark 1.35. The delay in the nonlinear delay PDE (1.7.1.2) can be an arbitrary function of time, τ = τ (t). In this case, equation (1.7.1.2) also has exact solutions of the form (1.7.1.3) and (1.7.1.6), where ψ = ψ(t) satisfies equations (1.7.1.4) and (1.7.1.7), respectively, with ψ¯ = ψ(t − τ (t)).

◮ Example 1.48. Consider the equation

ut = auxx + bu + f (u − u ¯),

u ¯ = u(x, t − τ ),

(1.7.1.8)

where f (. . .) is an arbitrary function. 1◦ . If ab > 0, equation (1.7.1.8) admits the additive separable solution p u = C1 cos(λx) + C2 sin(λx) + ψ(t), λ = b/a, (1.7.1.9)

where C1 and C2 are arbitrary constants and ψ = ψ(t) is a function satisfying the delay ordinary differential equation ¯ ψt′ = bψ + f (ψ − ψ).

(1.7.1.10)

2◦ . If ab < 0, equation (1.7.1.8) admits the additive separable solution p u = C1 exp(−λx) + C2 exp(λx) + ψ(t), λ = −b/a, (1.7.1.11)

where C1 and C2 are arbitrary constants and ψ = ψ(t) is a function satisfying the delay ordinary differential equation (1.7.1.10). 3◦ . If b = 0, equation (1.7.1.8) admits the additive separable solution u = C1 x2 + C2 x + ψ(t),

(1.7.1.12)

where C1 and C2 are arbitrary constants, and the function ψ(t) is described by the delay ordinary differential equation ¯ ψt′ = 2aC1 + f (ψ − ψ).

(1.7.1.13)

The delay ODE (1.7.1.13) has a particular solution linear in t: ψ(t) = kt + C3 , where C3 is an arbitrary constant, and k is a root of the algebraic (transcendental) equation k = 2aC1 + f (kτ ). ◭

74

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Table 1.5. Some nonlinear delay PDEs that admit simple separable solutions. Notation: u ¯ = u(x, t − τ ), ψ¯ = ψ(t − τ ), f (. . .) is an arbitrary function, and C is an arbitrary constant. No.

Nonlinear delay PDEs

Form of solutions

1

ut = auxx + uf (¯ u/u)

u = ϕ(x)ψ(t)

2

ut = auxx + bu ln u + uf (¯ u/u)

u = ϕ(x)ψ(t)

3

ut = a(unux )x + uf (¯ u/u)

u = ϕ(x)ψ(t)

4

ut = auxx + bu + f (u − u ¯)

u = ϕ(x) + ψ(t)

5

ut = a(eλuux )x + f (u − u ¯)

u = ϕ(x) + ψ(t)

6

utt = auxx + uf (¯ u/u)

u = ϕ(x)ψ(t)

7

utt = auxx + bu ln u + uf (¯ u/u)

u = ϕ(x)ψ(t)

8

utt = a(unux )x + uf (¯ u/u)

u = ϕ(x)ψ(t)

9

utt = auxx + bu + f (u − u ¯)

u = ϕ(x) + ψ(t)

10

utt = a(eλuux )x + f (u − u ¯)

u = ϕ(x) + ψ(t)

Determining equations aϕ′′xx = Cϕ, ¯ ψt′ = Cψ + ψf (ψ/ψ) aϕ′′xx = Cϕ − bϕ ln ϕ,

¯ ψt′ = Cψ + bψ ln ψ + ψf (ψ/ψ) a(ϕnϕ′x )′x = Cϕ, ¯ ψt′ = Cψ n+1 + ψf (ψ/ψ) aϕ′′xx + bϕ = C, ¯ ψt′ = bψ + C + f (ψ − ψ) a(eλϕϕ′x )′x = C, ¯ ψt′ = Ceλψ + f (ψ − ψ) aϕ′′xx = Cϕ, ′′ ¯ ψtt = Cψ + ψf (ψ/ψ)

aϕ′′xx = Cϕ − bϕ ln ϕ,

′′ ¯ ψtt = Cψ + bψ ln ψ + ψf (ψ/ψ)

a(ϕnϕ′x )′x = Cϕ, ′′ ¯ ψtt = Cψ n+1 + ψf (ψ/ψ)

aϕ′′xx + bϕ = C, ′′ ¯ ψtt = bψ + C + f (ψ − ψ)

a(eλϕϕ′x )′x = C, ′′ ¯ ψtt = Ceλψ + f (ψ − ψ)

Remark 1.36. The delay in the nonlinear delay PDE (1.7.1.8) can be an arbitrary function of time, τ = τ (t). Then, equation (1.7.1.8) also has exact solutions (1.7.1.9), (1.7.1.11) and (1.7.1.12), where ψ = ψ(t) satisfies equations (1.7.1.10) and (1.7.1.13) with ψ¯ = ψ(t − τ (t)).

Table 1.5 gathers the above and some other examples of simple, additive or multiplicative, separable solutions for several nonlinear delay PDEs. For all nonlinear delay PDEs listed in Table 1.5, the functions ϕ(x) appearing in the solution satisfy autonomous second-order ordinary differential equations, while the functions ψ(t) are described by first- or second-order ordinary differential equations with delay. Instead of constant delay time τ , a given function of time, τ (t), can appear in all nonlinear delay PDEs, in which case all equations for ψ = ψ(t) must have ψ¯ = ψ(t − τ (t)). Generalized separable solutions. The methods described previously in Sec-

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

75

tions 1.3–1.6 are also suitable for the construction of generalized separable solutions to some nonlinear delay PDEs. Let us look at the nonlinear PDE with a linear delay term ut = F [u] + s¯ u,

u¯ = u(x, t − τ ),

(1.7.1.14)

where s is a constant and F [u] is an nth-order nonlinear differential operator with respect to x of the form (1.6.1.2). Proposition 1. Suppose the linear subspace (1.6.1.3) is invariant under the operator F , implying that relation (1.6.1.4) holds. Then, equation (1.7.1.14) possesses generalized separable solutions of the form u(x, t) =

k X

ψi (t)ϕi (x),

(1.7.1.15)

i=1

with the functions ψ1 = ψ1 (t), . . . , ψk = ψk (t) described by the system of delay ordinary differential equations ψi′ = fi (ψ1 , . . . , ψn ) + sψ¯i ,

i = 1, . . . , k.

(1.7.1.16)

The prime denotes a derivative with respect to t and ψ¯i = ψi (t − τ ). Note that the delay τ in equations (1.7.1.14) and (1.7.1.16) can be time dependent, τ = τ (t). Let us look at a few examples of utilizing Proposition 1 to construct exact solutions to nonlinear reaction–diffusion equations of the form (1.7.1.1). ◮ Example 1.49. Consider the delay reaction–diffusion equation of the form (1.7.1.14) with a quadratic nonlinearity

ut = [(a1 u + a0 )ux ]x + b1 u + b2 u ¯ + c.

(1.7.1.17)

The differential operator on the right-hand side of this equation with b2 = 0 admits the invariant linear subspace L3 = {1, x, x2 } (since the polynomial C1 + C2 x+ C3 x2 reduces to a quadratic polynomial with other coefficients). From Proposition 1 it follows that the original equation (1.7.1.17) has a generalized separable solution of the form u = ψ1 (t) + ψ2 (t)x + ψ3 (t)x2 . (1.7.1.18) The functions ψi = ψi (t) (i = 1, 2, 3) are described by the delay ODEs ψ1′ = 2a1 ψ1 ψ3 + a1 ψ22 + 2a0 ψ3 + b1 ψ1 + b2 ψ¯1 + c, ψ2′ = 6a1 ψ2 ψ3 + b1 ψ + b2 ψ¯2 , ψ ′ = 6a1 ψ 2 + b1 ψ3 + b2 ψ¯3 , 3

where ψ¯i = ψi (t − τ ).

(1.7.1.19)

3

◭

76

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

◮ Example 1.50. Let us look at the more complicated delay reaction–diffusion equation of the form (1.7.1.14) with a quadratic nonlinearity

ut = [(a1 u + a0 )ux ]x + ku2 + b1 u + b2 u ¯ + c,

k 6= 0.

(1.7.1.20)

1◦ . For a1 k < 0, the differential operator on the right-hand side of equation −λx λx (1.7.1.20) with , e }, p b2 = 0 admits the invariant linear subspace L3 = {1, e where λ = −k/(2a1 ). In this case, it follows from Proposition 1 that equation (1.7.1.20) admits a generalized separable solution of the form r k u = ψ1 (t) + ψ2 (t) exp(−λx) + ψ3 (t) exp(λx), λ = − , (1.7.1.21) 2a1 where the functions ψ = ψn (t) are described by the delay ODEs ψ1′ = kψ12 + 2kψ2 ψ3 + b1 ψ1 + b2 ψ¯1 + c, ψ ′ = ( 3 kψ1 + a0 λ2 + b1 )ψ2 + b2 ψ¯2 , 2 ψ3′

2

= ( 32 kψ1 + a0 λ2 + b1 )ψ3 + b2 ψ¯3 ,

where ψ¯i = ψi (t − τ ) (i = 1, 2, 3).

2◦ . For a1 k > 0, it can be shown likewise that equation (1.7.1.20) admits the generalized separable solution r k u = ψ1 (t) + ψ2 (t) cos(λx) + ψ3 (t) sin(λx), λ = , (1.7.1.22) 2a1

where the functions ψ = ψn (t) are described by the system of delay ODEs k(ψ22 + ψ32 ) + b1 ψ1 + b2 ψ¯1 + c, ψ2′ = ( 32 kψ1 + b1 − a0 λ2 )ψ2 + b2 ψ¯2 , ψ3′ = ( 32 kψ1 + b1 − a0 λ2 )ψ3 + b2 ψ¯3 . ψ1′ = kψ12 +

1 2

◭

Below are two more-general propositions that allow one to obtain generalized separable solutions to certain partial differential equations with delay. Let us look at a more complicated, than (1.7.1.14), nonlinear delay PDE of the form p X ut = F [u] + sj u¯j , u ¯j = u(x, t − τj ), (1.7.1.23) j=1

where F [u] is an nth-order nonlinear differential operator with respect to x of the form (1.6.1.2) and τj are delay times (j = 1, . . . , p), which are assumed to be independent arbitrary constants. PDEs with several delay times are not uncommon in the literature (e.g., see [57, 122, 147, 360]). Proposition 2. Suppose the linear subspace (1.6.1.3) is invariant under the operator F , implying that relation (1.6.1.4) holds. Then, equation (1.7.1.23) possesses

77

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

generalized separable solutions of the form (1.7.1.15), with the functions ψ1 (t), . . . , ψk (t) described by the system of ODEs with p delay times p
X ψi′ (t) = fi ψ1 (t), . . . , ψk (t) + sj ψi (t − τj ),

i = 1, . . . , k.

(1.7.1.24)

j=1

Let us look at a more complicated, than (1.7.1.14), nonlinear delay PDEs of the form L[u] = F [u; u ¯], u¯ = u(x, t − τ ), (1.7.1.25) where L[u] is an arbitrary linear differential operator with respect to t, L[u] ≡

q X

(j)

aj (t)ut ,

(1.7.1.26)

j=1

and H[u; u ¯] is a nonlinear differential operator with respect to x, containing u and u¯,
F [u; u ¯] ≡ F u, ux, uxx . . . , u(m) ¯, u ¯x , u ¯xx , . . . , u ¯(r) . x ;u x

(1.7.1.27)

Suppose that linearly independent functions ϕ1 (x), . . . , ϕk (x) form a finitedimensional linear subspace Lk . Proposition 3. Let C1 , . . . , Ck and C¯1 , . . . , C¯k be two sets of arbitrary real constants and let there exist functions f1 , . . . , fk such that F

X k i=1

Ci ϕi (x);

k X i=1

 X k ¯ Ci ϕi (x) = fi (C1 , . . . , Cn ; C¯1 , . . . , C¯n )ϕi (x). i=1

(1.7.1.28) Then, equation (1.7.1.25) has generalized separable solutions of the form (1.7.1.15), with the functions ψ1 (t), . . . , ψk (t) described by the system of delay ordinary differential equations
L[ψi (t)] = fi ψ1 (t), . . . , ψk (t); ψ1 (t − τ ), . . . , ψk (t − τ ) ,

i = 1, . . . , k. (1.7.1.29) Proposition 3 can be used to construct generalized separable solutions to nonlinear delay PDEs other than those discussed above, including delay hyperbolic equations. The delay τ in equations (1.7.1.25) and (1.7.1.29) can be time-dependent, τ = τ (t). Remark 1.37. Many delay reaction–diffusion equations of the form (1.7.1.1) as well as other related nonlinear delay PDEs admit generalized and functional separable solutions [207, 250, 264, 266–268, 282, 284–286, 288–292] (see also [193, 269, 282, 284, 287]). Apart from the methods described above for the construction of exact solutions to such equations, one can employ various modifications of the method of differential constraints [286, 291], which are not discussed here.

78

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Differential-difference equations with a finite relaxation time. Nonlinear heat equations with a source ut = [f (u)ux ]x + g(u)

(f > 0),

(1.7.1.30)

which are derived from Fourier’s law for heat flux density, q = −λ(u)∇u, are parabolic equations, which have the physically paradoxical property that the propagation speed of disturbances is infinite. This is not observed in practice, which indicates a limited range of applicability of such equations. This circumstance led to the need to develop other thermal conductivity models that would give a finite speed of propagation of disturbances. As a result, a more sophisticated thermal conductivity model based on the Cattaneo–Vernotte differential law was suggested [55, 356] (see also [103, 166, 335]): q = −λ(u)∇u − τ qt . This model leads to a hyperbolic equation of thermal conductivity: τ utt + [1 − τ gu′ (u)]ut = [f (u)ux ]x + g(u),

(1.7.1.31)

where τ is the relaxation time (delay), which is considered small. For τ = 0, equation (1.7.1.31) becomes (1.7.1.30). For theoretical justification of the Cattaneo–Vernotte differential model, one often (but not always) uses the differential-difference relation for heat flux density [103, 335, 349]: q|t+τ = −λ(u)∇u, in which the left-hand side is evaluated at time t + τ , while the right-hand side at time t. This results in the following nonlinear differential-difference heat equation with a finite relaxation time: vt = [f (u)ux ]x + g(v),

v = u(x, t + τ ).

(1.7.1.32)

With the change of variable ζ = t + τ , it is reduced to a delay PDE. If one expands formally both sides of (1.7.1.32) in a Taylor series in small τ and retains the first two leading terms, then one arrives at the hyperbolic heat equation (1.7.1.31). Table 1.6 lists several nonlinear differential-difference equations of the form (1.7.1.32) that admit additive or multiplicative separable solutions [282]. It was shown in [282] that the equation vt = [(a1 u + a2 )ux ]x + b1 v + b2 ,

v = u(x, t + τ ),

(1.7.1.33)

has generalized separable solutions of the forms u = ψ1 (t)x + ψ2 (t) and u = ψ1 (t)x2 + ψ2 (t)x + ψ3 (t). In particular, for a2 = b2 = 0, a1 = a, and b1 = b,

79

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

Table 1.6. Differential-difference PDEs of the form vt = [f (u)ux ]x +g(v) that admit additive or multiplicative separable solutions. Notation: v = u(x, t + τ ), ψ¯ = ψ(t + τ ), and C1 , C2 , and C3 are arbitrary constants. No.

f (u)

g(v)

Form of solution ¯ u = − (b/a)(x + C1 )2 + ψ(t), ψ¯t′ = − 31 bψ + bψ; 2 ′ 2 ¯ ¯ u = (x + C1 ) ψ(t), ψt = 6aψ + bψ; ¯ u = ϕ(x)ψ(t), (ϕϕ′x )′x = C1 ϕ, ψ¯t′ = aC1 ψ 2 + bψ; 1 6

1

au

bv

u = C1 x + aC12 t + C2 at b = 0 ¯ u = (x + C1 )2/n ψ(t), ψ¯t′ = 2an−2 (n + 2)ψ n+1 + bψ; n ′ ′ ′ n+1 ¯ ¯ u = ϕ(x)ψ(t), (ϕ ϕx )x = C1 ϕ, ψt = aC1 ψ + bψ;

2

aun

bv

3

aun

bv n+1 + cv

u = ect ϕ(x), a(ϕn ϕ′x )′x + bec(n+1)τ ϕn+1 = 0 2aC12 t + C2 2aC12 t + C2 ; ; u= 2 (C1 x + C3 ) sinh2 (C1 x + C3 ) C2 − 2aC12 t 2aC12 t + C2 u= ; u= 2 cos2 (C1 x + C3 ) cosh (C1 x + C3 ) u=

4

au−1

0

5

au−1

b

6

au−1

bv + c

7

aeβu

b

u = ϕ(x)(C1 t + C2 ), a(ϕ′x /ϕ)′x − C1 ϕ + b = 0
c u = C1 exp bt − 2a x 2 + C2 x ; u = ϕ(x)(C1 ebt + C2 ), a(ϕ′x /ϕ)′x + bC2 ϕ + c = 0 u=

u=

1 β

1 β

ln |C1 x + C2 | + bt + C3 ; ln(C1 x + C2 x + C3 ) + ψ(t); ψ¯t′ = 2C1 eβψ + bβ 2

u = ln[C1 cos(kx) + C2 sin(kx)] + bt + C3 , 8

eu

aev + b

aebτ = k2 > 0; u = ln[C1 cosh(kx) + C2 sinh(kx)] + bt + C3 , aebτ = −k2 < 0

equation (1.7.1.33) has solutions that can be written in explicit form: u = C1 xebt + C2 ebt + C12 (a/b)e2b(t−τ ), b 1 b , u = − (x + C1 )2 + C2 eλt , λ = ln 6a τ 3(b − 1) where C1 and C2 are arbitrary constants (in the second solution, b < 0 or b > 1). Exact solutions to several more-complicated equations of the form (1.7.1.32) as well as systems of such equations were also obtained in [282]. It is noteworthy that the study [284] constructed exact solutions to the following differential-difference hydrodynamic equations with a finite relaxation time: 1 vt + (v · ∇)v = − ∇p + ν∆u, ρ ∇ · u = 0,

(1.7.1.34)

80

1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

where v = u(x, t + τ ). Equations (1.7.1.34) coincides with the Navier–Stokes equations at τ = 0 as well as for t → ∞ (steady-state case). Remark 1.38. Note that initial-boundary value problems for equation (1.7.1.33) and system (1.7.1.34) may be ill-posed in the sense of Hadamard [160, 284].

1.7.2. Nonlinear Integro-Differential Equations The linear operators in t that appear in (1.6.2.1) and (1.6.2.11) can be not only differential but also integral or integro-differential. If this is the case, Propositions 2 and 3 from Subsection 1.6.2 remain valid and still allow one to obtain generalized separable solutions of the form (1.6.1.5) with ψ1 (t), . . . , ψk (t) described by the system of equations (1.6.2.5). We will illustrate this with a specific class of integrodifferential equations. ◮ Example 1.51. Consider the integro-differential equation with a quadratic non-

linearity L[u] = [(au + b)ux ]x + cu,

(1.7.2.1)

which is a special case of equation (1.6.2.1). For now, we will not specify the expression of the linear operator L[u] with respect to t. The subspace L3 = {1, x, x2 } is invariant under the nonlinear differential operator F [u] = [(au + b)ux]x + cu. Therefore, equation (1.7.2.1) admits exact solutions of the form u = ψ1 (t) + ψ2 (t)x + ψ3 (t)x2 ,

(1.7.2.2)

where the functions ψi = ψi (t) satisfy the nonlinear system of equations L[ψ1 ] = 2(aψ1 + b)ψ3 + aψ22 + cψ1 , L[ψ2 ] = 6aψ2 ψ3 + cψ2 , L[ψ3 ] =

6aψ32

(1.7.2.3)

+ cψ3 .

We look for solutions to system (1.7.2.3) in the form ψ1 = θ + Aψ3 ,

ψ2 = Bψ3 ,

(1.7.2.4)

where B is an arbitrary constant; the function θ = θ(t) and constant A are to be determined. The second equation in (1.7.2.3) holds identically by virtue of the second relation in (1.7.2.4) and the third equation in (1.7.2.3). Eliminating ψ1 and ψ2 from the first equation in (1.7.2.3) with the help of (1.7.2.4) and replacing L[ψ3 ] with the right-hand side of the last equation in (1.7.2.3), we obtain L[θ] = (2aψ3 + c)θ + 2bψ3 + a(B 2 − 4A)ψ32 .

(1.7.2.5)

We choose the constant A so as to ensure that the coefficient of ψ32 vanishes. We get A=

1 4

B2.

(1.7.2.6)

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1.7. Other Nonlinear Equations Having Generalized Separable Solutions

Ultimately, we arrive at a linear nonhomogeneous equation for θ = θ(t): L[θ] = (2aψ3 + c)θ + 2bψ3 .

(1.7.2.7)

In what follows, to be specific, we will use the linear integral operator with a variable limit of integration Z t L[ψ] = (t − ξ)ψ(ξ) dξ. (1.7.2.8) 0

Then, the integro-differential equation (1.7.2.1) can be rewritten in the expanded form Z t (t − ξ)u(x, ξ) dξ = [(au + b)ux ]x + cu. (1.7.2.9) 0

Let us find an exact solution to the nonlinear system of integral equations (1.7.2.3) with operator (1.7.2.8) at c = 0. The third equation of system (1.7.2.3) with c = 0 becomes Z t (t − ξ)ψ3 (ξ) dξ = 6aψ32 . (1.7.2.10) 0

One can check by direct verification that it admits the exact solution ψ3 =

1 2 t . 72a

(1.7.2.11)

On substituting (1.7.2.11) and c = 0 into equation (1.7.2.7), we will look for its solution in the form θ = Ctk + m, where C is an arbitrary constant. We get the quadratic equation (k+1)(k+2) = 36 for k. Discarding the negative root, we find that θ = Ctk +

b , 17a

k=

√
1 −3 + 145 . 2

(1.7.2.12)

Using relations (1.7.2.4), (1.7.2.6), (1.7.2.11), and (1.7.2.12), we obtain a twoparameter family of exact solutions to the nonlinear system of integral equations (1.7.2.3) at c = 0 with operator (1.7.2.8): ψ1 = Ctk +

B2 2 b t + , 288 a 17a

ψ2 =

B 2 t , 72 a

where B and C are arbitrary constants, while k =

1 2

1 2 t , 72 a √
−3 + 145 . ψ3 =

(1.7.2.13) ◭

1.7.3. Nonlinear Equations with a Fractional Derivative Fractional integrals and fractional derivatives. Below we give brief introductory information about fractional integrals and fractional derivatives. For more details on this topic, see the books [139, 171, 213, 240, 261, 328].

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Definition of fractional integrals. Let ϕ(t) be a continuous function on a closed interval [a, b]. Then the integral Iµa ϕ(t)

1 ≡ Γ(µ)

Z

t

a

ϕ(ξ) dξ, (t − ξ)1−µ

µ > 0,

t > a,

(1.7.3.1)

is called the Riemann–Liouville fractional integral or the integral of fractional orR∞ der µ; Γ(µ) = 0 ξ µ−1 e−ξ dξ is the gamma function. The operator Iµa is called the operator of fractional integration. Fractional integration has the property Iµa Iβa ϕ(t) = Iµ+β ϕ(t), a

µ > 0,

β > 0.

Definition of fractional derivatives. It is natural to introduce fractional differentiation as the inverse of fractional integration. For a function f (t) defined on a closed interval [a, b], the expression Dµa f (t)

d 1 = Γ(1 − µ) dt

Z

t

a

f (ξ) dξ (t − ξ)µ

(1.7.3.2)

is called the Riemann–Liouville fractional derivative or fractional derivative of order µ. It is assumed here that 0 < µ < 1. Note that the fractional integral is defined for any order µ > 0, but the fractional derivative has so far been defined only for 0 < µ < 1. If the function f (t) is continuously differentiable on the interval [a, b], then the fractional derivative (1.7.3.2) is evaluated by the formula Dµa f (t) =

  Z t ′ fξ (ξ) f (a) 1 + dξ . µ Γ(1 − µ) (t − a)µ a (t − ξ)

Now let us proceed to define a fractional derivative of order µ ≥ 1. We will use the following notation: [µ] stands for the integral part of a real number µ and {µ} is the fractional part of µ, 0 ≤ {µ} < 1, so that µ = [µ] + {µ}. If µ is an integer, then by the fractional derivative of order µ we mean the classical ordinary derivative  µ d Dµa = , µ = 1, 2, . . . . dt If µ is not an integer, then Dµa f is introduced by the formulas Dµa f (t) ≡



d dt

[µ]

D{µ} a f (t) =



d dt

[µ]+1

Ia1−{µ} f (t),

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

83

Thus, for µ > 0, we have Dµa f (t)

1 dn = Γ(n − µ) dtn

Z

t

a

f (ξ) dξ, (t − ξ)µ−n+1

n = [µ] + 1.

(1.7.3.3)

Sufficient conditions for the existence of the fractional derivatives (1.7.3.3) are discussed in [139, 171, 213, 240, 261, 328]. The operators Dµa appearing in formulas (1.7.3.2) and (1.7.3.3) are called operators of fractional differentiation. Main properties of fractional integrals and fractional derivatives. Linearity of fractional operators: Iµa [C1 f1 (t) + C2 f2 (t)] = C1 Iµa f1 (t) + C2 Iµa f2 (t), Dµa [C1 f1 (t) + C2 f2 (t)] = C1 Dµa f1 (t) + C2 Dµa f2 (t), where f1 (t) and f2 (t) are continuous functions that have fractional derivatives of order µ, while C1 and C2 are arbitrary constants. For µ > 0, the relation Dµa Iµa ϕ(t) = ϕ(t)

(1.7.3.4)

holds for any integrable function ϕ(t). Let the function f (t) be continuous on the interval [a, b] and have an integrable derivative Dµa f (t). Then, the formula Iµa Dµa f (t) = f (t) −

n−1 X k=0

(x − a)µ−k−1 (n−k−1) fn−µ (a) Γ(µ − k)

holds. Here, n = [µ]+1 and fn−µ (t) = In−µ f (t). In particular, for 0 < µ < 1, we get a Iµa Dµa f (t) = f (t) −

f1−µ (a) (t − a)µ−1 . Γ(µ)

(1.7.3.5)

For the power-law function tλ , we obtain Γ(λ + 1) λ+µ t , µ > 0, λ > −1, t > 0; Γ(λ + µ + 1) Γ(λ + 1) Dµ0 tλ = tλ−µ , 0 < µ ≤ 1, λ 6∈ Z, t > 0. Γ(λ − µ + 1)

Iµ0 tλ =

(1.7.3.6)

It is apparent from the second formula in (1.7.3.6) that, unlike usual differentiation, the fractional differentiation of a constant quantity leads to a power-law function proportional to t−µ rather than zero. For a table of fractional derivatives for some elementary functions, see [13]. It is noteworthy that the fractional derivatives of the exponential and simple trigonometric functions are not elementary functions.

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

Remark 1.39. There are other definitions of a fractional derivative [139, 171, 213, 240, 328]. For example, in applications, the Caputo fractional derivative is fairly common. It is defined as [53]:

Dµ a f (t)

  

1 = Γ(n − µ)   f (n) (t) t

Z

t a

(n)

fξ (ξ) dξ (t − ξ)µ−n+1

if n − 1 < µ < n,

(1.7.3.7)

if µ = n, n ∈ N.

Linear fractional diffusion equation. Sometimes, to model anomalous diffusion processes, one uses the fractional diffusion equation of the form (µ)

ut

= auxx ,

0 < µ < 1.

(1.7.3.8)

It describes slow linear diffusion known as subdiffusion. On the left-hand side of (µ) equation (1.7.3.8), there is a fractional derivative with respect to t; that is, ut = µ D0 u. The notation used here coincides with the standard notation for a partial derivative at integer µ. The article [221] gave a physical interpretation of equation (1.7.3.8) with µ = 1/2 within the percolation (comb) model. For the relationship between µ and Hausdorff fractal dimension in this mathematical model of diffusion, see [120, 210]. Theoretical justifications for the fractional kinetic equations of diffusion and diffusion– advection as well as Fokker–Planck type equations can be found in [211]. The studies [198–200] used the Laplace transform in time and Fourier transform in the space coordinate to analyze solutions of the fractional linear equation (1.7.3.8) (see also [121]). Nonlinear fractional diffusion equations. Recently, quite a few publications have appeared that analyze symmetries and construct exact solutions to nonlinear partial differential equations involving fractional derivatives (e.g., see [66, 67, 118, 322, 324–326]). Since Dµ0 is a linear operator in t, Propositions 2 and 3 from Subsection 1.6.2 for equations (1.6.2.1) and (1.6.2.11) remain valid even though some of the linear operators Li (or all of them) are fractional differential operators. Below we illustrate the application of the method of invariant subspaces to the construction of exact solutions to nonlinear PDEs with fractional derivatives. ◮ Example 1.52. Consider the nonlinear diffusion equation with a fractional derivative with respect to time (µ)

ut

= [(au + b)ux ]x + cu.

(1.7.3.9) (µ)

This equation is a special case of equation (1.7.2.1) with L[u] = ut and c = 0. Therefore, the formulas and equations obtained in Example 1.51 can be applied to construct its exact solutions. Considering the above, one can argue that equation (1.7.3.9) admits generalized separable solutions (1.7.2.2) in the form a quadratic polynomial in x whose functional coefficients ψi = ψi (t) satisfy the following system of ordinary differential

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

85

equations with fractional derivatives: (µ)

= 2(aψ1 + b)ψ3 + aψ22 ,

(ψ2 )t

(µ)

= 6aψ2 ψ3 ,

(µ) (ψ3 )t

= 6aψ32 .

(ψ1 )t

(1.7.3.10)

We seek solutions to system (1.7.3.10) in the form 1 4

ψ1 = θ +

B 2 ψ3 ,

ψ2 = Bψ3 ,

(1.7.3.11)

where B is an arbitrary constant. We arrive at the following linear nonhomogeneous equation for θ = θ(t): (µ)

θt

= 2aψ3 θ + 2bψ3 .

(1.7.3.12)

We look for a solution to the third equation in (1.7.3.10) as a power-law function, ψ3 = ptλ . Using the second formula in (1.7.3.6), we find that ψ3 =

Γ(1 − µ) −µ t , 6a Γ(1 − 2µ)

µ 6=

1 . 2

(1.7.3.13)

On substituting (1.7.3.13) into equation (1.7.3.12), we look for its solution in the form θ = Ctk + m,

(1.7.3.14)

where C is an arbitrary constant. After simple calculations, we get b m= a



3 Γ(1 − 2µ) −1 Γ2 (1 − µ)

−1

.

(1.7.3.15)

Moreover, we arrive at the following transcendental equation for k: Γ(1 − µ) Γ(k + 1) = . Γ(k − µ + 1) 3 Γ(1 − 2µ)

(1.7.3.16)

Formulas (1.7.2.2), (1.7.3.11), (1.7.3.13), and (1.7.3.14) as well as the transcendental equation (1.7.3.16) define a generalized separable solution to the nonlinear ◭ diffusion equation with a fractional derivative (1.7.3.9).

1.7.4. Pseudo-Differential Equations 1◦ . Suppose f = f (x) is a function that can be represented as a finite or infinite power series, ∞ X f (x) = βn xn . (1.7.4.1) n=0

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

For simplicity, we assume that the radius of convergence of the series is infinitely large. Let ∞ X ∂ f (Dx ) ≡ βn (Dx )n , Dx = , (1.7.4.2) ∂x n=0 be a linear differential operator corresponding to the function (1.7.4.1). Such operators are called pseudo-differential [88]. They possess the properties f (Dx )(C1 u1 + C2 u2 ) = C1 f (Dx )u1 + C2 f (Dx )u2 , [f1 (Dx ) + f2 (Dx )]u = f1 (Dx )u + f2 (Dx )u, where u1 , u2 , and u are arbitrary functions, while C1 and C2 are arbitrary constants. Note also the useful properties (Dx )n E = λn E,

f (Dx )E = f (λ)E,

where E = eλx ,

which will be required later on. Considered below are a few nonlinear differential equations that involve the operator (1.7.4.2). 2◦ . The nonlinear “parabolic” equation ut = f (Dx )(u2 ) + au2 + bu + c

(1.7.4.3)

admits a generalized separable solution of the form u = ϕ(t) + eλx ψ(t).

(1.7.4.4)

This can be shown as follows. With the representation (1.7.4.2) and formula (1.7.4.4), we have u2 = ϕ2 + 2ϕψE + ψ 2 E 2 , E = eλx , ∞ ∞ X X f (Dx )(u2 ) = β0 ϕ2 + 2ϕψE βn λn + ψ 2 E 2 βn (2λ)n n=0

n=0 2

= β0 ϕ2 + 2f (λ)ϕψE + f (2λ)ψ 2 E ,

(1.7.4.5)

β0 = f (0).

Substituting (1.7.4.4) into (1.7.4.3) and taking into account (1.7.4.5), we arrive at a system of ordinary differential equations for ϕ = ϕ(t) and ψ = ψ(t): ϕ′t = [a + f (0)]ϕ2 + bϕ + c, ψt′ = 2[a + f (λ)]ϕψ + bψ.

(1.7.4.6)

We also obtain the following transcendental equation for the constant λ: f (2λ) + a = 0.

(1.7.4.7)

Remark 1.40. Equation (1.7.4.3) and the similar equations considered below are called nonlinear pseudo-differential equations.

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1.7. Other Nonlinear Equations Having Generalized Separable Solutions

Remark 1.41. The transcendental equation (1.7.4.7) can have more than one root or no

roots at all. ◮ Example 1.53. Consider the equation

ut = cos(σDx )(u2 ) + au2 + bu + c.

(1.7.4.8)

It has an exact solution of the form (1.7.4.4), where the functions ϕ1 and ϕ2 are described by the system of ordinary differential equations (1.7.4.6) with f (0) = 1 and f (λ) = cos(σλ), and the constant λ is determined by solving the transcendental equation cos(2σλ) + a = 0. (1.7.4.9) For −1 ≤ a ≤ 1, equation (1.7.4.9) has infinitely many real roots (in particular, 1 for a = 0, we have λ = 2σ ( π2 + πm) with m = 0, ±1, ±2, . . . ), which generate infinitely many exact solutions of the form (1.7.4.4). For |a| > 1, equation (1.7.4.9) does not have real roots, and hence, equation (1.7.4.8) does not admit real exact ◭ solutions of the form (1.7.4.4). Likewise, it can be shown that equation (1.7.4.3) admits a more complicated solution of the form u = ϕ1 (t) + ϕ2 (t)eλx + ϕ3 (t)e−λx ,

(1.7.4.10)

provided that the real constant λ 6= 0 satisfies the two transcendental equations f (2λ) + a = 0,

f (−2λ) + a = 0

(1.7.4.11)

simultaneously. In this case, the functions ϕm = ϕm (t) are described by the system of ordinary differential equations ϕ′1 = [a + f (0)](ϕ21 + 2ϕ2 ϕ3 ) + bϕ1 + c, ϕ′2 = [a + f (λ)]ϕ1 ϕ2 + bϕ2 , ϕ′3 = [a + f (−λ)]ϕ1 ϕ3 + bϕ3 .

(1.7.4.12)

The overdetermined system of equations (1.7.4.11) reduces to a single equation in many cases, two of which include: (i) a is any number and the function f (x) is even, f (x) = f (−x). In this case, the overdetermined system is reduced to a single equation (1.7.4.7), and it follows from the last two equations in (1.7.4.12) that ϕ2 = Cϕ3 , where C is an arbitrary constant. (ii) a = 0 and the function f (x) is odd, f (x) = −f (−x). ◮ Example 1.54. Consider equation (1.7.4.8) again. It is determined by an even function, f (x) = cos(σx) with σ = const and, hence, corresponds to case (i). Therefore, the equation admits an exact solution of the form

u = ϕ1 (t) + ϕ2 (t)(C1 eλx + C2 e−λx ),

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1. M ETHODS OF G ENERALIZED S EPARATION OF VARIABLES

where the constant λ is determined by solving the transcendental equation (1.7.4.9); it has been taken into account that ϕ3 /ϕ2 = const. As in Example 1.53, for −1 ≤ a ≤ 1, there are infinitely many real roots λ, which generate infinitely many exact ◭ solutions of the above form. Let f (x) be an even function, which corresponds to case (i), and let equation (1.7.4.7) have a purely imaginary root, λ = ik, where i2 = −1 and k is a real number. Then, equation (1.7.4.3) admits a solution involving trigonometric functions: u = ψ1 (t) + ψ2 (t) cos(kx) + ψ3 (t) sin(kx),

(1.7.4.13)

where the functions ψm = ψm (t) are described by the system of ordinary differential equations ψ1′ = [a + f (0)][ψ12 + 12 (ψ22 + ψ32 )] + bψ1 + c, ψ2′ = [a + f (ik)]ψ1 ψ2 + bψ2 , ψ3′

(1.7.4.14)

= [a + f (ik)]ψ1 ψ3 + bψ3 .

This fact can be proved by substituting eλx = eikx = cos(kx) + i sin(kx), e−λx = e−ikx = cos(kx) − i sin(kx), ψ2 − iψ3 ψ2 + iψ3 ϕ1 = ψ1 , ϕ2 = , ϕ3 = 2 2 into solution (1.7.4.10) and equations (1.7.4.12) followed by separating the real and imaginary parts. It should be noted that f (ik) = f (−ik) is a real number, and the last equation in (1.7.4.14) can be replaced by the simpler equation ϕ2 = Cϕ3 , where C is an arbitrary constant. ◮ Example 1.55. Consider equation (1.7.4.8), determined by the even function f (x) = cos(σx) and corresponding to the case a < −1. The transcendental equation (1.7.4.9) has two purely imaginary roots, λ = ±ik, where the real number k is determined from the equation cosh(k) = −a. Hence, equation (1.7.4.8) has a single solution of the form (1.7.4.13), and the two roots λ = ±ik generate identical ◭ solutions. ◮ Example 1.56. Likewise, it can be shown that the equation

ut = f (Dx )(u2 ) + g(Dx )u

(1.7.4.15)

has an exact solution of the form (1.7.4.4), where the functions ϕ = ϕ(t) and ψ = ψ(t) are described by the system of ordinary differential equations ϕ′t = f (0)ϕ2 + g(0)ϕ, ψt′ = 2f (λ)ϕψ + g(λ)ψ,

(1.7.4.16)

and the constant λ is determined from the transcendental equation f (2λ) = 0.

◭

89

1.7. Other Nonlinear Equations Having Generalized Separable Solutions

3◦ . The more complicated equation ut = f (Dx )u g(Dx )u + h(Dx )u,

(1.7.4.17)

where f (Dx )u g(Dx )u stands for the product of the functions f (Dx )u and g(Dx )u, also admits an exact solution of the form (1.7.4.4), where the functions ϕ = ϕ(t) and ψ = ψ(t) are described by the system of ordinary differential equations ϕ′ = f (0)g(0)ϕ2 + h(0)ϕ, ψ ′ = [f (0)g(λ) + g(0)f (λ)]ϕψ + h(λ)ψ,

(1.7.4.18)

and the parameter λ is determined by solving either transcendental equation f (λ) = 0 or g(λ) = 0. 4◦ . Instead of the “parabolic” equations (1.7.4.3), (1.7.4.15), and (1.7.4.17), one could treat the corresponding “hyperbolic” equations, in which the first derivative ut is substituted for by the second derivative utt . As an example, let us consider the equation utt = f (Dx )(u2 ) + au2 + bu + c, (1.7.4.19) whose parabolic analogue is equation (1.7.4.3). Equation (1.7.4.19) has a solution of the form (1.7.4.4), where the functions ϕ = ϕ(t) and ψ = ψ(t) are described by the system of ordinary differential equations (1.7.4.6) in which the first derivatives ϕ′t and ψt′ must be replaced by the second ′′ derivatives ϕ′′tt and ψtt , respectively, and the equation for the parameter λ remains the same, (1.7.4.7). This also applies to solutions (1.7.4.10) and (1.7.4.13); in systems (1.7.4.12) and (1.7.4.14), the first derivatives must be replaced with the respective second derivatives, and the equations for λ remain unchanged. 5◦ . Nonlinear equations of the form f (Dt )u = g(Dx )(up ) admit a multiplicative separable solution u = ϕ(t)ψ 1/p (x), where the functions ϕ = ϕ(t) and ψ = ψ(x) are determined by the ordinary differential equations f (Dt )ϕ = Cϕp , with C being an arbitrary constant.

g(Dx )ψ = Cψ 1/p ,

2. Methods of Functional Separation of Variables 2.1. Preliminary Remarks 2.1.1. Structure of Functional Separable Solutions 1◦ . Suppose a nonlinear PDE for u = u(x, t) is obtained from a linear PDE for z = z(x, t) with a change of variable u = U (z). Then, if the linear equation for z admits separable solutions, the transformed nonlinear equation for u will have exact solutions of the form n X u(x, t) = U (z), where z = ϕm (x)ψm (t). (2.1.1.1) m=1

Many nonlinear partial differential equations that are not reducible to linear equations also have exact solutions of the form (2.1.1.1). Such solutions will be referred to as functional separable solutions. In the general case, the functions ϕm (x), ψm (t), and U (z) of (2.1.1.1) are unknown a priori and are to be determined in the analysis. The function U will be called the outer function, while ϕm and ψm , inner functions. Remark 2.1. A generalized separable solution (see Chapter 1) is a functional separable solution of the special form U (z) = z . The presence of the outer function U in (2.1.1.1), which is unknown, is a complicating factor in constructing functional separable solutions.

The main idea behind the method of functional separations of variables is that the differential-functional equation resulting from the substitution of expression (2.1.1.1) into the partial differential equation in question has to be reduced, if possible, to the standard bilinear functional equation (1.5.1.1) from Subsection 1.5.1 (or the differential functional equation (1.2.2.1)–(1.2.2.2) from Subsection 1.2.2). 2◦ . Often (in a narrow sense) the term ‘solution with functional separation of variables’ (or ‘functional separable solution’) is used for simpler exact solutions of the following form (see, for example, [12, 86, 127, 159, 214, 215, 278, 385]): u = U (z),

z = ϕ(x) + ψ(t),

(2.1.1.2)

where the functions U (z), ϕ(x), and ψ(t) are all unknown. In searching for solutions (2.1.1.2), it is assumed that ϕ 6= const and ψ 6= const.

Remark 2.2. With functional separation of variables, the search for simple solutions of the form u = U (ϕ(x) + ψ(t)) or u = U (ϕ(x)ψ(t)) leads to the same results, since the representation U (ϕ(x)ψ(t)) = U1 (ϕ1 (x) + ψ1 (t)) holds true, where U1 (z) = U (ez ), ϕ1 (x) = ln ϕ(x), and ψ1 (t) = ln ψ(t).

DOI: 10.1201/9781003042297-2

91

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

3◦ . For linear functions, ϕ(x) = κx and ψ(t) = λt, where κ and λ are constants, solution (2.1.1.2) becomes [125, 150, 275, 278]: u = U (z),

z = κx + λt.

(2.1.1.3)

It is known as a traveling wave solution. Partial differential equations that do not involve the independent variables x and t explicitly and have the form F (u, ux , ut , uxx , uxt , utt , . . . ) = 0

(2.1.1.4)

admit, as a rule, traveling wave solutions (2.1.1.3), where κ and λ are arbitrary constants, with U (z) satisfying the ordinary differential equation ′′ ′′ ′′ F (U, κUz′ , λUz′ , κ2 Uzz , κλUzz , λ2 Uzz , . . . ) = 0.

(2.1.1.5)

Remark 2.3. Very rarely, one may come across an equation of the form (2.1.1.4) that does not have a traveling wave solution (2.1.1.3). In such cases, the left-hand side of equation (2.1.1.5) will be nonzero for any κ, λ, and U (z).

◮ Example 2.1. The nonhomogeneous Monge–Amp`ere equation

u2xt − uxx utt + 1 = 0, which is of the form (2.1.1.4), does not have traveling wave solutions. Indeed, substituting expression (2.1.1.3) into the original equation leads to the false equality ◭ 1 = 0. ◮ Example 2.2. It is easy to verify that the nonlinear equation

u2x utt − u2t uxx + a = 0 with a 6= 0, which is also of the form (2.1.1.4), does not have traveling wave solutions ◭ either. 4◦ . Sometimes, the implicit representation of a functional separable solution Z(u) = ϕ(x) + ψ(t)

(2.1.1.6)

is used, where the functions Z(u), ϕ(x), and ψ(t) are to be determined in a subsequent analysis (e.g., see [142, 159, 315, 383]). A more general representation, than (2.1.1.6), of functional separable solutions in implicit form, will be described below in Section 2.5. This representation was used in [252, 253, 298]. 5◦ . In general, the term functional separable solution in regard to PDEs with two independent variables is used for exact solutions that can be represented as a nonlinear superposition of functions [251, 257, 300], u = U (z),

z = ϕ(x, t),

(2.1.1.7)

where the unknown functions U (z) and ϕ(x, t) are described by overdetermined systems of ODEs and PDEs, respectively. In the simplest cases, either function satisfies a single equation.

2.2. Simplified Method for Constructing Functional Separable Solutions

93

A method for seeking functional separable solutions of the form (2.1.1.7) is outlined in Section 2.6. Section 2.7 presents an even more effective method, which is based on constructing exact solutions in implicit form, Z ϑ = ζ(u) du, where the functions ϑ = ϑ(x, t) and ζ = ζ(u) are determined in a subsequent analysis that employs a generalized splitting method.

2.1.2. Direct and Indirect Functional Separation of Variables One should distinguish between direct functional separation of variables and indirect functional separation of variables. Both are based on the same representation of solutions in the form (2.1.1.1), (2.1.1.2), (2.1.1.6), or (2.1.1.7). However, there are significant differences. With direct functional separation of variables, one first substitutes the solution representation into the PDE in question and then analyzes the resulting functional differential equation (e.g., see [250–254, 275]). With indirect functional separation of variables, one replaces the solution representation with one or more equivalent differential constraints,∗ and then performs a compatibility analysis of the overdetermined system consisting of the equation in question and the differential constraints (e.g., see [142, 159, 256, 310]). The current chapter employs direct methods for constructing exact solutions to nonlinear partial differential equations, which are based on direct functional separation of variables. These methods result in fewer intermediate calculations and are technically simpler than indirect methods based on using equivalent differential constraints. Remark 2.4. Section 4.5.3 will discuss differential constraints that are equivalent to functional separable solutions. It will also give a comparison of the effectiveness of direct and indirect methods for constructing functional separable solutions.

2.2. Simplified Method for Constructing Functional Separable Solutions 2.2.1. Description of the Simplified Method Based on Transformations of the Unknown Function In certain cases, finding a solution in the form (2.1.1.1) can be carried out in two stages. First, one uses a transformation that reduces the original equation to one with ∗ Any differential equation that depends on the same variables as the PDE in question can be treated as a differential constraint. See Chapter 4 for details.

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

a quadratic nonlinearity (sometimes a power-law nonlinearity). Then, one seeks a solution to the resulting equation using the methods described in Sections 1.3–1.6. Unfortunately, there are no regular methods that would reduce any given PDE to a PDE with a quadratic nonlinearity. Sometimes, equations with a quadratic nonlinearity can be obtained using a transformation u = U (z) of the desired function u. The most common transformations include: u = zλ

(for PDEs with power-law nonlinearity),

u = λ ln z

(for PDEs with exponential nonlinearity),

u=e

λz

(for PDEs with logarithmic nonlinearity),

where λ is a constant to be determined. This approach is equivalent to a priori setting of the form of the outer function U (z) in (2.1.1.1); whether this is successful or not will depend on the experience and intuition of the researcher. Many nonlinear equations of mathematical physics of various forms that can be reduced with suitable transformations to equations with a quadratic nonlinearity are described in [106, 108, 110, 112–114, 272, 274, 275]. Some of the results of these studies are outlined in the next subsection.

2.2.2. Examples of Constructing Exact Solutions to Nonlinear PDEs Below are a few examples that illustrate the use of the simplified method to construct functional separable solutions of nonlinear second-order partial differential equations. ◮ Example 2.3. Let us look at the five-parameter family of heat-type equations with power-law nonlinearities

ut = a(un ux )x + bun+1 + cu + ku1−n ,

(2.2.2.1) n

where a, b, c, k, and n are free parameters. The change of variable z = u converts (2.2.2.1) to an equation with a quadratic nonlinearity a zt = azzxx + zx2 + bnz 2 + cnz + kn. (2.2.2.2) n This equation admits various generalized separable solutions that depend on the coefficients of the nonlinear terms on the right-hand side of (2.2.2.2). Solutions to equation (2.2.2.2) are not difficult to find with the help of Table 1.3 (see rows 3–6). In addition, it should be taken into account that the last two terms on the right-hand side of (2.2.2.2) do not affect the solution structure. In particular, if the inequality ab(n + 1) > 0 holds, there will exist solutions with trigonometric functions, while if ab(n + 1) < 0, there will exist solutions with exponential functions. See also Table 2.1. The above approach allows us to obtain solutions of the form    1/n u = ϕ(t) C1 cos(βx) + C2 sin(βx) + ψ(t) if ab(n + 1) > 0,    1/n u = ϕ(t) C1 cosh(βx) + C2 sinh(βx) + ψ(t) if ab(n + 1) < 0, (2.2.2.3)

2.2. Simplified Method for Constructing Functional Separable Solutions

95

where C1 and C2 are arbitrary constants, s β=

|b|n2 , |a(n + 1)|

while ϕ = ϕ(t) and ψ = ψ(t) are functions that satisfy the system of ordinary differential equations ϕ′t =

bn(n + 2) ϕψ + cnϕ, n+1

(2.2.2.4) bn (C12 ± C22 )ϕ2 . n+1 The upper sign in the second equation corresponds to the first solution in (2.2.2.3), and the lower sign to the second solution in (2.2.2.3). For C1 = C2 , the last equation in (2.2.2.4) (lower sign) can be satisfied if we set ψ = const, where ψ is a root of the quadratic equation bψ 2 + cψ + k = 0. In this case, the general solution to the first equation of (2.2.2.4) is given by   bn(n + 2) ϕ = C3 exp ψ + cn t , n+1 ψt′ = n(bψ 2 + cψ + k) +

◭

where C3 is an arbitrary constant.

◮ Example 2.4. Now we look at the five-parameter family of heat-type equations with exponential nonlinearities

ut = a(eλu ux )x + beλu + c + ke−λu .

(2.2.2.5)

λu

The change of variable z = e converts (2.2.2.5) to an equation with a quadratic nonlinearity zt = azzxx + bλz 2 + cλz + kλ. (2.2.2.6) One can find solutions to equation (2.2.2.6) using Table 1.3 (see row 3). It should be taken into account that the last two terms on the right-hand side of (2.2.2.6) do not affect the solution structure. It is apparent that if the inequality abλ > 0 holds, equation (2.2.2.6) has a solution with trigonometric functions, if abλ < 0, it has a solution with exponential functions, and if b = 0, its solution has the form of a quadratic polynomial in x (see also Table 2.1). With the above change of variable, one can find, in particular, the following functional separable solutions to equation (2.2.2.5) [110]: p p  1   u = ln eαt C1 cos(x β ) + C2 sin(x β ) + γ if abλ > 0, λ p p    1 u = ln eαt C1 cosh(x −β ) + C2 sinh(x −β ) + γ if abλ < 0, λ where C1 and C2 are arbitrary constants, while α = λ(bγ + c),

β = bλ/a,

with γ = γ1,2 being roots of the quadratic equation bγ 2 + cγ + k = 0.

◭

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

◮ Example 2.5. The four-parameter heat equation with a source containing logarithmic nonlinearities

ut = auxx + bu ln2 u + cu ln u + ku

(2.2.2.7)

can be reduced with the substitution u = ez to an equation with a quadratic nonlinearity zt = azxx + azx2 + bz 2 + cz + k. (2.2.2.8) One can find solutions to equation (2.2.2.8) with the help of Table 1.3 (see equations No. 1 and No. 2). It is apparent that if the inequality ab > 0 holds, equation (2.2.2.8) has a solution with trigonometric functions, and if ab < 0, it has a solution with exponential functions. For b = 0, the equation admits a generalized separable solution as a quadratic polynomial in the space variable, z = ψ1 (t)x2 + ψ2 (t)x + ψ3 (t), where the functions ψi = ψi (t), i = 1, 2, 3, are described by the system of ordinary differential equations ψ1′ = 4aψ12 + cψ1 , ψ2′ = 4aψ1 ψ2 + cψ2 , ψ3′ = cψ3 + 2aψ1 + aψ22 + k. These are easy to integrate sequentially. The first equation is integrable, since it is a Bernoulli equation [273]. It follows from the comparison of the first and second equations that ψ2 = C1 ψ1 , where C1 is an arbitrary constant. The last equation is linear in ψ3 with a known (from the solution of the first two equations) nonhomoge◭ neous term. Table 2.1 gives examples of nonlinear heat equations with power-law, exponential, and logarithmic nonlinearities reducible, by simple substitutions of the form u = U (z), to equations with a quadratic or cubic nonlinearity. The last PDE in the table is a hyperbolic-type heat equation.

2.3. Functional Separable Solutions of Special Form 2.3.1. Generalized Traveling Wave Solutions and Other Solutions of Special Form To facilitate the analysis, one can set some of the inner functions in (2.1.1.1) a priori and determine the other ones, including the outer function, U , in the course of the solution. Such solutions will be referred to as functional separable solutions of special form.

2.3. Functional Separable Solutions of Special Form

97

Table 2.1. Some nonlinear heat equations reducible to equations with a quadratic or cubic nonlinearity with a substitution of the form u = U (z); the constant σ is expressed in terms of the coefficients of the original equation. Original equation

Transformation

Transformed equation

Form of solutions for z

ut = a(un ux )x + bu

u = z 1/n

zt = azzxx + an−1zx2 + bnz

z = ϕ(t)x2 + ψ(t)x + χ(t), z = ϕ(t)x2 + ψ(t)xn/(n+1)

ut = a(un ux )x + bu + cu1−n

u = z 1/n

zt = azzxx + an−1zx2 + bnz + cn

z = ϕ(t)x2 + ψ(t)x + χ(t)

ut = a(un ux )x + bun+1 + cu

u = z 1/n

zt = azzxx + an−1zx2 + bnz 2 + cnz

z = ϕ(t)eσx + ψ(t)e−σx + χ(t), z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)

ut = a(u2n ux )x + bu1−n

u = z 1/n

zt = az 2zxx + a(1 + n−1 )zzx2 + bn

z = ϕ(t)x + ψ(t)

ut = a(un ux )x + (bun + c)ux

u = z 1/n

zt = azzxx + an−1zx2 + (bz + c)zx

z = ϕ(t)x + ψ(t)

ut = a(u2n ux )x + bunux

u = z 1/n

zt = az 2zxx a(1 + n−1 )zzx2 + bzzx

z = ϕ(t)x + ψ(t)

ut = a(eλu ux )x + b + ce−λu

u = λ−1 ln z

zt = azzxx + bλz + cλ

z = ϕ(t)x2 + ψ(t)x + χ(t)

ut = a(eλu ux )x + beλu + c

u = λ−1 ln z

z = ϕ(t)eσx + ψ(t)e−σx + χ(t), zt = azzxx + bz 2 + cλz z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)

ut = a(e2λu ux)x + b + ce−λu

u = λ−1 ln z

zt = az 2 zxx + azzx2 + bλz + cλ

z = ϕ(t)x + ψ(t)

ut = auxx + bu ln u + cu

u = ez

zt = azxx + azx2 + bz + c

z = ϕ(t)x2 + ψ(t)x + χ(t)

ut = auxx + bu ln2 u + cu

u = ez

zt = azxx + azx2 + bz 2 + c

z = ϕ(t)eσx + ψ(t)e−σx + χ(t), z = ϕ(t) sin(σx) + ψ(t) cos(σx) + χ(t)

ut = auxx + b(ln u)ux + cu

u = ez

zt = azxx + azx2 + bzzx + c

z = ϕ(t)x + ψ(t)

ut = [(a ln u + b)ux ]x + cu

u = ez

zt = (az + b)zxx + (az + a + b)zx2 + c

z = ϕ(t)x + ψ(t)

utt + kut = auxx + bu ln u + cu

u = ez

ztt + zt2 + kz = azxx + azx2 + bz + c

z = ϕ(x) + ψ(t)

Below we list a few simple functional separable solutions of special form (x and t can be swapped): u = U (z), z = ϕ(t)x+ψ(t) 2

u = U (z), z = ϕ(t)x +ψ(t) λx

(argument z linear in x); (argument z quadratic in x);

u = U (z), z = ϕ(t)e +ψ(t) (z contains an exponential of x).

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The first one will be called a generalized traveling wave solution; in the special case of ψ(t) = k1 and ψ(t) = k2 t, where k1 and k2 are arbitrary constants, this is a traveling wave solution. The last formula can have a simple hyperbolic or trigonometric function instead of eλx . After substituting any of the above expressions in the equation in question, one has to eliminate x using the expression of z. This will result in a functional differential equation with two variables, t and z. In certain cases, its solutions can be obtained using the methods described in Chapter 1. For clarity, a schematic representing the general procedure of constructing generalized traveling wave solutions is shown in Figure 2.1. Original equation: ut = F(t, u, ux , uxx , ..., ux(n) ) Seeking generalized traveling wave solutions

Set the form of solutions: u = U(z), where z = j (t) x + y(t) Substitute it into original equation and replace x with (z - y)/j

Arrive at a functional differential equation in two arguments, t and z Use the splitting method

Get the bilinear functional equation: å Φn Ψn = 0 Solve the bilinear equation. Find linear relations for {Φn} and {Ψn}

Obtain the determining system of ODEs by the splitting method Solve the determining system of ODEs

Find the functions j, y, and U Find the argument z = j (t) x + y(t)

Obtain exact solutions to the original equation Figure 2.1. Algorithm for constructing generalized traveling wave solutions for evolution equations.

Remark 2.5. The algorithm depicted in Figure 2.1 can also be used to construct more general exact solutions∗ u = σ(t)F (z) + ϕ2 (t)x + ψ2 (t), where z = ϕ1 (t)x + ψ1 (t). An example is considered later in Subsection 3.1.1 (see Example 3.3). ∗ The class of solutions considered contains all most common solutions as special cases: traveling wave solutions, generalized self-similar solutions, additive separable solutions, and multiplicative separable solutions, as well as many invariant solutions.

99

2.3. Functional Separable Solutions of Special Form

2.3.2. Examples of Constructing Generalized Traveling Wave Solutions Below we give a few examples of nonlinear PDEs that admit functional separable solutions of special form where the complex argument z is linear or quadratic in one of the independent variables. ◮ Example 2.6. Let us look as the nonlinear third-order equation

uy uxy − ux uyy = a(uyy )n−1 uyyy ,

(2.3.2.1)

which describes a boundary layer of a power-law fluid on a flat plate; u is the stream function, x and y are coordinates along and normal to the plate, and n is a rheological parameter (the value n = 1 corresponds to a Newtonian fluid). We seek exact solutions to this equation as a generalized traveling wave u = U (z),

z = ϕ(x)y + ψ(x).

(2.3.2.2)

′′ n−1 ′′′ Substituting (2.3.2.2) into (2.3.2.1) yields ϕ′x (Uz′ )2 = aϕ2n (Uzz ) Uzzz . This equation is independent of ψ. Separating the variables and integrating, we find that ( (ax + C)1/(1−2n) if n = 6 1/2, ϕ(x) = ψ(x) is an arbitrary function. C exp(ax/λ) if n = 1/2,

The function U (z) satisfies the ordinary differential equations ′′ n−1 ′′′ (Uz′ )2 = (1 − 2n)(Uzz ) Uzzz

(Uz′ )2 For

1 2

=

if n = 6 1/2,

′′ −1/2 ′′′ λ(Uzz ) Uzzz

if n = 1/2.

< n < 2, the first equation admits a power-law solution U (z) = Az

2n−1 n−2

,

A=



3(2 − n) 2n − 1



1 2−n



(2n − 1)(n + 1) (2 − n)2



n 2−n

. ◭

◮ Example 2.7. Consider the unsteady heat equation with a nonlinear source

ut = uxx + f (u).

(2.3.2.3)

We seek its exact solutions as a generalized traveling wave∗ u = u(z),

z = ϕ(t)x + ψ(t).

(2.3.2.4)

It is required to determine the functions u(z), ϕ(t), and ψ(t) as well as the term f (u) on the right-hand side. ∗ Here and in what follows, whenever the equation involves a function f (u), which is undefined and needs to be found in the course of the solution by the method of functional separation of variables, we will denote the solution u = u(z) instead of u = U (z). The form of the argument z = z(x, t) can be chosen in different ways; see Subsection 2.1.1 for details.

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting (2.3.2.4) into (2.3.2.3) and dividing by u′z gives ϕ′t x + ψt′ = ϕ2

u′′zz f (u) + ′ . u′z uz

(2.3.2.5)

On expressing x in terms of z in (2.3.2.4), we get x = (z − ψ)/ϕ. In view of this, we eliminate x from (2.3.2.5) to arrive at the following functional differential equation with two independent variables, t and z: −ψt′ +

ϕ′ u′′ f (u) ψ ′ ϕt − t z + ϕ2 zz + ′ = 0. ′ ϕ ϕ uz uz

It can be conveniently represented in the bilinear form with four terms Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 = 0,

(2.3.2.6)

where Φ1 = −ψt′ + Ψ1 = 1,

ψ ′ ϕ′ ϕt , Φ2 = − t , Φ3 = ϕ2 , Φ4 = 1, ϕ ϕ u′′ f (u) Ψ2 = z, Ψ3 = zz , Ψ4 = . u′z u′z

(2.3.2.7)

Substituting (2.3.2.7) into formulas (1.5.2.7), which satisfy relation (2.3.2.6) identically, we obtain the system of ordinary differential equations −ψt′ +

ϕ′ ψ ′ ϕt = A1 ϕ2 + A2 , − t = A3 ϕ2 + A4 , ϕ ϕ u′′zz f (u) = −A1 − A3 z, = −A2 − A4 z, u′z u′z

where A1 , . . . , A4 are arbitrary constants. Case 1. If A4 6= 0, the solution of system (2.3.2.8) is −1/2  A3 2A4 t , ϕ(t) = ± C1 e − A4  Z  Z dt ψ(t) = −ϕ(t) A1 ϕ(t) dt + A2 + C2 , ϕ(t) Z
u(z) = C3 exp − 12 A3 z 2 − A1 z dz + C4 ,
f (u) = −C3 (A4 z + A2 ) exp − 21 A3 z 2 − A1 z ,

(2.3.2.8)

(2.3.2.9)

where C1 , . . . , C4 are arbitrary constants. The function f = f (u) is defined by the last two relations in parametric form, with z being the parameter. If A3 6= 0, the source function f (u) in (2.3.2.9) can be expressed in terms of elementary functions and the inverse error function. In the special case of A3 = C4 = 0, A1 = −1, and C3 = 1, the source function can be represented in explicit form: f (u) = −u(A4 ln u + A2 ).

(2.3.2.10)

2.3. Functional Separable Solutions of Special Form

101

In this case, the solution to equation (2.3.2.3) can also be obtained using Lie group analysis [85]. Case 2. If A4 = 0 and A3 6= 0, the solutions of the first two equations in (2.3.2.8) are 1 ϕ(t) = ± √ , 2A3 t + C1

C2 A1 A2 − − (2A3 t + C1 ). A3 3A3 2A3 t + C1 (2.3.2.11) The solutions to the other equations are defined by the last two formulas in (2.3.2.9) with A4 = 0. If A4 = A3 = 0, the solutions to the first two equations in (2.3.2.8) are given by ϕ = C1 ,

ψ(t) = √

ψ(t) = −(A1 C12 + A2 )t + C2 ,

where C1 and C2 are arbitrary constants.

(2.3.2.12) ◭

◮ Example 2.8. Let us look at the nonlinear convective heat equation with variable coefficient and a source

ut = a(t)uxx + b(t)ux + c(t)f (u),

(2.3.2.13)

where a = a(t), b = b(t), and c = c(t) are arbitrary functions. It converts to equation (2.3.2.3) for a = c = 1 and b = 0. We look for solutions in the form (2.3.2.4). Reasoning in a similar way to that in Example 2.7, we arrive at the system of ordinary differential equations −ψt′ +

ϕ′ ψ ′ ϕt + bϕ = A1 aϕ2 + A2 c, − t = A3 aϕ2 + A4 c, ϕ ϕ f (u) u′′zz = −A1 − A3 z, = −A2 − A4 z. u′z u′z

(2.3.2.14)

It is apparent that the first two equations, for ϕ = ϕ(t) and ψ = ψ(t), have only changed in system (2.3.2.14) as compared to system (2.3.2.8). Despite becoming slightly more complex, these equations can be fully integrated, since the second equation is a Bernoulli equation and the first one is linear in ψ. As a result, we obtain 

−1/2   Z a(t) ϕ(t) = ± C1 E(t) + 2A3 E(t) dt , E(t) = exp 2A4 c(t) dt , E(t)  Z  Z Z c(t) ψ(t) = −ϕ(t) A1 a(t)ϕ(t) dt + A2 dt − b(t) dt + C2 . ϕ(t) Z

As before, u(z) and f (u) are defined by the last two formulas in (2.3.2.9).

◭

◮ Example 2.9. The nonlinear heat equation with variable diffusivity and a non-

linear source ut = [g(u)ux ]x + f (u)

(2.3.2.15)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

also has solutions of the form (2.3.2.4). In this case, the unknown quantities are described by the system of ODEs ψ ′ ϕ′ ϕt = A1 ϕ2 + A2 , − t = A3 ϕ2 + A4 , ϕ ϕ f (u) [g(u)u′z ]′z = −A1 − A3 z, = −A2 − A4 z. u′z u′z

−ψt′ +

(2.3.2.16)

It is apparent that the third equation has only changed in system (2.3.2.16) as compared to system (2.3.2.8). Therefore, the functions ϕ(t) and ψ(t) are defined by the first two formulas in (2.3.2.9). Either g(u) or f (u) in equation (2.3.2.15) can be set arbitrarily, while the other is found from the last two ODEs of (2.3.2.16). The last two equations in (2.3.2.16) involve three unknown functions: f = f (u), g = g(u), and u = u(z). Hence, one of them can be set arbitrarily. In what follows, we assume that u = u(z) is defined in implicit form: z = Z(u), where Z(u) is any twice continuously differentiable function. Using relation u′z = 1/Zu′ , we obtain the following formulas from the last two equations of (2.3.2.16): 1 f (u) = −[A2 + A4 Z(u)] ′ , Zu (u)   Z g(u) = − A1 u + A3 Z(u) du + C Zu′ (u).

(2.3.2.17)

These define parametrically the admissible forms of the functions f = f (u) and g = g(u) that enter equation (2.3.2.15). This equation has a solution of the form (2.3.2.4), where u = u(z) is defined implicitly as z = Z(u), while ϕ(t) and ψ(t) are defined by the first two formulas in (2.3.2.9) (with A4 6= 0) or by formulas (2.3.2.11) (with A4 = 0, A3 6= 0) and (2.3.2.12) (with A4 = A3 = 0). Consider two examples of utilizing formulas (2.3.2.17). 1◦ . Setting a2 (k + 1) a1 , A2 = −b2 k, A3 = − , A4 = −b1 k, C = 0 k k in (2.3.2.17), we get an equation of the form (2.3.2.15) with power-law nonlinearities: ut = [(a1 uk + a2 u2k )ux ]x + b1 u + b2 u1−k . Z(u) = uk , A1 = −

It admits the generalized traveling wave solution (2.3.2.4) with u(z) = z 1/k , where ϕ(t) and ψ(t) are defined by the first two formulas of (2.3.2.9) (with b1 6= 0) or formulas (2.3.2.11) (with b1 = 0, a2 6= 0) and (2.3.2.12) (with b1 = a2 = 0). 2◦ . Setting

a1 , A2 = −b2 λ, A3 = −a2 , A4 = −b1 λ, C = 0 λ in (2.3.2.17), we get an equation of the form (2.3.2.15) with exponential and powerexponential nonlinearities: Z(u) = eλu , A1 = −

ut = [(a1 ueλu + a2 e2λu )ux ]x + b1 + b2 e−λu .

103

2.3. Functional Separable Solutions of Special Form

It admits the generalized traveling wave solution (2.3.2.4) with u(z) = (ln u)/λ, where ϕ(t) and ψ(t) are defined by the first two formulas of (2.3.2.9) (with b1 6= 0) or formulas (2.3.2.11) (with b1 = 0, a2 6= 0) and (2.3.2.12) (with b1 = a2 = 0). ◭ ◮ Example 2.10. The nth-order nonlinear equation

ut = u(n) x + f (u)

(2.3.2.18)

can be treated in a similar way. As above, we look for generalized traveling wave solutions (2.3.2.4). Reasoning as in Example 2.7, we arrive at the system of ordinary differential equations −ψt′ +

ψ ′ ϕ = A1 ϕn + A2 , ϕ t (n)

uz = −A1 − A3 z, u′z

−

ϕ′t = A3 ϕn + A4 , ϕ

f (u) = −A2 − A4 z, u′z

(2.3.2.19)

If A4 6= 0, the general solution of the first two equations of system (2.3.2.19) is given by −1/n  A3 A4 nt ϕ(t) = C1 e − , A  Z 4 Z n−1 ψ(t) = −ϕ(t) A1 ϕ (t) dt + A2

 dt + C2 , ϕ(t)

(2.3.2.20)

If A4 = 0, the solution of the second equation in (2.3.2.19) is ϕ(t) = (A3 nt + C1 )−1/n . Substituting this into the second formula of (2.3.2.20) gives the function ψ(t). Let us give two equations of the form (2.3.2.18) and their solutions. 1◦ . Setting A1 = −1, A3 = 0, and u(z) = ez in the last two ODEs of (2.3.2.19), we get the equation with a logarithmic nonlinearity ut = u(n) x − A4 u ln u − A2 u. It has a solution of the form (2.3.2.4) with u(z) = ez . 2◦ . For odd n = 2m + 1 (m = 1, 2, . . .), we set A1 = (−1)m+1 , A3 = 0, and u(z) = sin z to obtain the equation p ut = ux(2m+1) − (A2 + A4 arcsin u) 1 − u2 .

It has a solution of the form (2.3.2.4) with u(z) = sin z.

◭

◮ Example 2.11. Let us look at the nth-order generalized Korteweg–de Vries

equation ut = u(n) x + f (u)ux .

(2.3.2.21)

104

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Seeking its solution in the form (2.3.2.4) leads to the following system of equations for ϕ(t), ψ(t), u(z), and f (u): −ψt′ +

ψ ′ ϕ = A1 ϕn + A2 ϕ, ϕ t

−

ϕ′t = A3 ϕn + A4 ϕ, ϕ

(n)

uz = −A1 − A3 z, u′z

(2.3.2.22)

f (u) = −A2 − A4 z.

As compared to (2.3.2.19), the first two equations and the last one have changed in system (2.3.2.22). The first two equations can be fully integrated, since the second equation is separable, while the first equation is linear in ψ. Below we consider the case A3 = 0, in which f (u) can be represented in explicit form. The general solution of the first two equations of (2.3.2.22) with A3 = 0 is expressed as ϕ(t) =

1 , A4 t + C1

ψ(t) =

A1 A2 t + C2 (A4 t + C1 )1−n − . A4 (n − 2) A4 t + C1

Let us give two equations of the form (2.3.2.21) with solutions. 1◦ . Setting A1 = −1, A3 = 0, and u(z) = ez in the last two ODEs of (2.3.2.22), we get an equation with a logarithmic nonlinearity: ut = u(n) x − (A4 ln u + A2 )ux . It has a solution of the form (2.3.2.4) with u(z) = ez . 2◦ . For odd n = 2m + 1 (m = 1, 2, . . .), we set A1 = (−1)m+1 , A3 = 0, and u(z) = sin z to obtain the equation ut = ux(2m+1) − (A2 + A4 arcsin u)ux . It has a generalized traveling wave solution (2.3.2.4) with u(z) = sin z.

◭

◮ Example 2.12. Consider the equation

∂ n+1 u = f (u). ∂xn ∂y

(2.3.2.23)

We look for a generalized traveling wave solution u = u(z),

z = ϕ(y)x + ψ(y).

(2.3.2.24)

Substituting (2.3.2.24) into (2.3.2.23), eliminating x using the expression of z, di(n+1) viding by uz , and rearranging the terms, we arrive at the functional differential equation in two variables  (n)  uz f (u) ϕn ψy′ − ϕn−1 ψϕ′y + ϕn−1 ϕ′y z + n (n+1) − (n+1) = 0. (2.3.2.25) uz uz It reduces to a three-term bilinear functional equation of the form (1.5.2.4), which has two solutions (1.5.2.5); see Section 1.5. Accordingly, we consider two cases.

105

2.3. Functional Separable Solutions of Special Form

1◦ . In the first case, we equate the expression in parentheses and the last fraction in (2.3.2.25) with constants. On rearranging, we obtain (z − C1 )u(n+1) + nu(n) = 0, z z

C2 u(n+1) − f (u) = 0, z

ϕn ψy′ − ϕn−1 ψϕ′y + C1 ϕn−1 ϕ′y − C2 = 0, where C1 and C2 are arbitrary constants. Setting C1 = 0, which correspond to a translation in z and renaming ψ, and integrating, we get u = A ln |z| + Bn−1 z n−1 + · · · + B1 z + B0 ,

f (u) = AC2 n! (−1)n z −n−1 , Z dy + C3 ϕ(y), ψ(y) = C2 ϕ(y) [ϕ(y)]n+1

(2.3.2.26)

where A, Bm , and C3 are arbitrary constants, while ϕ(y) is an arbitrary function. The first two formulas in (2.3.2.26) give a parametric representation of f (u). In the special case Bn−1 = · · · = B0 = 0, on eliminating z, we arrive at the exponential dependence f (u) = αeβu ,

α = AC2 n! (−1)n ,

β = −(n + 1)/A.

By virtue of (2.3.2.26), the corresponding exact solution to equation (2.3.2.23) with an exponential right-hand side will have a functional arbitrariness. 2◦ . In the second case, we equate in (2.3.2.25) the difference between the first two terms and the functional coefficient of the expression in parentheses with constants to obtain three ordinary differential equations ϕn−1 ϕ′y = C1 , (C1 z +

C2 )u(n+1) z

ϕn ψy′ − ϕn−1 ψϕ′y = C2 ,

+

C1 nu(n) z

(2.3.2.27)

− f (u) = 0,

where C1 and C2 are arbitrary constants. The solution of the first two equations is ϕ = (C1 nt + C3 )1/n ,

ψ = C4 (C1 nt + C3 )1/n −

C2 . C1

Together with the last ODE of (2.3.2.27), these formulas define a generalized traveling wave solution (2.3.2.24) to the nonlinear equation (2.3.2.23) with arbitrary ◭ f (u).

2.3.3. Construction of Other Functional Separable Solutions of Special Form Below we look at a specific example to illustrate the procedure for constructing more complicated functional separable solutions than generalized traveling wave solutions.

106

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

◮ Example 2.13. Consider the class of nonlinear heat equations with a source

ut = x1−n [xn−1 g(u)ux ]x + f (u).

(2.3.3.1)

To n = 2 in (2.3.3.1) there correspond two-dimensional problems with axial symmetry. To n = 3 there correspond spatial problems with radial symmetry. For n = 1, equation (2.3.3.1) coincides with (2.3.2.15). We will seek exact solutions to equation (2.3.3.1) with a quadratic dependence of the argument z on x: u = u(z),

z = ϕ(t)x2 + ψ(t).

(2.3.3.2)

Substituting this into (2.3.3.1) gives an equation that involves terms with x2 , but does not contain linear terms in x. Eliminating x2 from this equation using (2.3.3.2), we obtain   [g(u)u′z ]′z f (u) ϕ′t [g(u)u′z ]′z ψ ′ ′ z + 2ϕ ng + 2z + ′ = 0. −ψt + ϕt − − 4ϕψ ′ ϕ ϕ uz u′z uz (2.3.3.3) To solve this functional differential equation in two arguments, we apply the splitting method described in Section 1.5. To this end, we rewrite equation (2.3.3.3) in the bilinear form Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 + Φ5 Ψ5 = 0, (2.3.3.4) where ϕ′ ψ ′ ϕt , Φ2 = − t , Φ3 = 2ϕ, Φ4 = −4ϕψ, Φ5 = 1; ϕ ϕ [g(u)u′z ]′z f (u) [g(u)u′z ]′z , Ψ4 = , Ψ5 = . Ψ1 = 1, Ψ2 = z, Ψ3 = ng + 2z ′ uz u′z u′z (2.3.3.5) Nondegenerate cases. It can be verified by direct check that the bilinear equation (2.3.3.4) holds identically if we put Φ1 = −ψt′ +

Φ1 = −A1 Φ3 − A3 Φ4 − A5 Φ5 , Φ2 = −A2 Φ3 − A4 Φ4 − A6 Φ5 ; Ψ3 = A1 Ψ1 + A2 Ψ2 , Ψ4 = A3 Ψ1 + A4 Ψ2 , Ψ3 = A5 Ψ1 + A6 Ψ2 , (2.3.3.6) where Ai are arbitrary constants. Substituting (2.3.3.5) into (2.3.3.6) gives the system of ODEs ψ ′ ϕ′ ϕt = −2A1 ϕ + 4A3 ϕψ − A5 , − t = −2A2 ϕ + 4A4 ϕψ − A6 ; ϕ ϕ [g(u)u′z ]′z [g(u)u′z ]′z f (u) ng + 2z = A1 + A2 z, = A3 + A4 z, = A5 + A6 z. u′z u′z u′z (2.3.3.7) It follows from comparing the third and fourth equations of (2.3.3.7) that the function g is quadratic in z: − ψt′ +

g=−

2A4 2 1 A1 z + (A2 − 2A3 )z + . n n n

(2.3.3.8)

107

2.3. Functional Separable Solutions of Special Form

Integrating the fourth equation in (2.3.3.7) gives  Z Z dz A3 + A4 z dz + C2 , u = C1 exp g g

(2.3.3.9)

where C1 and C2 are arbitrary constants, while g is defined by (2.3.3.8). Formulas (2.3.3.8) and (2.3.3.9) define g = g(u) in parametric form. Taking into account formula (2.3.3.9), we find from the fifth equation of (2.3.3.7) that Z  A3 + A4 z A5 + A6 z exp dz . (2.3.3.10) f = C1 g g Formulas (2.3.3.9) and (2.3.3.10) define f = f (u) in parametric form. Consider a few examples of utilizing formulas (2.3.3.8)–(2.3.3.10). 1◦ . Setting A1 = n,

A2 = 2,

A3 = 1,

A4 = 0,

C1 = 1,

C2 = 0,

(2.3.3.11)

we get g = 1,

u = ez ,

z = ln u,

f = (A5 + A6 z)ez = A6 u ln u + A5 u,

which results in an equation of the form (2.3.3.1) with a logarithmic source: ut = uxx + (n − 1)x−1 ux + A6 u ln u + A5 u. The functions ϕ = ϕ(t) and ψ = ψ(t) in solution (2.3.3.2) are determined from the first two equations of system (2.3.3.7) with coefficients (2.3.3.11), which gives −1  4 , ϕ = C3 e−A6 t − A6    Z   Z Z  A5 exp 4 ϕ dt dt + C4 , ψ = ϕ exp −4 ϕ dt 2n + ϕ where C3 and C4 are arbitrary constants. 2◦ . Setting A1 = A3 = A4 = 0,

A2 = an,

A5 = cλ,

A6 = bλ,

C1 = a/λ,

C2 = 0,

we get 1 ln z, λ which leads to the equation g = az = aeλu ,

u=

z = eλu ,

f = (A5 + A6 z)

1 = b + ce−λu , λz

ut = ax1−n (xn−1 eλu ux )x + b + ce−λu . The functions ϕ = ϕ(t) and ψ = ψ(t) in solution (2.3.3.2) to this equation are given by −1  c 2an , ψ = − ϕ C3 e−bλt + 2ant + C4 ). ϕ = C3 e−bλt − bλ b

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

3◦ . Setting 2a , A2 = an + k a C1 = , C2 = 0, k

A3 =

A1 = A4 = 0, A6 = bk,

a , k

A5 = ck,

we get g = az = auk ,

u = z 1/k ,

z = uk ,

f = (A5 + A6 z)

1 (1−k)/k z = bu + cu1−k . k

This results in the equation ut = ax1−n (xn−1 uk ux )x + bu + cu1−k , which has a functional separable solution of the form (2.3.3.2) in which −1  4a(nk + 2) −bkt , ϕ(t) = C3 e − bk 2   Z    Z Z 4a dt 4a + C4 . ϕ(t) dt ck exp ϕ(t) dt ψ(t) = ϕ(t) exp − k k ϕ(t) 4◦ . Setting A1 = A2 = A3 = 0, A6 = −

2b , n+2

A4 = −

an , 2

A5 = −

1 C1 = − a(n + 2), 2

2c , n+2

(2.3.3.12)

C2 = 0,

we get g = az 2 = au f = cz −

n+4 2

4 − n+2

+

u = z−

,

n+2 bz − 2

n+2 2

= bu + cu

,

z=u

n+4 n+2

2 − n+2

,

.

This results in the equation ut = ax1−n xn−1 u

4 − n+2

ux




x

n+4

+ bu + cu n+2 ,

which admits a functional separable solution of the form (2.3.3.2), where ϕ = ϕ(t) and ψ = ψ(t) are described by the first two equations of system (2.3.3.7) with coefficients (2.3.3.12). Degenerate cases. Now consider two degenerate cases where one or more functions Φi vanish. 1◦ . For ψ = 0, which corresponds to Φ1 = Φ4 = 0, there remain only three terms in equation (2.3.3.4). It holds identically if we put Φ2 + A1 Φ3 + A2 Φ5 = 0,

Ψ3 = A1 Ψ2 ,

Ψ5 = A2 Ψ2 .

(2.3.3.13)

2.3. Functional Separable Solutions of Special Form

109

Substituting (2.3.3.5) into (2.3.3.13) yields the system of equations ϕ′t = 2A1 ϕ2 + A2 ϕ, [g(u)u′z ]′z ng + 2z = A1 z, u′z f (u) = A2 z. u′z The general solution to the first equation in (2.3.3.14) with A2 6= 0 is  −1 2A1 ϕ(t) = C1 e−A2 t − . A2

(2.3.3.14)

(2.3.3.15)

The last two equations of (2.3.3.14) involve three unknown functions, f = f (u), g = g(u), and u = u(z); therefore, one of them can be set arbitrarily. In what follows, we assume that the function u = u(z) is defined implicitly as z = Z(u), where Z(u) is any twice continuously differentiable function. Considering the relation u′z = 1/Zu′ , we find from the last two equations of (2.3.3.14) that A2 Z , Zu′

Z = Z(u),   Z A1 g(u) = Z −n/2 Zu′ C2 + Z n/2 du . 2

f (u) =

(2.3.3.16)

These relations define parametrically admissible forms of the functions f = f (u) and g = g(u), which enter equation (2.3.3.1). The solution of this equation admits an implicit representation −1  2A1 x2 = Z(u). C1 e−A2 t − A2 2◦ . For ϕ = 12 a = const, which corresponds to Φ2 = 0 and Φ3 = a = const, equation (2.3.3.3) simplifies significantly:   [g(u)u′z ]′z [g(u)u′z ]′z f (u) −ψt′ − 2aψ + a ng + 2z (2.3.3.17) + ′ = 0. ′ ′ uz uz uz It can be represented as a three-term bilinear equation (1.5.2.4), which holds identically if we put −ψt′ − 2aA1 ψ + A2 = 0, [g(u)u′z ]′z = A1 , (2.3.3.18) u′z   f (u) [g(u)u′z ]′z + ′ = A2 . a ng + 2z u′z uz The general solution to the first equation in (2.3.3.18) is expressed as ψ = C1 e−2aA1 t +

A2 . 2aA1

(2.3.3.19)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The analysis of the last two equations in (2.3.3.18) shows that the following statement holds true. Let two functions f (u) and g(u) be expressed in terms of one twice continuously differentiable function Z(u), which can be set arbitrarily, as follows: f (u) = [A2 − 2A1 aZ(u)]

1 Zu′ (u)

− an(A1 u + C2 ),

(2.3.3.20)

g(u) = (A1 u + C2 )Zu′ (u). Then, equation (2.3.3.1) has a solution which can be written in the implicit form 2 1 2 ax

+ C1 e−2aA1 t +

A2 = Z(u). 2aA1

◭

2.4. Method of Differentiation. Using Nonlinear Functional Equations 2.4.1. Brief Description of the Method of Differentiation Usually, the procedure for constructing functional separable solutions based on the method of differentiation consists of four stages. These are briefly outlined below. 1◦ . Expression (2.1.1.2) is substituted into the nonlinear partial differential equation in question. This results in a functional differential equation with three arguments (the first two, x and t, are simple and the third one, z, is composite). 2◦ . The three-argument functional differential equation is multiplied by suitable functions and differentiated with respect to x or/and t to eliminate either x or t and obtain a standard two-argument functional differential equation, which was discussed in Section 1.2. 3◦ . A solution to the two-argument functional differential equation is constructed using the splitting method (the formulas from Section 1.5 are used). 4◦ . The solution of Item 3◦ is substituted into the three-argument functional differential equation of Item 1◦ . This defines constraints for the constants of integration. The extra constants, which may arise from the differentiation in Item 2◦ , are eliminated and all unknowns are determined. 5◦ . Possible degenerate cases are treated separately, which may arise when the assumptions used in the solution are violated. Remark 2.6. The second stage is the most complicated. It cannot always be realized.

The above procedure for constructing functional separable solutions can, in practice, be implemented in the following way. On substituting u = U (z),

z = ϕ(x) + ψ(t)

111

2.4. Method of Differentiation. Using Nonlinear Functional Equations

into the nonlinear partial differential equation, one arrives at a functional differential equation of the form Φ1 (x)Ψ1 (t, z) + Φ2 (x)Ψ2 (t, z) + · · · + Φk (x)Ψk (t, z)

+ Ψk+1 (t, z) + Ψk+2 (t, z) + · · · + Ψn (t, z) = 0,

(2.4.1.1)

with the functionals Φj (x) and Ψj (t, z) dependent on the variables x and t, z, respectively:
′′ ′′ Φj (x) ≡ Φj x, ϕ, ϕ′x , ϕ′′xx ), Ψj (t, z) ≡ Ψj t, ψ, ψt′ , ψtt , U, Uz′ , Uzz . (2.4.1.2)

To be specific, it has been assumed that the original equation is of the second order. To solve equation (2.4.1.1), we use the method of differentiation. We divide equation (2.4.1.1) by Ψ1 and then differentiate with respect to t to arrive at a similar equation but with fewer terms containing the functions Φm . Specifically, we get (2)

(2)

(2)

Φ2 (x)Ψ2 (t, z) + · · · + Φk (x)Ψk (t, z) + Ψk+1 (t, z) + · · · + Ψ(2) n (t, z) = 0, (2.4.1.3) ′ where Ψ(2) = (Ψ /Ψ ) + ψ (Ψ /Ψ ) . Using the same approach several times, m 1 t m 1 z m t we eventually arrive at an equation independent explicitly of x: (k+1)

Ψk+1 (t, z) + · · · + Ψ(k+1) (t, z) = 0, n

(2.4.1.4)

(k)
(k)
where Ψ(k+1) = Ψ(k) + ψt′ Ψ(k) . m m /Ψk m /Ψk t z Equation (2.4.1.4) can be treated as an equation with two independent variables, P (k+1) (k+1) (k+1) t and z. If Ψm (t, z) = s Qm,s (t)Rm,s (z) for all m = k + 1, . . . , n, then the splitting method described in Section 1.5 can be used to solve equation (2.4.1.4). It is noteworthy that the independent variables x and t in formulas (2.4.1.1)– (2.4.1.4) can be swapped.

2.4.2. Examples of Constructing Functional Separable Solutions by the Method of Differentiation Below we give a few examples of using the method of differentiation to construct functional separable solutions to nonlinear diffusion and wave type equations. The focus is on the methodology of obtaining a standard bilinear functional equation and its use; therefore, some solutions may be omitted or not detailed. ◮ Example 2.14. Consider the nonlinear heat equation

ut = [f (u)ux ]x .

(2.4.2.1)

In general, equation (2.4.2.1) admits a traveling wave solution u = U (kx − λt) [85] as well as a self-similar solution u = U (xt−1/2 ) [229]. We seek functional separable solutions to equation (2.4.2.1) in the form u = u(z),

z = ϕ(x) + ψ(t).

(2.4.2.2)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting (2.4.2.2) into (2.4.2.1) and dividing by u′z , we obtain a functional differential equation with three variables ψt′ = ϕ′′xx f (u) + (ϕ′x )2 H(z),

(2.4.2.3)

u′′zz + fz′ (u), u′z

(2.4.2.4)

where H(z) = f (u)

u = u(z).

Differentiating (2.4.2.3) with respect to x gives ′ ′′ ′ ′ 3 ′ ϕ′′′ xxx f (u) + ϕx ϕxx [fz (u) + 2H(z)] + (ϕx ) Hz = 0.

(2.4.2.5)

This functional differential equation with two variables can be treated as the threeterm bilinear functional equation (1.5.2.4) from Section 1.5, which has two different solutions. Hence, we should consider two cases. Case 1. Solutions of the functional differential equation (2.4.2.5) are determined from the system of ordinary differential equations fz′ + 2H = 2A1 f,

Hz′ = A2 f,

′ ′′ ′ 3 ϕ′′′ xxx + 2A1 ϕx ϕxx + A2 (ϕx ) = 0,

(2.4.2.6)

where A1 and A2 are arbitrary constants. The first two equations of (2.4.2.6) are linear and independent of the third equation. Their general solution is  A1 z kz −kz ) if A21 > 2A2 ,  e (B1 e + B2 e A z 1 f = e (B1 + B2 z) k = |A21 − 2A2 |1/2 , if A21 = 2A2 ,  Az e 1 [B1 sin(kz) + B2 cos(kz)] if A21 < 2A2 , H = A1 f − 12 fz′ , (2.4.2.7) where B1 and B2 are constants of integration. Substituting the expression of H from (2.4.2.7) into (2.4.2.4), we obtain a differential equation for u = u(z). Integrating it yields Z u = C1 eA1 z |f (z)|−3/2 dz + C2 , (2.4.2.8) where C1 and C2 are arbitrary constants. Formula (2.4.2.7) for f in conjunction with expression (2.4.2.8) define f = f (u) in parametric form. Let us look at the case of A2 = 0 and A1 6= 0 (k = |A1 |). From formulas (2.4.2.7) and (2.4.2.8) we get f (z) = B1 e2A1 z + B2 , u(z) = C3 (B1 + B2 e

H = A1 B2 ,

−2A1 z −1/2

)

Eliminating z yields f (u) =

+ C2

(C1 = A1 B2 C3 ).

B2 C32 . − B1 u 2

C32

(2.4.2.9)

(2.4.2.10)

2.4. Method of Differentiation. Using Nonlinear Functional Equations

113

The first integral of the last equation in (2.4.2.6) with A2 = 0 is ϕ′′xx + A1 (ϕ′x )2 = const . Its general solution is given by   1 1 D2
ϕ(x) = − ln √ 2A1 D1 sinh2 A1 D2 x + D3   1 D2 1
√ ϕ(x) = − ln − 2A1 D1 cos2 A1 −D2 x + D3   1 D2 1
ϕ(x) = − ln − √ 2A1 D1 cosh2 A1 D2 x + D3

if

D1 > 0, D2 > 0;

if

D1 > 0, D2 < 0;

if

D1 < 0, D2 > 0,

(2.4.2.11) where D1 , D2 , and D3 are constants of integration. In all three cases, the following relations hold: (ϕ′x )2 = D1 e−2A1 ϕ + D2 ,

ϕ′′xx = −A1 D1 e−2A1 ϕ .

(2.4.2.12)

Substituting (2.4.2.9) and (2.4.2.12) into the original functional differential equation (2.4.2.3) and taking into account the form of z from (2.4.2.2), we obtain an equation for ψ = ψ(t): ψt′ = −A1 B1 D1 e2A1 ψ + A1 B2 D2 . Integrating gives the solution ψ(t) =

B2 D 2 1 ln , 2A1 D4 exp(−2A21 B2 D2 t) + B1 D1

(2.4.2.13)

where D4 is an arbitrary constant. Formulas (2.4.2.2), (2.4.2.9) (for u), (2.4.2.11), and (2.4.2.13) define three solutions of the nonlinear equation (2.4.2.1) with f (u) of the form (2.4.2.10) (recall that these solutions correspond to the special case A2 = 0 in (2.4.2.7) and (2.4.2.8)). Note that the above solutions were obtained in [86] by a different method. Case 2. Solutions to the functional differential equation (2.4.2.5) are determined from the system of ordinary differential equations ′ 3 ϕ′′′ ϕ′x ϕ′′xx = A2 (ϕ′x )3 , xxx = A1 (ϕx ) , ′ A1 f + A2 (fz + 2H) + Hz′ = 0.

(2.4.2.14)

The first two equations of (2.4.2.14) are compatible in two cases: A1 = A2 = 0 A1 = 2A22

=⇒ ϕ(x) = B1 x + B2 , 1 =⇒ ϕ(x) = − ln |B1 x + B2 |. A2

(2.4.2.15)

The first solution in (2.4.2.15) leads eventually to a traveling wave solution u = u(B1 x + B2 t) of equation (2.4.2.1), while the second solution leads to a self-similar solution of the form u = u e(x2/t). In these cases, the function f (u) in equation ◭ (2.4.2.1) is arbitrary.

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Remark 2.7. The more general nonlinear heat equation

ut = [f (u)ux ]x + g(u)

(2.4.2.16)

also has a solution of the form (2.4.2.2). For the unknown functions ϕ(x) and ψ(t), we arrive at the functional differential equation with three independent variables ψt′ = ϕ′′xx f (u) + (ϕ′x )2 H(z) + g(u)/u′z ,

with H(z) defined by (2.4.2.4). Differentiating with respect to x gives ′ ′′ ′ ′ 3 ′ ′ ′ ′ ϕ′′′ xxx f (u) + ϕx ϕxx [fz (u) + 2H(z)] + (ϕx ) Hz + ϕx [g(u)/uz ]z = 0.

This functional differential equation with two independent variables can be treated as the four′ ′′ term bilinear functional equation (1.5.2.6) from Section 1.5 with Φ1 = ϕ′′′ xxx , Φ2 = ϕx ϕxx , Φ3 = (ϕ′x )3 , and Φ4 = ϕ′x . The study [97] dealt with a more general equation than (2.4.2.16). ◮ Example 2.15. Consider the nonlinear Klein–Gordon equation

utt − uxx = f (u).

(2.4.2.17)

We seek exact solutions in the form u = u(z),

z = ϕ(x) + ψ(t).

On substituting (2.4.2.18) into (2.4.2.17), we get   ′′ ψtt − ϕ′′xx + (ψt′ )2 − (ϕ′x )2 g(z) = h(z),

where

g(z) = u′′zz /u′z ,


h(z) = f u(z) /u′z .

(2.4.2.18)

(2.4.2.19) (2.4.2.20)

On differentiating equation (2.4.2.19) first with respect to t and then with respect to x and on dividing by ψt′ ϕ′x , we obtain  ′′  ′′ = h′′zz . 2(ψtt − ϕ′′xx ) gz′ + (ψt′ )2 − (ϕ′x )2 gzz

′′ On eliminating the difference ψtt − ϕ′′xx from this equation using (2.4.2.19), we find that  ′ 2  ′′ (ψt ) − (ϕ′x )2 (gzz − 2ggz′ ) = h′′zz − 2gz′ h. (2.4.2.21)

This relation only holds in two cases: (i) (ii)

′′ gzz − 2ggz′ = 0,

h′′zz − 2gz′ h = 0;

(ψt′ )2 = Aψ + B, (ϕ′x )2 = −Aϕ + B − C, ′′ h′′zz − 2gz′ h = (Az + C)(gzz − 2ggz′ ),

(2.4.2.22)

where A, B, and C are arbitrary constants∗ . Let us look at these cases in order. ∗ In case (ii), the first two ODEs arise from solving the functional differential equation (ψ ′ )2 −(ϕ′ )2 = x t R(z), which reduces to equation (2.4.3.1).

2.4. Method of Differentiation. Using Nonlinear Functional Equations

115

Case 1. The first two equations in (2.4.2.22) allow us to find g(z) and h(z). Integrating the first equation gives gz′ = g 2 + const. Integrating further, we obtain g g g g g

= k, = −1/(z + C1 ), = −k tanh(kz + C1 ), = −k coth(kz + C1 ), = k tan(kz + C1 ),

(2.4.2.23) (2.4.2.24) (2.4.2.25) (2.4.2.26) (2.4.2.27)

where C1 and k are arbitrary constants. The second equation of (2.4.2.22) is linear in h and has a particular solution h = g(z). Hence, its general solution is expressed as follows [273]: Z dz h = C2 g(z) + C3 g(z) , (2.4.2.28) 2 g (z) where C2 and C3 are arbitrary constants. The functions u(z) and f (u) are determined from relations (2.4.2.20): Z  Z u(z) = B1 G(z) dz + B2 , f (u) = B1 h(z)G(z), G(z) = exp g(z) dz ,

(2.4.2.29) where B1 and B2 are arbitrary constants (the function f is defined in parametric form). Let us study case (2.4.2.24) in more detail. Using formula (2.4.2.28), we find that h = A1 (z + C1 )2 +

A2 , z + C1

(2.4.2.30)

where A1 = −C3 /3 and A2 = −C2 are arbitrary constants. Substituting (2.4.2.24) and (2.4.2.30) into (2.4.2.29) gives u = B1 ln |z + C1 | + B2 ,

f = A1 B1 (z + C1 ) +

A2 B1 . (z + C1 )2

Eliminating z from these relations, we find the right-hand side of equation (2.4.2.17) in explicit form: f (u) = A1 B1 ew + A2 B1 e−2w ,

where w =

u − B2 . B1

(2.4.2.31)

For clarity, we set C1 = 0, B1 = 1, and B2 = 0 and denote A1 = a and A2 = b. Then, we get h(z) = az 2 + b/z. (2.4.2.32) It remains to determine ψ(t) and ϕ(x). We substitute (2.4.2.32) into the functional differential equation (2.4.2.19), take into account (2.4.2.18), and rearrange to obtain u(z) = ln |z|,

f (u) = aeu + be−2u ,

g(z) = −1/z,

′′ ′′ [ψψtt −(ψt′ )2 −aψ 3 −b]−[ϕϕ′′xx −(ϕ′x )2 +aϕ3 ]+(ψtt −3aψ 2 )ϕ−ψ(ϕ′′xx +3aϕ2 ) = 0. (2.4.2.33)

116

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

By differentiating (2.4.2.33) with respect to t and x, we kill the terms in square brackets. As a result, we arrive at a separable equation∗ ′′′ ′ ′ (ψttt − 6aψψt′ )ϕ′x − (ϕ′′′ xxx + 6aϕϕx )ψt = 0,

whose solution is described by the autonomous ordinary differential equations ′′′ ψttt − 6aψψt′ = Aψt′ , ′′′ ϕxxx + 6aϕϕ′x = Aϕ′x ,

where A is the separation constant. Either equation can be integrated twice to obtain (ψt′ )2 = 2aψ 3 + Aψ 2 + C1 ψ + C2 , (ϕ′x )2 = −2aϕ3 + Aϕ2 + C3 ϕ + C4 ,

(2.4.2.34)

where C1 , . . . , C4 are arbitrary constants. Using (2.4.2.34) to eliminate the derivatives from equation (2.4.2.33), we find the following constraints for the constants: C3 = −C1 and C4 = C2 +b. Consequently, the functions ψ(t) and ϕ(x) are described by the first-order autonomous ODEs with a cubic nonlinearity (ψt′ )2 = 2aψ 3 + Aψ 2 + C1 ψ + C2 , (ϕ′x )2 = −2aϕ3 + Aϕ2 − C1 ϕ + C2 + b. The solutions of these equations are expressed in terms of elliptic functions. The other cases in (2.4.2.23)–(2.4.2.27) are investigated likewise. The results are summarized in Table 2.2 (these solutions were first obtained in [127] by a different method; see also [12, 385]). Case 2. Integrating the first two equations of (2.4.2.22) (second case), we obtain two solutions √ √ ψ = ± B t + D1 , ϕ = ± B − C t + D2 if A = 0; 1 B 1 B−C ψ= (At + D1 )2 − , ϕ = − (Ax + D2 )2 + if A 6= 0; 4A A 4A A (2.4.2.35) where D1 and D2 are arbitrary constants. In both cases, the function f (u) in equation (2.4.2.17) is arbitrary. The first solution (2.4.2.35) corresponds to a traveling wave solution u = u(kx + λt), while the second one leads to a solution of the form u = ◭ u(x2 − t2 ). ◮ Example 2.16. The nonlinear steady-state heat (diffusion) equation with a

source uxx + uyy = f (u) is investigated in exactly the same way as the nonlinear Klein–Gordon equation (see Example 2.15). The main results are summarized in Table 2.3 (these equations were obtained for the first time in [215] by a different method). For arbitrary f (u), there are also a traveling wave solution u = u(k1 x + k2 y) and a solution with radial ◭ symmetry of the form u = u(x2 + y 2 ). ∗ The easiest way to solve equation (2.4.2.33) is to use the results of solving the functional equation (1.5.2.6) from Section 1.5 (see formulas (1.5.2.7) and (1.5.2.8)).

2.4. Method of Differentiation. Using Nonlinear Functional Equations

117

Table 2.2. Nonlinear wave-type equations utt − uxx = f (u) that admit functional separable solutions of the form u = u(z), where z = ϕ(x) + ψ(t). No. Right-hand side of equation f (u) Solution u(z) 1

au ln u+bu

ez

2

aeu+be−2u

ln |z|

a sin u+b  u u 4 arctan ez × sin u ln tan +2 sin 4 4 a sinh u+b z   2 ln coth 4 × sinh u ln tanh u +2 sinh u 2 4 2 a sinh u+2b z   2 ln tan 5 × sinh u arctan eu/2+cosh u 2 2 3

Equations for ψ(t) and ϕ(x) (ψt′ )2 =C1e−2ψ+aψ− 12 a+b+A, (ϕ′x)2 =C2e−2ϕ−aϕ+ 12 a+A (ψt′ )2 =2aψ3+Aψ2+C1ψ+C2, (ϕ′x)2 =−2aϕ3+Aϕ2−C1ϕ+C2+b (ψt′ )2 =C1e2ψ+C2e−2ψ +bψ+a+A, (ϕ′x)2 =−C2e2ϕ−C1e−2ϕ−bϕ+A (ψt′ )2 =C1e2ψ+C2e−2ψ −σbψ+a+A, (ϕ′x)2 =C2e2ϕ+C1e−2ϕ+σbϕ+A (ψt′ )2 =C1 sin 2ψ+C2 cos 2ψ+σbψ+a+A, (ϕ′x)2 =−C1 sin 2ϕ+C2 cos 2ϕ−σbϕ+A

Notation: A, C1, C2 are arbitrary constants; σ=1 for z >0 and σ=−1 for z 0 and σ=−1 for z 0, A3 B1 = A4 /A3 , B2 = 0; p
2 ln sinh 12 −A3 A4 x + C1 + C2 if A1 = 12 A23 , A2 = 21 A3 A4 < 0, ϕ=− A3 B1 = −A4 /A3 , B2 = 0; p
2 ϕ=− ln cosh 12 −A3 A4 x + C1 + C2 if A1 = 12 A23 , A2 = 21 A3 A4 < 0, A3 B1 = A4 /A3 , B2 = 0,

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2.4. Method of Differentiation. Using Nonlinear Functional Equations

where C1 and C2 are arbitrary constants. These solutions correspond to a right-hand side of equation (2.4.3.10) defined in parametric form. Case 3. To the degenerate solutions (2.4.3.8) and (2.4.3.9) of the functional equation there correspond traveling wave solutions of the heat equation (2.4.3.10) with arbitrary f (u) and solutions to the linear equation (2.4.3.10) with the source ◭ f (u) = k1 u + k2 . Remark 2.8. One can seek more complicated functional separable solutions of equation (2.4.3.10) in the form

u = u(z),

z = ϕ(ξ) + ψ(t),

ξ = x + at.

Substituting these expressions into equation (2.4.3.10) also leads to the functional equation (2.4.3.3), in which x must be renamed ξ and the following formulas must be used: p(t) = −ψt′ , q(ξ) = ϕ′′ξξ − aϕ′ξ , h(ξ) = (ϕ′ξ )2 , R(z) = u′′zz /u′z , S(z) = f (u(z))/u′z .

The subsequent procedure for constructing the solution is the same as in Example 2.17. ◮ Example 2.18. The more general equation

ut = a(x)uxx + b(x)ux + f (u)

(2.4.3.14)

is treated in a similar manner. This equation arises in problems of convective heat and mass exchange (a = const and b = const), heat transfer problems in anisotropic media (b = a′x ), and spatial problems of heat conduction with axial or central symmetry (a = const and b = const/x). Searching for exact solutions to equation (2.4.3.14) in the form (2.4.3.11) leads to the functional equation (2.4.3.3) in which p(t) = −ψt′ ,

q(x) = a(x)ϕ′′xx + b(x)ϕ′x (x),

h(x) = a(x)(ϕ′x )2 ,

R(z) = u′′zz /u′z ,

S(z) = f (u(z))/u′z .

On substituting these expressions into (2.4.3.6)–(2.4.3.9), one can obtain systems of ◭ ordinary differential equations for the unknown quantities. Remark 2.9. In Examples 2.17 and 2.18, the construction of exact solutions to different equations of mathematical physics reduced to the same functional equation. This clearly demonstrates the usefulness of isolating and independently studying individual nonlinear functional equations with a composite argument.

3◦ . Now let us consider the functional equation p(t) + q(x)R(z) + h(x)S(z) = 0,

where z = ϕ(x) + ψ(t).

(2.4.3.15)

Differentiating (2.4.3.15) with respect to x results in a functional differential equation with two variables, x and z: qx′ R + qϕ′x Rz′ + h′x S + hϕ′x Sz′ = 0.

(2.4.3.16)

Up to obvious renaming, this equation can be rewritten as the bilinear equation (1.5.2.6).

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Case 1. A solution to equation (2.4.3.16) can be obtained using formulas (1.5.2.7) from Section 1.5. In doing so, we arrive at the system of ordinary differential equations qx′ = (A1 g + A2 h)ϕ′x , h′x = (A3 g + A4 h)ϕ′x , Rz′ = −A1 R − A3 S,

(2.4.3.17)

Sz′ = −A2 R − A4 S,

where A1 , . . . , A4 are arbitrary constants. The solution of system (2.4.3.17) is given by q(x) = A2 B1 ek1 ϕ + A2 B2 ek2 ϕ ,

h(x) = (k1 − A1 )B1 ek1 ϕ + (k2 − A1 )B2 ek2 ϕ ,

R(z) = A3 B3 e−k1 z + A3 B4 e−k2 z ,

(2.4.3.18)

S(z) = (k1 − A1 )B3 e−k1 z + (k2 − A1 )B4 e−k2 z , where B1 , . . . , B4 are arbitrary constants, while k1 and k2 are roots of the quadratic equation (k − A1 )(k − A4 ) − A2 A3 = 0. (2.4.3.19) If the roots are multiple, k1 = k2 , the terms ek2 ϕ and e−k2 z in (2.4.3.18) must be replaced with ϕek1 ϕ and ze−k1 z , respectively. In the case of pure imaginary or complex roots, one must separate the real (or imaginary) part in solution (2.4.3.18). On substituting (2.4.3.18) into the original functional equation (2.4.3.15), we obtain constraints for the free coefficients and find the function p(t): B2 = B4 = 0 B1 = B3 = 0

=⇒ p(t) = [A2 A3 + (k1 − A1 )2 ]B1 B3 e−k1 ψ ,

=⇒ p(t) = [A2 A3 + (k2 − A1 )2 ]B2 B4 e−k2 ψ ,

=⇒ p(t) = (A2 A3 + k12 )B1 B3 e−k1 ψ + (A2 A3 + k22 )B2 B4 e−k2 ψ . (2.4.3.20) Solutions (2.4.3.18) and (2.4.3.20) involve arbitrary functions ϕ = ϕ(x) and ψ = ψ(t). Case 2. The functional equation (2.4.3.15) also admits the degenerate solution A1 = 0

p = B1 B2 e A1 ψ ,

q = A2 B1 e−A1 ϕ ,

h = B1 e−A1 ϕ ,

S = −B2 eA1 z − A2 R,

where ϕ = ϕ(x), ψ = ψ(t), and R = R(z) are arbitrary functions, while A1 , A2 , B1 , and B2 are arbitrary constants. There is another degenerate solution, which is expressed as p = B1 B2 e A1 ψ ,

h = −B1 e−A1 ϕ − A2 q,

R = A2 B2 eA1 z ,

S = B2 e A1 z ,

where ϕ = ϕ(x), ψ = ψ(t), and q = q(x) are arbitrary functions, while A1 , A2 , B1 , and B2 are arbitrary constants. The degenerate solutions can be obtained from the original equation (2.4.3.15) and its corollary (2.4.3.16) using formulas (1.5.2.8) from Section 1.5.

2.4. Method of Differentiation. Using Nonlinear Functional Equations

123

◮ Example 2.19. For the nonlinear heat equation (2.4.2.1) (see Example 2.14 from Subsection 2.4.2), searching for exact solutions in the form u = u(z) with z = ϕ(x) + ψ(t) leads to the functional differential equation (2.4.2.3), which can be rewritten as the functional equation (2.4.3.15) if we set

p(t) = −ψt′ , q(x) = ϕ′′xx , h(x) = (ϕ′x )2 , R(z) = f (u), S(z) =

[f (u)u′z ]′z , u′z

where u = u(z).

◭

◮ Example 2.20. For the nonlinear first-order equation

ut = f (u)u2x + g(x), searching for exact solutions in the form (2.4.3.11) leads to the functional equation (2.4.3.15) with p(t) = −ψt′ , q(x) = (ϕ′x )2 , h(x) = g(x), R(z) = f (u)u′z , S(z) = 1/u′z , where u = u(z).

◭

4◦ . Let us look at the functional equation p1 (x) + p2 (t) + q1 (x)P (z) + q2 (t)R(z) + S(z) = 0,

(2.4.3.21)

where z = ϕ(x) + ψ(t). We differentiate (2.4.3.21) with respect to t, divide the resulting expression by ψt′ Pz′ , and differentiate again with respect to t. As a result, we arrive at a functional differential equation with two arguments, t and z, which can be reduced to a bilinear functional equation; see equation (1.5.1.1) and its solutions (1.5.2.9). ◮ Example 2.21. Consider the steady-state heat equation in an inhomogeneous anisotropic medium with a nonlinear source

[a(x)ux ]x + [b(y)uy ]y = f (u).

(2.4.3.22)

Searching for exact solutions to equation (2.4.3.22) in the form u = u(z) with z = ϕ(x) + ψ(y) results in the functional equation (2.4.3.21) with t = y and p1(x) = a(x)ϕ′′xx + a′x(x)ϕ′x , q2 (y) = b(y)(ψy′ )2 ,

′′ p2(y) = b(y)ψyy + b′y (y)ψy′ ,

P (z) = R(z) = u′′zz/u′z ,

q1(x) = a(x)(ϕ′x )2,

S(z) = −f (u)/u′z,

u = u(z).

Without carrying out a comprehensive analysis of equation (2.4.3.22), we restrict ourselves to studying functional separable solutions, which exist for arbitrary kinetic function f (u). On making the change of variable z = ζ 2 , we seek solutions to equation (2.4.3.22) in the form u = u(ζ), ζ 2 = ϕ(x) + ψ(y). (2.4.3.23)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Taking into account that ζx = 

(aϕ′x )′x +(bψy′ )′y

ψy′ ϕ′x and ζy = , from (2.4.3.22) we obtain 2ζ 2ζ

 u′ζ   ζu′′ζζ − u′ζ + a(ϕ′x )2 +b(ψy′ )2 = f (u), 2ζ 4ζ 3


f (u) = f u(ζ) .

(2.4.3.24) For this functional equation to be solvable, we require that the expressions in square brackets are functions of ζ: (aϕ′x )′x + (bψy′ )′y = M (ζ),

a(ϕ′x )2 + b(ψy′ )2 = N (ζ).

On differentiating the first of these relations with respect to x and then with respect to y, we arrive at the equation (Mζ′ /ζ)′ζ = 0, whose general solution is M (ζ) = C1 ζ 2 +C2 . Likewise, we find that N (ζ) = C3 ζ 2 +C4 . Here, C1 , . . . , C4 are arbitrary constants. As a result, we get (aϕ′x )′x + (bψy′ )′y = C1 (ϕ + ψ) + C2 ,

a(ϕ′x )2 + b(ψy′ )2 = C3 (ϕ + ψ) + C4 .

The separation of variables leads to the following system of ordinary differential equations for ϕ(x), a(x), ψ(y), and b(y): (aϕ′x )′x − C1 ϕ − C2 = k1 ,

a(ϕ′x )2 − C3 ϕ − C4 = k2 ,

(bψy′ )′y − C1 ψ = −k1 , b(ψy′ )2 − C3 ψ = −k2 .

This system is always integrable by quadrature and can be rewritten as (C3 ϕ + C4 + k2 )ϕ′′xx + (C1 ϕ + C2 + k1 − C3 )(ϕ′x )2 = 0, ′′ (C3 ψ − k2 )ψyy + (C1 ψ − k1 − C3 )(ψy′ )2 = 0;

(2.4.3.25)

a = (C3 ϕ + C4 + k2 )(ϕ′x )−2 , b = (C3 ψ − k2 )(ψy′ )−2 .

The equations for ϕ and ψ are independent of a and b and can be solved individually. Without carrying out a comprehensive analysis of system (2.4.3.25), we note one simple special case where it is integrable in explicit form. For C1 = C2 = C4 = k1 = k2 = 0 and C3 = C 6= 0, we have a(x) = αeµx ,

b(y) = βeνy ,

ϕ(x) =

Ce−µx , αµ2

ψ(y) =

Ce−νy , βν 2

where α, β, µ, and ν are arbitrary constants. On substituting these expressions into (2.4.3.24) and taking into account the form of ζ from (2.4.3.23), we obtain the following equation for u(ζ): u′′ζζ −

1 ′ 4 u = f (u). ζ ζ C

System (2.4.3.25) also has other solutions, which lead to different expressions of a(x) and b(y). Table 2.4 lists the cases where these functions are expressible in explicit form [272, 274, 376] (the traveling wave solution corresponding to a = const and b = const is omitted). In general, solving system (2.4.3.25) leads to a(x) and b(y) ◭ representable in parametric form [279].

2.5. Construction of Functional Separable Solutions in Implicit Form

125

Table 2.4. Functional separable solutions of the form u = u(ζ) with ζ 2 = ϕ(x) + ψ(y) for heat equations in an inhomogeneous anisotropic medium with an arbitrary nonlinear source. Heat equation

Functions ϕ(x) and ψ(y) Cx2−n , α(2 − n)2 2−k Cy ψ(y)= β(2 − k)2

Equation for u=u(ζ)

ϕ(x)= (αxn ux)x + (βy k uy )y =f (u)

C −µx e , αµ2 C −νy ψ(y)= e βν 2

u′′ ζζ +

4 − nk 1 ′ 4 u = f (u) (2 − n)(2 − k) ζ ζ C

ϕ(x)= (αe

µx

ux )x + (βe

νy

uy )y =f (u)

u′′ ζζ −

1 ′ 4 u = f (u) ζ ζ C

(αeµx ux )x + (βy k uy )y =f (u)

C −µx e , αµ2 Cy 2−k ψ(y)= β(2 − k)2

(αx2 ux )x + (βy 2 uy )y =f (u)

ϕ(x)=µ ln |x|, ψ(y)=ν ln |y|

Equation (2.4.3.24), both expressions in square brackets are constant

αwxx + (βy 2 uy )y =f (u)

ϕ(x)=µx, ψ(y)=ν ln |y|

Equation (2.4.3.24), both expressions in square brackets are constant

ϕ(x)=

u′′ ζζ +

k 1 ′ 4 u = f (u) 2−k ζ ζ C

Notation: C, α, β, µ, ν, n, and k are free parameters (C 6= 0, µ6= 0, ν 6= 0, n6= 2, and k6= 2).

Remark 2.10. The results described in Example 2.21 are also applicable to nonlinear hyperbolic type equations, where the variable y = t can be treated as time and the functional coefficients a(x) and b(y) have unlike signs, implying that a(x)b(y) < 0. In this case, the functions ϕ(x) and ψ(y) appearing in the first three equations in Table 2.4 also have unlike signs, so that ϕ(x)ψ(y) < 0. The sign of C is chosen to ensure that the condition ϕ(x) + ψ(y) ≥ 0 holds in the domain of interest. Remark 2.11. The equations and solutions described in Example 2.21 admit various multidimensional generalizations, which are discussed below in Section 2.6.5.

2.5. Construction of Functional Separable Solutions in Implicit Form 2.5.1. Preliminary Remarks. Traveling Wave Solutions in Implicit Form The method for constructing functional separable solutions in implicit form is based on a generalization of traveling wave solutions to different partial differential equations. Prior to describing the method, we first give four simple examples, which illustrate the existence of solutions defined in implicit form for heat/diffusion and

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

wave equations. ◮ Example 2.22. Let us look at the nonlinear heat equation

ut = [f (u)ux ]x ,

(2.5.1.1)

which involves an arbitrary function f (u). The equation is explicitly independent of x and t and has a traveling wave solution u = u(z),

z = λt + κx,

(2.5.1.2)

where κ and λ are arbitrary constants. On substituting (2.5.1.2) into (2.5.1.1), we arrive at the ODE λu′z = κ2 [f (u)u′z ]′z . Integrating gives its solution in implicit form Z f (u) du κ2 = λt + κx + C2 , (2.5.1.3) λu + C1 where C1 and C2 are arbitrary constants. On the right-hand side of (2.5.1.3), z has been replaced with the original variables using (2.5.1.2). It is apparent that even for very simple functions such as f (u) = u, f (u) = eu , f (u) = sin u, or f (u) = cos u, solution (2.5.1.3) of equation (2.5.1.1) cannot be represented explicitly in terms of elementary functions. Therefore, looking for exact solutions to more complex heat/diffusion type equations in explicit form seems to be ◭ of little prospect. ◮ Example 2.23. Now let us look at the generalized Burgers equation

ut = uxx − f (u)ux ,

(2.5.1.4)

where f (u) is an arbitrary function. It has a traveling wave solution of the form (2.5.1.2), which can be written in implicit form as Z Z du = λt + κx + C2 , F (u) = f (u) du. (2.5.1.5) κ2 κF (u) + λu + C1 ◭ ◮ Example 2.24. The nonlinear wave equation

utt = [f (u)ux ]x ,

(2.5.1.6)

where f (u) is an arbitrary function, admits a traveling wave solution of the form (2.5.1.2). Its solution can be represented in implicit form as Z [κ2 f (u) − λ2 ] du = C1 (λt + κx) + C2 . (2.5.1.7) ◭ ◮ Example 2.25. The nonlinear Klein–Gordon equation

utt = uxx + f (u),

(2.5.1.8)

where f (u) is an arbitrary function, also admits traveling wave solutions u = u(z), z = λt + κx (with λ 6= ±κ), which can be represented in the implicit form −1/2 Z  Z 2 C1 + 2 F (u) du = C ± (λt + κx), F (u) = f (u) du. (2.5.1.9) 2 λ − κ2 ◭

2.5. Construction of Functional Separable Solutions in Implicit Form

127

Examples 2.22, 2.23, 2.24, and 2.25 show that the nonlinear equations (2.5.1.1), (2.5.1.4), (2.5.1.6), and (2.5.1.8), which often arise in applications, have traveling wave solutions that can be written in implicit form. Importantly, in the general case of arbitrary f (u), these solutions cannot be represented explicitly. More complicated examples of nonlinear PDEs that have solutions in implicit form can be found, for example, in [275]. Remark 2.12. In practice, to construct exact solutions to nonlinear partial differential equations, researchers often use methods based on a priori setting the explicit form of solutions with free parameters (e.g., the tanh-method [91, 99, 201, 232, 233, 367, 384], the Exp-function method [27, 68, 92, 136, 137, 182, 327, 380, 384], the sine-cosine method [9, 25, 365, 366], and some other). The values of the parameters are found by the method of undetermined coefficients, often using computer algebra techniques. These and similar simple direct methods have a narrow range of applicability, since the form of solution is specified in advance (blindly) without taking into account the properties of the nonlinear equations under consideration. This is well illustrated by examples 2.22–2.25, where exact solutions cannot generally be represented in explicit form at all. Importantly, the overwhelming majority of known general solutions to nonlinear ordinary differential equations are written in implicit or parametric form; this follows from a statistical analysis of the most comprehensive handbooks on exact solutions to ODEs [273, 276]. The above simple examples, 2.22 to 2.25, let us suggest a plausible hypothesis that nonlinear partial differential equations admit exact solutions in implicit or parametric form more frequently than in explicit form. In particular, the most general nonlinear PDEs that involve arbitrary functions of the unknown quantity do not have nondegenerate solutions that can be represented in explicit form. Therefore, it is very important to develop direct methods for constructing exact solutions in implicit form for nonlinear PDEs; at present there are no effective direct methods for constructing exact solutions in parametric form for nonlinear PDEs.

Subsection 2.5.2 will describe a method for constructing exact solutions to nonlinear equations of mathematical physics that relies on a significant generalization of the traveling wave solutions considered in Examples 2.22 to 2.25.

2.5.2. Direct Method for Constructing Functional Separable Solutions in Implicit Form. The Splitting Principle We will deal with nonlinear partial differential equations G(x, ux , ut , uxx , uxt , utt , . . . ) = 0. We will seek their exact solutions in implicit form [252, 253, 298]: Z h(u) du = ξ(x)ω(t) + η(x),

(2.5.2.1)

(2.5.2.2)

where h = h(u), ξ = ξ(x), η = η(x), and ω = ω(t) are functions to be determined in the subsequent analysis. Remark 2.13. The representation of solutions in the implicit form (2.5.2.2) substantially generalizes traveling wave solutions (2.5.1.3), (2.5.1.5), (2.5.1.7), and (2.5.1.9) to equations (2.5.1.1), (2.5.1.4), (2.5.1.6), and (2.5.1.8) considered above. For example, the representation

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

of solutions in the form (2.5.2.2) is based on a generalization of solution (2.5.1.3) to equation (2.5.1.1), which can be displayed as follows: κ2 f (u) =⇒ h(u), λu + C1

λ =⇒ ξ(x),

t =⇒ ω(t),

κx + C2 =⇒ η(x).

We will now outline the procedure for constructing exact solutions in the implicit form (2.5.2.2). First, using (2.5.2.2), one calculates the partial derivatives ux , ut , uxx , . . . , which are expressed in terms of the functions h, ξ, η, ω and their derivatives. Then, the partial derivatives must be substituted into equation (2.5.2.1) followed by eliminating t with the help of (2.5.2.2). As a result (with a suitable choice of ω), one arrives at a bilinear functional differential equation N X

Φj [x]Ψj [u] = 0,

j=1

′′ ′′ Φj [x] ≡ Φj (x, ξ, η, ξx′ , ηx′ , ξxx , ηxx . . . ),

(2.5.2.3)

Ψj [u] ≡ Ψj (u, h, h′u , h′′uu , . . . ).

Here, Φj [x] and Ψj [u] are differential forms (in some cases, functional coefficients) that only depend on x and u, respectively. The following statement holds true. The splitting principle. Functional differential equations of the form (2.5.2.3) can have solutions only if the forms Ψj [u] (j = 1, . . . , N ) are connected by linear relations mi X kij Ψj [u] = 0, i = 1, . . . , n, (2.5.2.4) j=1

where kij are some constants, 1 ≤ mi ≤ N − 1, and 1 ≤ n ≤ N − 1. It is necessary to consider also degenerate cases when, in addition to the linear relations, some individual differential forms Ψj [u] vanish. The splitting principle is also true for the forms Φj [x]. A more detailed description of the splitting principle is given in Subsections 1.5.1 and 1.5.2. The splitting principle will be used in Subsections 2.5.3–2.5.5 to construct solutions for some functional differential equations of the form (2.5.2.3), which arise while seeking exact solutions to diffusion and wave type equations. Remark 2.14. Constructing a solution in implicit form with an integral term on the lefthand side of (2.5.2.2) often leads to lower-order differential equations for the function h than when exact solutions are sought in explicit form. In addition, the implicit form of representation of solutions usually leads to simpler explicit representations of the functions f and g via h (when searching for exact solutions in explicit form, f and g are often expressed in terms of u in parametric form [275]). Note also that, in the generic case, different linear relations of the form (2.5.2.4) correspond to different solutions of the PDE under consideration.

It is noteworthy that solutions of the form (2.5.2.2) cannot usually be obtained by applying the classical Lie group analysis of PDEs [34, 150, 225, 231].

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2.5. Construction of Functional Separable Solutions in Implicit Form

2.5.3. Nonlinear Reaction–Diffusion Equations with Variable Coefficients A brief overview of exact solutions to nonlinear diffusion type PDEs. Transformations, symmetries, and exact solutions of various classes of nonlinear reaction– diffusion–convection equations that do not depend explicitly on the variables x and t, have been considered in many studies (e.g., see [43, 60–62, 64, 65, 75, 85, 86, 97, 107, 109, 114, 141, 150, 165, 173, 174, 176, 183, 229, 272, 275, 278, 310, 315, 321, 376, 381] and the literature cited therein). The methods most frequently used to construct exact solutions include the classical and nonclassical methods of symmetry reductions [61, 62, 65, 75, 85, 109, 141, 150, 229, 275, 278], the methods of generalized and functional separation of variables [86, 97, 107, 114, 275, 278, 310, 381], and the method of differential constraints [107, 114, 165, 275, 278, 310]. A number of studies (e.g., see [44, 254, 272, 275, 310, 350, 351, 353, 354, 376]) were devoted to nonlinear reaction–diffusion equations with variable coefficients dependent on the spatial variable x (from now on sometimes called autonomous coefficients). Table 2.5 lists the forms of exact solutions to some equations of this type with one or two arbitrary functions. Table 2.5. Nonlinear reaction–diffusion equations with variable coefficients and their exact solutions. No. Equation 1

Form of solution or remark

ut = [a(x)unux]x +b(x)un+1 λu

λu

u = ϕ(x)ψ(t)

[272, 275, 376]

u = ϕ(x)+ψ(t)

[272, 275, 376]

2

ut = [a(x)e ux]x +b(x)e

3

ut = [a(x)ux]x +b(x)u+cu ln u u = ϕ(x)ψ(t) −4/3

−1/3

4

ut = (u

5

ut = [xnf (u)ux]x +xn−2g(u)

6 7

ux)x +b(x)u

2

ut = [x f (u)ux]x +g(u) λx

λx

ut = [e f (u)ux]x +e g(u) 2

8

ut = [a(x)ux]x+[x /a(x)]g(u)

9

ut = uxx +tanh2(kx)g(u)

References

Reduces to vt = (v

[272, 275] −4/3

vz )z

[272, 275, 376]

u = U (z), z = xt1/(n−2), k 6= 2

[252]

u = U (z), z = λt+ln x

[252]

u = U (z), z = λx+ln t, λ 6= 0 Z u = U (z), z = t+ [x/a(x)] dx

[252]

u = U (z), z = t+k−2 ln cosh(kx)

[254] [254]

Here, a(x), b(x), f (u), and g(u) are arbitrary functions; c, n, and λ are free parameters. Remark 2.15. The equations and solutions given in Table 2.5 in rows 5–7 generalize reaction–diffusion equations with power and exponential nonlinearities and their invariant solutions, which were considered in [350, 351, 353].

Related and more complicated nonlinear equations of diffusion type were studies in [24, 159, 183, 185, 236, 275, 382]. Exact solutions to a number of reaction– diffusion type equations can be found in the books [59, 275] (these also include relevant literature sources).

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The class of nonlinear reaction–diffusion equations in question. We will consider one-dimensional nonlinear equations of the reaction–diffusion type with variable coefficients c(x)ut = [a(x)f (u)ux ]x + b(x)g(u).

(2.5.3.1)

Note that if a(x) = b(x) = c(x) = xn , equation (2.5.3.1) describes reaction– diffusion processes with radial symmetry in two-dimensional (n = 1) and threedimensional (n = 2) cases, where x is the radial coordinate. The argument of the functions a = a(x), b = b(x), c = c(x), f = f (u), g = g(u), h = h(u), ξ = ξ(x), η = η(x), and ω = ω(t), which appear in equation (2.5.3.1) and solution (2.5.2.2), will often be omitted. Remark 2.16. The study [310] considered a nonlinear reaction–diffusion equation of the form (2.5.3.1) with a(x) = c(x) = 1 and used the implicit representation (2.5.2.2) with ξ(x) = 1 to look for its exact solutions.

In what follows, we will focus on the construction of exact solutions to nonlinear reaction–diffusion equations of a reasonably general form (2.5.3.1), which depend on one or two arbitrary functions. Importantly, exact solutions of mathematical physics equations that involve arbitrary functions and therefore have a significant generality are of great practical interest for assessing the accuracy of various numerical and approximate analytical methods for solving related initial-boundary value problems. Derivation of the functional differential equation. We look for exact solutions to the reaction–diffusion equation (2.5.3.1) in the implicit form (2.5.2.2). Differentiating (2.5.2.2) with respect to t and x, we get hut = ξωt′ =⇒ ut =

ξωt′ ; h

hux = ξx′ ω + ηx′ =⇒ ux =   f (af ux )x = (aξx′ ω + aηx′ ) h x =

[(aξx′ )′x ω

+

(aηx′ )′x ]

ξx′ ω + ηx′ ; h

1 f + a(ξx′ ω + ηx′ )2 h h

(2.5.3.2) 

f h

′

.

u

Substituting these expressions into (2.5.3.1) yields the functional differential equation (2.5.3.3) ωt′ = Q1 (x, u)ω 2 + Q2 (x, u)ω + Q3 (x, u), where the functions Qn are not explicitly dependent on t and are defined by  ′ a(ξx′ )2 f Q1 (x, u) = , cξ h u  ′   f 1 ′ ′ ′ ′ (2.5.3.4) Q2 (x, u) = (aξx )x f + 2aξx ηx , cξ h u   ′  1 f Q3 (x, u) = (aηx′ )′x f + a(ηx′ )2 + bgh . cξ h u

2.5. Construction of Functional Separable Solutions in Implicit Form

131

Equation (2.5.3.3)–(2.5.3.4) depends on three variables, t, x, and u, which are connected by one additional relation (2.5.2.2); it contains unknown functions (and their derivatives) that have different arguments. This equation is more complicated than equations of the form (2.5.2.3). The functional differential equation (2.5.3.3)–(2.5.3.4) significantly simplifies in two cases: (i) ξx′ = 0 and (ii) (f /h)′u = 0. These cases are discussed below in order (as in [252]). Case ξ(x) = 1: generalized traveling wave solutions for ω(t) = kt. For ξx′ = 0, without loss of generality, we can set ξ = 1. On substituting ξ = 1 into (2.5.3.4), we get Φ1 (x, u) = Φ2 (x, u) = 0. As a result, equation (2.5.3.3) reduces to ωt′ =

  ′  1 f + bgh . (aηx′ )′x f + a(ηx′ )2 c h u

(2.5.3.5)

The functional differential equation (2.5.3.5) is an equation with separated variables (the left-hand side only depends on t, while the right-hand side depends on x and u). Therefore, we can set ωt′ = k = const, which gives ω(t) = kt. Substituting this function and ξ(x) = 1 into (2.5.2.2) yields the following generalized traveling wave solution defined in implicit form: Z Z h(u) du = kt + θ(x) dx. (2.5.3.6) The functions h(u) and θ(x) = ηx′ (x) will be determined in the subsequent analysis from the functional differential equation (aθ)′x f

+ aθ

2



f h

′

u

+ bgh − kc = 0,

(2.5.3.7)

which results from substituting ω(t) = kt and θ(x) = ηx′ (x) into (2.5.3.5). Equation (2.5.3.7) can be represented in the four-term bilinear form (2.5.2.3): Φ1 Ψ1 + Φ2 Ψ2 + Φ3 Ψ3 + Φ4 Ψ4 = 0,

(2.5.3.8)

where Φ1 = (aθ)′x , Φ2 = aθ2 , Φ3 = b, Φ4 = kc, ′ Ψ1 = f, Ψ2 = (f /h)u , Ψ3 = gh, Ψ4 = −1.

(2.5.3.9)

Further, taking into account relations (2.5.3.8) and (2.5.3.9), we use the splitting principle to construct exact solutions of the functional differential equation (2.5.3.7). Solution 1. First, let us look at the degenerate case where the differential form (f /h)′u is zero. Then, the bilinear functional equation (2.5.3.8) with Ψ2 = 0 has a solution Ψ2 = 0,

Φ1 = −AΦ3 ,

Φ4 = BΦ3 ,

Ψ3 = AΨ1 − BΨ4 ,

(2.5.3.10)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where A and B are arbitrary constants. By inserting (2.5.3.9) into (2.5.3.10), we arrive at the equations h = f,

(aθ)′x + Ab = 0,

Bb − kc = 0,

g =A+

B . f

(2.5.3.11)

From (2.5.3.11) with b(x) = c(x) = 1, it follows that the equation ut = [a(x)f (u)ux ]x + A +

k , f (u)

(2.5.3.12)

which involves two arbitrary functions, a(x) and f (u), has the generalized traveling wave solution Z Z Z x dx dx f (u) du = kt − A + C1 + C2 , (2.5.3.13) a(x) a(x) where C1 and C2 are arbitrary constants. In the special case a(x) = 1, equation (2.5.3.12) and its solution (2.5.3.13) become the equation and solution obtained in [107]. Remark 2.17. It is easy to verify that the nonlinear PDE

ut = [a(x)f (u)ux ]x + b(x) +

k , f (u)

which involves three arbitrary functions, a(x), b(x), and f (u), and generalizes equation (2.5.3.12), has the exact solution in implicit form Z  Z Z Z 1 dx f (u) du = kt − b(x) dx dx + C1 + C2 . a(x) a(x) The more complex nonlinear equation ut = c(x)[a(x)f (u)ux ]x + b(x) +

k(t) , f (u)

(2.5.3.14)

which involves five arbitrary functions, a(x), b(x), c(x), k(t), and f (u), has the exact solution in implicit form Z  Z Z Z Z 1 b(x) dx f (u) du = k(t) dt − dx dx + C1 + C2 . a(x) c(x) a(x)

Solution 2. It is easy to verify that the bilinear equation (2.5.3.8) holds identically if we put Ψ1 = −AΨ4 ,

Ψ2 = Ψ3 ,

AΦ1 = Φ4 ,

Φ2 = −Φ3 ,

(2.5.3.15)

where A is an arbitrary constant. Substituting (2.5.3.9) into (2.5.3.15) yields  ′ 1 f f = A, g = , A(aθ)′x − kc = 0, b = −aθ2 , (2.5.3.16) h h u where A is an arbitrary constant.

2.5. Construction of Functional Separable Solutions in Implicit Form

133

Using relations (2.5.3.16) with c(x) = 1 and A = k = 1, one can obtain the nonlinear reaction–diffusion equation ut = [a(x)ux ]x −

x2 g(u), a(x)

(2.5.3.17)

where a(x) is an arbitrary function. The function g(u) is expressed in terms of the arbitrary function h = h(u) as g(u) = −h−3 h′u .

(2.5.3.18)

Equation (2.5.3.17) admits under condition (2.5.3.18) the exact solution Z Z x dx h(u) du = t + + C1 . (2.5.3.19) a(x) Solving relation (2.5.3.18) for h, we get two functions  Z −1/2 h(u) = ± 2 g(u) du + C2 . Eliminating h from (2.5.3.19) using this expression, we write the solutions to equation (2.5.3.17) in the form −1/2 Z  Z Z x dx ± 2 g(u) du + C2 du = t + + C1 . (2.5.3.20) a(x) Here, a(x) and g(u) are arbitrary functions, while C1 and C2 are arbitrary constants. ◮ Example 2.26. On substituting a(x) = xn into (2.5.3.17), we arrive at the

equation ut = (xn ux )x − x2−n g(u),

whose solutions are defined by formulas (2.5.3.20) with a(x) = xn .

◭

Note that equation (2.5.3.17) and its solutions were derived in [250] from other considerations. In what follows, we will only write out the final determining equations resulting from substituting expressions (2.5.3.9) into intermediate linear relations between Φi and Ψi similar to (2.5.3.10) and (2.5.3.15). Solution 3. Equation (2.5.3.7) can be satisfied by setting  ′ f f = A, g = − , Aaθ2 = kc, (aθ)′x = b, h u h

(2.5.3.21)

where A is an arbitrary constant. If we put c(x) = 1, we get two reaction–diffusion equations r 1 k a′x (x) p ut = [a(x)f (u)ux ]x ∓ (Au + B). (2.5.3.22) 2 A a(x)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The function f (u) is expressed in terms of the arbitrary function h = h(u) as f (u) = (Au + B)h(u).

(2.5.3.23)

Equation (2.5.3.22) with (2.5.3.23) admits the exact solutions Z

h(u) du = kt ±

r

k A

Z

dx p + C. a(x)

(2.5.3.24)

Formulas (2.5.3.23) and (2.5.3.24) involve arbitrary constants A, B, and C. Setting A = 4k and B = 0 in (2.5.3.22)–(2.5.3.24), we get the equations a′ (x) u, ut = [a(x)f (u)ux ]x ∓ k px a(x)

(2.5.3.25)

which involve two arbitrary functions, a(x) and f (u), as well as one arbitrary constant, k. Taking into account that h(u) = f (u)/(4ku), we can represent the exact solutions to these equations in the implicit form Z

f (u) du = 4k 2 t ± 2k u

where C1 = 4Ck is an arbitrary constant.

Z

dx p + C1 , a(x)

(2.5.3.26)

◮ Example 2.27. On substituting a(x) = x2β and k = ∓α/(2β) into (2.5.3.25)

and (2.5.3.26), we arrive at the equation

ut = [x2β f (u)ux ]x + αxβ−1 u,

β 6= 0,

which depends on the arbitrary function f (u) and admits the exact solutions in implicit form  2 α Z α t− x1−β + C1 if β 6= 1, f (u) 2 β(1 − β) du = β  2 u α t − α ln |x| + C1 if β = 1. ◭

◮ Example 2.28. On substituting a(x) = e2βx and k = ∓α/(2β) into (2.5.3.25)

and (2.5.3.26), we arrive at the equation

ut = [e2βx f (u)ux ]x + αeβx u,

β 6= 0,

which depends on the arbitrary functions f (u) and admits the exact solution in implicit form Z f (u) α α2 du = 2 t + 2 e−βx + C1 . u β β ◭

2.5. Construction of Functional Separable Solutions in Implicit Form

Solution 4. Equation (2.5.3.7) holds if we set  ′ k f , g = , (aθ)′x + Aaθ2 = 0, Af = h u h

b = c,

135

(2.5.3.27)

where A is an arbitrary constant. The general solution to equation (2.5.3.27) is  Z   Z −1  Z −1 k dx 1 g= A f du + B , h = f A f du + B A , θ= , + C1 f a a (2.5.3.28) where f = f (u) and a = a(x) are arbitrary functions, while B and C1 are arbitrary constants. On substituting c(x) = 1 and A = 1 into (2.5.3.27) and (2.5.3.28), we obtain the nonlinear reaction–diffusion equation Z  k ut = [a(x)f (u)ux ]x + f (u) du + B , (2.5.3.29) f (u) which has the exact solution Z Z kt f (u) du + B = C2 e

 dx + C1 . a(x)

(2.5.3.30)

Equation (2.5.3.29) and solution (2.5.3.30) involve two arbitrary functions, a(x) and f (u), and four arbitrary constants, B,R C1 , C2 , and k. Formula (2.5.3.30) was derived R 1 ln(A f du + B). using the relation h du = A It is noteworthy that solution (2.5.3.30) is degenerate in the sense that it nullifies the diffusion term [a(x)f (u)ux ]x of equation (2.5.3.29) (however, uxx 6≡ 0). ◮ Example 2.29. On setting f (u) = um , B = 0, and k = (m + 1)β (m 6= −1)

in (2.5.3.29) and (2.5.3.30), we obtain the reaction–diffusion equation with a powerlaw nonlinearity ut = [a(x)um ux ]x + βu, which has the exact solution  Z C¯1 u=e βt

dx + C¯2 a(x)



1 m+1

,

where C¯1 = C2 (m + 1) and C¯2 = C1 C2 (m + 1) are arbitrary constants.

◭

◮ Example 2.30. Setting f (u) = eλu , B = 0, and k = βλ, in (2.5.3.29) and

(2.5.3.30), we get the reaction–diffusion equation with an exponential nonlinearity ut = [a(x)eλu ux ]x + β, which has the exact solution

  Z 1 dx ¯ ¯ u = βt + ln C1 + C2 , λ a(x)

where C¯1 = C2 λ and C¯2 = C1 C2 λ are arbitrary constants.

◭

136

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 5. Equation (2.5.3.7) also has exact solutions provided that  ′ f ′ 2 (aθ)x = Ac, aθ = Bc, b = c, Af + B + gh − k = 0, (2.5.3.31) h u where A and B are arbitrary constants. 1◦ . On substituting a(x) = b(x) = c(x) = 1, θ(x) = κ, A = 0, B = κ2 , and k = λ into (2.5.3.31), we obtain a traveling wave solution of the form (2.5.1.2), which is not written out here. 2◦ . Setting c(x) = 1 and A = B = 1 in the first three equations of (2.5.3.31), we get (2.5.3.32) a(x) = x2 , b(x) = 1, θ(x) = 1/x. As a result, we arrive at the equation

where g(u) =

ut = [x2 f (u)ux ]x + g(u),

(2.5.3.33)

  k f (u) 1 d f (u) − − , h(u) h(u) h(u) du h(u)

(2.5.3.34)

which admits the exact solution in implicit form Z h(u) du = kt + ln x.

(2.5.3.35)

Notably, equation (2.5.3.33) with (2.5.3.34) involves two arbitrary functions, f = f (u) and h = h(u). Remark 2.18. The invariant solution (2.5.3.35) to equation (2.5.3.33) can be sought in the usual form u = U (z) with z = kt + ln x, in which case formula (2.5.3.34) that expresses g in terms of f and h is not used. The function U (z) satisfies the ODE

kUz′ = [f (U )Uz′ ]′z + f (U )Uz′ + g(U ). Remark 2.19. Some other generalized traveling wave solutions of the form (2.5.3.6) to equation (2.5.3.1) can be found in [252].

Case ξ(x) = 1. Functional separable solutions for ω(t) = keλt . Let us return to equation (2.5.3.5). The function ω(t) enters formula (2.5.2.2) linearly. If we put ω(t) = keλt (k is an arbitrary constant), the solution will become Z H(u) = keλt + η(x), H(u) = h(u) du, (2.5.3.36) and the function eλt can be eliminated from equation (2.5.3.5) using (2.5.3.36). As a result, we arrive at a functional differential equation of the form (2.5.2.3) with N = 5:  ′ (aηx′ )′x a(ηx′ )2 f b λη − λH + f+ + gh = 0. (2.5.3.37) c c h u c

2.5. Construction of Functional Separable Solutions in Implicit Form

137

Remark 2.20. Equation (2.5.3.37) can be derived from other considerations. Indeed, rewriting relation (2.5.2.2) as ξω/(H − η) = 1, (2.5.3.38)

multiplying the right-hand side of (2.5.3.5) by ξω/(H − η), and rearranging while taking into account that ξ = 1, we obtain   ′  1 f ωt′ = (aηx′ )′x f + a(ηx′ )2 + bgh . ω c(H − η) h u

(2.5.3.39)

The variables in equation (2.5.3.39) are separated: the left-hand side only depends on t and the right-hand side depends on x and u. On equating both sides of (2.5.3.39) with a constant λ, we obtain two equations. The left-hand side of (2.5.3.39) gives the equation ωt′ /ω = λ, whose solution is ω = keλt . The right-hand side of (2.5.3.39) leads to equation (2.5.3.37).

Solution 6. Equation (2.5.3.37) holds identically if we set f = C1 uh + C2 h, b = c,

g=λ

(aηx′ )′x = C3 c,

H − C1 C3 u − C2 C3 , h C1 a(ηx′ )2 + λcη = 0,

(2.5.3.40)

where C1 , C2 , and C3 are arbitrary constants. Relations (2.5.3.40) involve two arbitrary functions, h and c, while the other functions, f , g, a, b, and η, are expressed in terms of them. The general solution to the system of the last two equations in (2.5.3.40) is 2+λ/(C1 C3 )  Z C5 C3 c(x) dx + C4 , a(x) = c(x) −λ/(C1 C3 )  Z C1 η(x) = − C3 c(x) dx + C4 , C5 λ

(2.5.3.41)

where C4 and C5 are arbitrary constants. In particular, setting c(x) = 1, C1 = C3 = C5 = 1, C2 = C4 = 0, and λ = n − 2 in (2.5.3.41), we find that a(x) = xn , b(x) = 1, and η(x) = x2−n/(2 − n). In view of the first two relations of (2.5.3.40), we get the reaction–diffusion equation ut = [xn f (u)ux ]x + g(u), Z (n − 2) f (u) = uh(u), g(u) = h(u) du − u, h(u)

(2.5.3.42)

where h(u) is an arbitrary function and n 6= 2 is an arbitrary constant. It admits the functional separable solution in implicit form Z

h(u) du = ke(n−2)t +

where k is an arbitrary constant.

x2−n , 2−n

(2.5.3.43)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Taking into account that h = f /u, we can rewrite equation (2.5.3.42) in the explicit form Z (n − 2)u f (u) n ut = [x f (u)ux ]x − u + du. f (u) u Its solution is expressed as Z

x2−n f (u) du = ke(n−2)t + . u 2−n

Case ξ(x) = 1. Functional separable solutions for ω(t) = k ln t. On substituting ξ = 1 and ω(t) = k ln t into (2.5.2.2), we seek solutions in the form Z h(u) du = k ln t + η(x). (2.5.3.44) Eliminating t from (2.5.3.5) (for ω = k ln t) and (2.5.3.44), we obtain the functional differential equation  ′ Z ′ ′ ′ 2 f η/k −H/k (aηx )x f + a(ηx ) + bgh− kce e = 0, H = h(u) du. (2.5.3.45) h u Remark 2.21. Equation (2.5.3.45) can be derived from other considerations. Indeed, let us rewrite formula (2.5.2.2) with ξ = 1 in the form

e(H−η−ω)/k = 1,

where k is some constant, and then multiply the right-hand side of (2.5.3.5) by e(H−η−ω)/k . On rearranging, we get   ′  e(H−η)/k f eω/k ωt′ = (aηx′ )′x f + a(ηx′ )2 + bgh . (2.5.3.46) c h u The variables in equation (2.5.3.46) are separated: the left-hand side only depends on t, while the right-hand side depends on x and u. Equating both sides of (2.5.3.46) with a constant λ, we obtain two equations. The left-hand side of (2.5.3.46) gives the equation eω/k ωt′ = λ, whose solution is ω = k ln(t + t0 ) + k ln(λ/k). The right-hand side of (2.5.3.46) with λ = k leads to equation (2.5.3.45).

Solution 7. Equation (2.5.3.45) holds if we set (aηx′ )′x = Aceη/k , a(ηx′ )2 = Bceη/k , b = ceη/k ,  ′ (2.5.3.47) f Af + B + gh − k exp(−H/k) = 0, h u R where A and B are arbitrary constants and H = h(u) du. On substituting c(x) = 1, A = 1/k, and B = 1 into the first three equations of (2.5.3.47), we get a(x) = b(x) = eλx ,

η(x) = x,

λ=

1 . k

(2.5.3.48)

2.5. Construction of Functional Separable Solutions in Implicit Form

139

This results in the equation ut = [eλx f (u)ux ]x + eλx g(u),

(2.5.3.49)

where     Z 1 f (u) 1 d f (u) g(u) = . (2.5.3.50) exp −λ h(u) du − λ − λh(u) h(u) h(u) du h(u) The equation admits the exact solution in implicit form Z 1 h(u) du = x + ln t. λ

(2.5.3.51)

Note that equation (2.5.3.49) with (2.5.3.50) involves two arbitrary functions, f = f (u) and h = h(u). Remark 2.22. The invariant solution (2.5.3.51) to equation (2.5.3.49) can be sought in the usual form u = U (z) with z = x + (1/λ) ln t, in which case relation (2.5.3.50) is not used. The function U (z) satisfies the ODE

1 ′ Uz = [eλz f (U )Uz′ ]′z + eλz g(U ). λ

Solution 8. Setting c(x) = 1, A = (1 + k)/k, and B = 1 in the first three equations of (2.5.3.47), we get a(x) = xn ,

b(x) = xn−2 ,

η(x) = ln x,

n = 2+

1 . k

(2.5.3.52)

As a result, we obtain the reaction–diffusion type equation ut = [xn f (u)ux ]x + xn−2 g(u),

n 6= 2,

(2.5.3.53)

where     Z 1 f (u) 1 d f (u) . exp −(n−2) h(u) du −(n−1) − (n − 2)h(u) h(u) h(u) du h(u) (2.5.3.54) It admits the exact solution in implicit form Z 1 ln t. (2.5.3.55) h(u) du = ln x + n−2 g(u) =

Remark 2.23. The self-similar solution (2.5.3.55) of equation (2.5.3.53) can be sought in the usual form u = U (z) with z = xt1/(n−2) , in which case relation (2.5.3.54) is not required. The function U (z) is determined from the ODE

1 zUz′ = [z n f (U )Uz′ ]′z + z n−2 g(U ). n−2

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 9. Equation (2.5.3.45) holds if the relations  ′ f = Af, exp(−H/k) = Bf, gh = f, h u (aηx′ )′x

+

Aa(ηx′ )2

+ b − Bkce

η/k

(2.5.3.56)

=0

R hold. Here, A and B are arbitrary constants and H = h(u) du. For A = −1/k = λ and B = 1, the first three equations in (2.5.3.56) have the solutions f (u) = g(u) = eλu , h(u) = 1. (2.5.3.57) Then, we obtain the equation with an exponential nonlinearity c(x)ut = [a(x)eλu ux ]x + b(x)eλu ,

(2.5.3.58)

which admits the exact solution in explicit form u=−

1 ln t + η(x). λ

(2.5.3.59)

The function η = η(x) satisfies the ODE [a(x)eλη ηx′ ]′x + b(x)eλη +

1 c(x) = 0. λ

(2.5.3.60)

Equations (2.5.3.58) and (2.5.3.60) involve three arbitrary functions, a(x), b(x), and c(x). Note that equation (2.5.3.60) reduces to a linear ODE with the substitution ζ = eλη . Solution 10. Setting A = n+ 1, B = 1, and k = −1/n in the first three equations of (2.5.3.56), we get f (u) = un ,

g(u) = un+1 ,

h(u) = 1/u.

(2.5.3.61)

As a result, we arrive at the reaction–diffusion equation c(x)ut = [a(x)un ux ]x + b(x)un+1 ,

(2.5.3.62)

where a(x), b(x), and c(x) are arbitrary functions. It admits the exact solution ln u = −(1/n) ln t + η(x), which can be rewritten in the explicit form u = t−1/n ζ(x),

ζ(x) = eη(x) ,

(2.5.3.63)

where ζ is a function described by the ODE [a(x)ζ n ζx′ ]′x + b(x)ζ n+1 +

1 c(x)ζ = 0. n

(2.5.3.64)

Case h(u) = f (u). Generalized traveling wave solutions for ω(t) = t. If (f /h)′u = 0, one can set, without loss of generality, that h = f . On substituting h = f into (2.5.3.3)–(2.5.3.4), one obtains the equation ωt′ =

 (aξx′ )′x 1  ′ ′ (aηx )x f + bf g . fω + cξ cξ

(2.5.3.65)

2.5. Construction of Functional Separable Solutions in Implicit Form

141

Solution 11. In the degenerate case, which arises under the condition (aξx′ )′x = 0,

(2.5.3.66)

the variables in equation (2.5.3.65) separate, and hence, we can set ω(t) = t. As a result, we arrive at the functional differential equation (aηx′ )′x f + bf g − cξ = 0.

(2.5.3.67)

Integrating (2.5.3.66) gives the relation between a = a(x) and ξ = ξ(x): Z dx + C2 , (2.5.3.68) ξ = C1 a(x) where C1 and C2 are arbitrary constants. Equation (2.5.3.67) admits exact solutions if the conditions g = k1 + k2 f −1 ,

(aηx′ )′x + k1 b = 0,

k2 b − cξ = 0

(2.5.3.69)

hold, where k1 and k2 are arbitrary constants. It follows from relations (2.5.3.68) and (2.5.3.69) that the nonlinear reaction–diffusion equation   1 k1 c(x)ut = [a(x)f (u)ux ]x + c(x)ξ(x) k + , k= , (2.5.3.70) f (u) k2 where a(x), c(x), and f (u) are arbitrary functions, and ξ = ξ(x) is defined by (2.5.3.68), admits the generalized traveling wave solution in implicit form Z f (u) du = ξ(x)t + η(x), Z  Z Z (2.5.3.71) dx 1 + C4 . c(x)ξ(x) dx dx + C3 η(x) = −k a(x) a(x) Here, C3 and C4 are arbitrary constants. ◮ Example 2.31. We put a(x) = c(x) = 1 in equation (2.5.3.70) as well as C1 = 1 and C2 = 0 in formulas (2.5.3.68) and (2.5.3.71). This gives the equation   1 , (2.5.3.72) ut = [f (u)ux ]x + x k + f (u)

which involves an arbitrary function, f (u), and an arbitrary constant, k. It has the generalized traveling wave solution Z 1 f (u) du = xt − kx3 + C3 x + C4 . 6 In the special case f (u) = eu , equation (2.5.3.72) becomes ut = (eu ux )x + x(k + e−u ). Its solution is expressed explicitly as u = ln xt −

1 6


kx3 + C3 x + C4 .

◭

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 12. Substituting ω(t) = t into (2.5.2.2) and (2.5.3.65) and then eliminating t, we obtain the functional differential equation Z ξ(aηx′ )′x − η(aξx′ )′x + bξg − cξ 2 f −1 + (aξx′ )′x F = 0, F = f (u) du. (2.5.3.73) It has been taken into account that h = f . Equation (2.5.3.73) is a special case of equation (2.5.2.3) with N = 5. It admits exact solutions if, for example, the following conditions hold: Z −1 g = k1 + k2 f + k3 F, F = f (u) du, (2.5.3.74) ξ(aηx′ )′x − η(aξx′ )′x + k1 bξ = 0, k2 b − cξ = 0, (aξx′ )′x + k3 bξ = 0,

(2.5.3.75) (2.5.3.76) (2.5.3.77)

where f (u) is an arbitrary function, while k1 , k2 , and k3 are arbitrary constants. Assuming that a = a(x) and c = c(x) are given functions and eliminating b from (2.5.3.76) and (2.5.3.77), we arrive at an Emden–Fowler type equation for ξ: (aξx′ )′x +

k3 2 cξ = 0. k2

(2.5.3.78)

Equation (2.5.3.75) is a linear nonhomogeneous ODE in η, which has a particular solution ηp = −k1 /k3 ; after inserting this, equation (2.5.3.75) converts to (2.5.3.77). The truncated linear homogeneous equation (2.5.3.75) with k1 = 0 has the particular solution η0 = ξ. It follows that the order of this equation can be reduced [276]. Considering the aforesaid, we find the general solution to equation (2.5.3.75): Z dx k1 η = C1 ξ + C2 ξ − , (2.5.3.79) aξ 2 k3 where C1 and C2 are arbitrary constants. The functional coefficient b is determined from equation (2.5.3.76). Eventually, we arrive at the nonlinear reaction–diffusion type equation   Z 1 c(x)ut = [a(x)f (u)ux ]x + c(x)ξ(x) k1 + + k3 f (u) du , (2.5.3.80) f (u) where a(x), c(x), and f (u) are arbitrary functions and ξ = ξ(x) satisfies (2.5.3.78) with k2 = 1. Equation (2.5.3.80) has the generalized traveling wave solution in implicit form Z f (u) du = ξ(x)t + η(x),

with the function η(x) defined by (2.5.3.79). Note that (2.5.3.80) is a generalization of equation (2.5.3.70).

(2.5.3.81)

2.5. Construction of Functional Separable Solutions in Implicit Form

143

◮ Example 2.32. For a(x) = c(x) = k2 = 1, equation (2.5.3.78) has the exact solution ξ = −(6/k3 )x−2 . In this case, the function η, which is defined by formula (2.5.3.79), becomes η = A1 x−2 + A2 x3 − (k1 /k3 ), where A1 and A2 are arbitrary ◭ constants. These constants can be expressed in terms of C1 , C2 , and k3 .

Solution 13. It is easy to verify that the functional differential equation (2.5.3.73) also has solutions if the following conditions hold: g = k1 f −1 ,

F = k2 f −1 + k3 ,

(2.5.3.82)

where kn are arbitrary constants. Setting k1 = 1, k2 = 2, and k3 = 0 in (2.5.3.82), we find that f = u−1/2 , g = u1/2 , and F = 2u1/2 . The associated nonlinear reaction– diffusion type equation c(x)ut = [a(x)u−1/2 ux ]x + b(x)u1/2 ,

(2.5.3.83)

where a(x), b(x), and c(x) are arbitrary functions, has an exact solution that can be written explicitly as u = 14 [ξ(x)t + η(x)]2 . (2.5.3.84) The functions ξ = ξ(x) and η = η(x) are determined from the ordinary differential equations 2(aξx′ )′x + bξ − cξ 2 = 0, (2.5.3.85) ξ(aηx′ )′x − η(aξx′ )′x = 0. Suppose that ξ = ξ(x) solves the first equation in (2.5.3.85). Then the general solution to the second equation in (2.5.3.85) can be obtained by formula (2.5.3.79) with k1 = 0. ◮ Example 2.33. The first equation in (2.5.3.85) holds if we put ξ(x) = b(x) = k = const and c(x) = 1. Therefore, the equation

ut = [a(x)u−1/2 ux ]x + ku1/2 , which involves an arbitrary function, a(x), has the exact solution  2 Z dx 1 kt + C1 + C2 . u= 4 a(x)

◭

Remark 2.24. The more general equation

c(x)ut = [a(x)u−1/2 ux ]x + b(x)u1/2 + p(x),

(2.5.3.86)

than (2.5.3.83), which involves four arbitrary functions, a(x), b(x), c(x), and p(x), has an exact solution of the form (2.5.3.84). Equation (2.5.3.86) belongs to the class of equations in question, (2.5.3.1), when b(x) = 0 or p(x)/b(x) = const.

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Case h(u) = f (u). Functional separable solutions for ω(t) = eλt . We substitute ω(t) = eλt into (2.5.2.2) and (2.5.3.65) and then eliminate t to obtain the functional differential equation   Z λ (aξx′ )′x b (aηx′ )′x 1 = + + g , F = f du. (2.5.3.87) f cξ c c F −η Equation (2.5.3.87) holds identically if we set   λ ′ ′ + k1 (F + k2 ), (aξx )x = −k1 cξ, η = −k2 , b = c, g = f

(2.5.3.88)

where k1 and k2 are arbitrary constants. Solution 14. It follows from relations (2.5.3.88) that the nonlinear reaction– diffusion equation Z   λ (2.5.3.89) + k1 f (u) du + k2 , c(x)ut = [a(x)f (u)ux ]x + c(x) f (u) where a(x), c(x), f (u) are arbitrary functions and k1 , k2 , λ are arbitrary constants, admits the functional separable solution in implicit form Z f (u) du = ξ(x)eλt − k2 . (2.5.3.90) The function ξ = ξ(x) is found by solving the linear ordinary differential equation [a(x)ξx′ ]′x + k1 c(x)ξ = 0. In the degenerate case k1 = 0, the function ξ = ξ(x) is defined by (2.5.3.68). ◮ Example 2.34. Setting a(x) = c(x) = 1 in (2.5.3.89) and (2.5.3.90), we arrive

at the equation ut = [f (u)ux ]x +



λ + k1 f (u)

Z

 f (u) du + k2 ,

whose exact solutions are given by  λt  Z e [C1 cos(βx) + C2 sin(βx)] − k2 f (u) du = eλt (C1 e−βx + C2 eβx ) − k2   λt e (C1 + C2 x) − k2

(2.5.3.91)

if k1 = β 2 > 0, if k1 = −β 2 < 0, if k1 = 0. β

◭

Case h(u) = f (u). Functional separable solutions for ω(t) = t . We substitute ω(t) = t1/(1−n) into (2.5.2.2) and (2.5.3.65) and then eliminate t to obtain the functional differential equation   1 b (aξx′ )′x (aηx′ )′x 1 1 + + g , (2.5.3.92) = (n − 1)f cξ 2−n (F − η)n−1 cξ 1−n cξ 1−n (F − η)n

145

2.5. Construction of Functional Separable Solutions in Implicit Form

R where F = f du. Equation (2.5.3.92) holds identically if we set (aξx′ )′x = −k1 cξ 2−n ,

η = −k2 ,

b = cξ 1−n ,

g = k1 (F + k2 ) +

(F + k2 )n , (n − 1)f (2.5.3.93)

where k1 and k2 are arbitrary constants. Solution 15. It follows from relations (2.5.3.93) that the nonlinear reaction– diffusion equation   [F (u) + k2 ]n 1−n k1 [F (u)+k2 ]+ c(x)ut = [a(x)f (u)ux ]x +c(x)ξ , (2.5.3.94) (n − 1)f (u) where a(x), Rc(x), f (u) are arbitrary functions, k1 , k2 , n, λ are arbitrary constants, and F (u) = f (u) du, admits the functional separable solution in implicit form Z f (u) du = ξ(x)t1/(1−n) − k2 . (2.5.3.95) The function ξ = ξ(x) in (2.5.3.94) and (2.5.3.95) is described by the nonlinear ordinary differential equation [a(x)ξx′ ]′x + k1 c(x)ξ 2−n = 0.

(2.5.3.96)

Note that for n = 2, the general solution of equation (2.5.3.96) is expressed as Z  Z Z dx 1 c(x) dx dx + C1 + C2 , (2.5.3.97) ξ = −k1 a(x) a(x) where C1 and C2 are arbitrary constants. ◮ Example 2.35. On substituting a(x) = c(x) = 1, k1 = 0, C1 = 1, and C2 = 0 into (2.5.3.94)–(2.5.3.96), we get the equation Z n 1−n [F (u) + k2 ] ut = [f (u)ux ]x + x , F (u) = f (u) du, (2.5.3.98) (n − 1)f (u) R which admits the exact solution in implicit form f (u) du = xt1/(1−n) − k2 . This is a noninvariant self-similar solution which nullifies the diffusion term [f (u)ux ]x of ◭ equation (2.5.3.98); however, uxx 6= 0.

2.5.4. Nonlinear Convection–Diffusion Equations with Variable Coefficients The class of nonlinear convection–diffusion equations in question. We will deal with one-dimensional nonlinear convection–diffusion type equations with variable coefficients of the form c(x)ut = [a(x)f (u)ux ]x + b(x)g(u)ux .

(2.5.4.1)

Some exact solutions to nonlinear convection–diffusion equations with variable coefficients that belong to the class of equations (2.5.4.1) were obtained in [115, 154, 155, 249, 304, 352].

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Remark 2.25. The convection–diffusion equation (2.5.4.1) with a(x) = 1, b(x) = −1, and c(x) = 1 is also known as Richards’ equation. It is used to model the seepage of water through unsaturated soils. In thisR equation, u is the volumetric water content, f = f (u) is the soil-water diffusivity, and K = g(u) du is the hydraulic conductivity. For exact solutions to Richards’ equation, see, for example, [134, 237, 338, 370].

In the sequel, our main focus will be on constructing exact solutions to nonlinear convection–diffusion equations of the reasonably general form (2.5.4.1) that involve one or two arbitrary functions. In addition, a number of exact solutions to equation (2.5.4.1) with f (u) = 1 are described in Subsection 2.6.3. Derivation of the functional differential equation. We will seek exact solutions to the convection–diffusion equations (2.5.4.1) in the implicit form (2.5.2.2) with ξ(x) = 1. On substituting the expressions of the derivatives (2.5.3.2) into (2.5.4.1), we obtain the functional differential equation ωt′

   ′ 1 ′ ′ ′ ′ 2 f = (aηx )x f + a(ηx ) + bηx g . c h u

(2.5.4.2)

Here and henceforth, the arguments of the functions a = a(x), b = b(x), c = c(x), f = f (u), g = g(u), h = h(u), η = η(x), and ω = ω(t), which appear in equation (2.5.4.1) and solution (2.5.2.2) with ξ(x) = 1, will often be omitted. Equation (2.5.4.2), which depends on x, t, and u, can be reduced with the differentiation with respect to x to a bilinear functional differential equation of the form (2.5.2.3) with N = 6. Then, the solution method described in Section 1.5 can be applied. However, this approach is technically quite difficult to implement, since differentiation increases the order of the derivatives in the reduced equation. Furthermore, at the final stage, one still has to return to the analysis of equation (2.5.4.2) to determine ω(t). In the current subsection, to solve equation (2.5.4.2), we will employ the direct method relying on the splitting principle (see Subsection 2.5.2) and using the functions ω(t) = kt, ω(t) = keλt , and ω(t) = k ln t, which were obtained in [249] for equations (2.5.4.1) of special form, with f (u) = 1 and arbitrary g(u). Generalized traveling wave solutions for ω(t) = kt. The variables in the functional differential equation (2.5.4.2) are separated: the left-hand side depends on t alone and the right-hand side depends on x and u. Therefore, we can set ωt′ = k = const, which gives ω(t) = kt. This situation corresponds to generalized traveling wave solutions specified in implicit form: Z Z h(u) du = kt + θ(x) dx. (2.5.4.3) The integrands h(u) and θ(x) = ηx′ (x) will be determined in the subsequent analysis from the functional differential equation (aθ)′x f + aθ2



f h

′

u

+ bθg − kc = 0,

(2.5.4.4)

147

2.5. Construction of Functional Separable Solutions in Implicit Form

which results from substituting ω(t) = kt and θ(x) = ηx′ (x) into (2.5.4.2). Equation (2.5.4.4) is a bilinear functional differential equation of the form (2.5.2.3) with N = 4. Solution 1. We first consider the degenerate case where the differential form (f /h)′u in (2.5.4.4) vanishes. In this case, equation (2.5.4.4) has solutions if the following relations hold: h = f,

(aθ)′x + Bbθ = 0,

g = A + Bf,

Abθ − kc = 0,

(2.5.4.5)

where A and B are arbitrary constants. Setting A = k in (2.5.4.5), we arrive at the equation c(x)ut = [a(x)f (u)ux ]x +

c(x) [k + Bf (u)]ux , θ(x)

which, for arbitrary functions a(x), c(x), and f (u) and Z B C1 θ(x) = − , c(x) dx − a(x) a(x) has the generalized traveling wave solution Z  Z Z Z 1 dx + C2 . f (u) du = kt − B c(x) dx dx − C1 a(x) a(x)

(2.5.4.6)

(2.5.4.7)

(2.5.4.8)

Here, C1 and C2 are arbitrary constants. ◮ Example 2.36. Setting c(x) = 1, B = 1, and C1 = 0 in (2.5.4.6)–(2.5.4.8) and then replacing a(x) with xa(x), we obtain the equation

ut = [xa(x)f (u)ux ]x − a(x)[k + f (u)]ux . It admits the generalized traveling wave solution in implicit form Z Z dx f (u) du = kt − + C2 , a(x) where a(x) and f (u) are arbitrary functions; k and C2 are arbitrary constants.

◭

Solution 2. Equation (2.5.4.4) holds identically if f = A,

g=



f h

′

u

,

A(aθ)′x − kc = 0,

b = −aθ,

(2.5.4.9)

where A is an arbitrary constant. Using (2.5.4.9) with c(x) = 1, A = 1, and k = 1, we get the nonlinear convection– diffusion equation ut = [a(x)ux ]x − xg(u)ux , (2.5.4.10)

148

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where a(x) is an arbitrary function and g(u) is expressed in terms of the arbitrary function h = h(u) as g(u) = −h−2 h′u . (2.5.4.11) Under condition (2.5.4.11), equation (2.5.4.10) has the exact solution Z Z x dx h(u) du = t + + C1 . a(x)

(2.5.4.12)

On solving (2.5.4.11) for h, we get Z −1 h(u) = g(u) du + C2 . Eliminating h from (2.5.4.12) using this expression, we arrive at the following solution to equation (2.5.4.10) in implicit form: −1 Z Z Z x dx g(u) du + C2 du = t + + C1 . (2.5.4.13) a(x) Here, a(x) and g(u) are arbitrary functions, while C1 and C2 are arbitrary constants. ◮ Example 2.37. An exact solution to the equation

ut = (xn ux )x − xg(u)ux is defined by (2.5.4.13) with a(x) = xn .

◭

Remark 2.26. Equation (2.5.4.10) and its solution were obtained in [249] in a different

way.

Solution 3. Equation (2.5.4.4) can be satisfied if we put  ′ f = A, g = f, (aθ)′x + bθ = 0, Aaθ2 − kc = 0, h u

(2.5.4.14)

where A is an arbitrary constant. The first relation in (2.5.4.14) gives h(u) = f (u)/(Au + C1 ).

(2.5.4.15)

Assuming that c(x) = 1, A = 1, and C1 = 0 in (2.5.4.14) and (2.5.4.15), we obtain the nonlinear convection–diffusion equation ut = [a(x)f (u)ux ]x −

1 ′ 2 ax (x)f (u)ux .

It has two generalized traveling wave solutions in implicit form Z √ Z f (u) dx du = kt ± k p + C2 . u a(x)

(2.5.4.16)

(2.5.4.17)

Equation (2.5.4.16) and formulas (2.5.4.17) involve two arbitrary functions, a(x) and f (u), as well as two arbitrary constants, k and C2 .

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2.5. Construction of Functional Separable Solutions in Implicit Form

Solution 4. Equation (2.5.4.4) holds identically if  ′ f Af = , g = 1, (aθ)′x + Aaθ2 = 0, h u

bθ = kc,

(2.5.4.18)

where A is an arbitrary constant. It follows from equations (2.5.4.18) that  Z −1  Z  1 d h(u) = f (u) A f (u) du + C1 = ln A f (u) du + C1 , A du Z −1 1 kc(x) dx θ(x) = + C2 , b(x) = , Aa(x) a(x) θ(x) where C1 and C2 are arbitrary constants. Setting c(x) = 1 and A = 1, we arrive at the nonlinear convection–diffusion equation  Z dx + C2 ux , (2.5.4.19) ut = [a(x)f (u)ux ]x + ka(x) a(x) which admits the exact solution Z Z kt f (u) du + C1 = e

 dx + C2 . a(x)

(2.5.4.20)

Equation (2.5.4.19) and formula (2.5.4.20) involve two arbitrary functions, a(x) and f (u). Solution (2.5.4.20) is degenerate in the sense that it nullifies the diffusion term [a(x)f (u)ux ]x of equation (2.5.4.19); however, uxx 6≡ 0. Remark 2.27. The right–hand side of solution (2.5.4.20) can additionally be multiplied by a new arbitrary constant, C3 .

◮ Example 2.38. On substituting a(x) = 1 and C1 = C2 = 0 into (2.5.4.19) and (2.5.4.20), we arrive at the equation [275]

R

ut = [f (u)ux ]x + kxux ,

which has the solution f (u) du = xekt .

◭

◮ Example 2.39. Setting a(x) = ex , C1 = 0, and C2 = 1 in (2.5.4.19) and (2.5.4.20), we obtain the convection–diffusion equation

ut = [ex f (u)ux ]x − kux . It has the noninvariant traveling wave solution in implicit form Z f (u) du = −ekt−x .

◭

Solution 5. Equation (2.5.4.4) also admits solutions if  ′ f (aθ)′x = Ac, aθ2 = Bc, bθ = c, Af + B + g − k = 0, (2.5.4.21) h u where A and B are arbitrary constants.

150

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

1◦ . On substituting a(x) = b(x) = c(x) = θ(x) = 1, A = 0, and B = 1 into (2.5.4.21), we obtain a traveling wave solution (2.5.1.2), which is omitted here. 2◦ . Setting c(x) = 1 and A = B = 1 in the first three equations of (2.5.4.21), we get a(x) = x2 , b(x) = x, θ(x) = 1/x. (2.5.4.22) As a result, we arrive at the equation

where

ut = [x2 f (u)ux ]x + xg(u)ux ,

(2.5.4.23)

  d f (u) . g(u) = k − f (u) − du h(u)

(2.5.4.24)

It admits the solution in implicit form Z h(u) du = kt + ln x.

(2.5.4.25)

Note that equation (2.5.4.23) with (2.5.4.24) involves two arbitrary functions, f = f (u) and h = h(u). The function h(u) can be expressed from (2.5.4.24) in terms of f (u) and g(u). Remark 2.28. The invariant solution (2.5.4.25) of equation (2.5.4.23) can be sought in the explicit form u = U (z) with z = kt + ln x, in which case relation (2.5.4.24) for f , g , and h is not required. The function U (z) is determined from the autonomous ODE

[f (U )Uz′ ]′z + [f (U ) + g(U ) − k]Uz′ = 0,

which is easy to integrate.

Functional separable solutions for ω(t) = keλt . Let us return to equation (2.5.4.2). We considered above the simplest case of linear dependence, ω(t) = kt, which led immediately to a functional differential equation with two variables of the form (2.5.4.4). The function ω(t) enters formula (2.5.2.2) in a linear fashion. If we choose ω(t) = keλt (k is an arbitrary constant), then solution (2.5.2.2) with ξ(x) = 1 becomes Z λt H(u) = ke + η(x), H(u) = h(u) du. (2.5.4.26) The exponential eλt can be eliminated from equation (2.5.4.2) using (2.5.4.26). As a result, we arrive at a functional differential equation of the form (2.5.2.3) with N = 5:  ′ (aηx′ )′x a(ηx′ )2 f bη ′ λη − λH + f+ (2.5.4.27) + x g = 0. c c h u c Remark 2.29. Equation (2.5.4.27) can be derived from other considerations. Indeed, on rewriting (2.5.2.2) with ξ(x) = 1 as

ω/(H − η) = 1,

(2.5.4.28)

2.5. Construction of Functional Separable Solutions in Implicit Form

151

we multiply the right-hand side of equation (2.5.4.2) by ω/(H −η). On rearranging, we obtain   ′  ωt′ 1 f ′ ′ ′ 2 ′ = (aηx )x f + a(ηx ) + bηx g . (2.5.4.29) ω c(H − η) h u The variables in equation (2.5.4.29) are separated: the left-hand side depends on t alone, while the right-hand side depends on x and u. Equating both sides of (2.5.4.29) with a constant λ, we get two equations. The left-hand side of (2.5.4.29) gives the equation ωt′ /ω = λ, whose solution is ω = keλt . The right-hand side of (2.5.4.29) leads to equation (2.5.4.27).

Solution 6. Equation (2.5.4.27) can be satisfied if we put f = C1 uh + C2 h, bηx′

= c,

(aηx′ )′x

g = λH − C1 C3 uh − C2 C3 h,

= C3 c,

C1 a(ηx′ )2

(2.5.4.30)

+ λcη = 0,

(2.5.4.31)

where C1 , C2 , and C3 are arbitrary constants. Relations (2.5.4.30) and (2.5.4.31) involve two arbitrary functions, h and c, and the other functions, f , g, a, b, and η, are expressed in terms of them. The general solution to the system of the last two equations in (2.5.4.31) is expressed as 2+λ/(C1 C3 )  Z C5 a(x) = C3 c(x) dx + C4 , c(x) −λ/(C1 C3 )  Z C1 C3 c(x) dx + C4 , η(x) = − C5 λ

(2.5.4.32)

where C4 and C5 are arbitrary constants. Relations (2.5.4.30) define the admissible form of the functions f and g in terms of the single arbitrary function h. ◮ Example 2.40. On substituting c(x) = 1, C1 = C3 = C5 = 1, C2 = C4 = 0, and λ = n − 2 into (2.5.4.31) and (2.5.4.32), we get

a(x) = xn ,

b(x) = xn−1 ,

η(x) = x2−n /(2 − n),

n 6= 2.

In view of (2.5.4.30), we arrive at the nonlinear equation ut = [xn f (u)ux ]x + xn−1 g(u)ux , Z f (u) = uh(u), g(u) = (n − 2) h(u) du − uh(u),

(2.5.4.33)

where h(u) is an arbitrary function. It admits the functional separable solution in implicit form Z x2−n h(u) du = ke(n−2)t + ; (2.5.4.34) 2−n where k is an arbitrary constant. On substituting h = f /u into (2.5.4.33), we obtain the equation   Z f (u) ut = [xn f (u)ux ]x + xn−1 (n − 2) du − f (u) ux , u

152

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

whose solution is written as Z

x2−n f (u) du = ke(n−2)t + . u 2−n

◭

Functional separable solutions for ω(t) = k ln t. On substituting the logarithmic function ω(t) = k ln t into (2.5.2.2) with ξ(x) = 1, we seek exact solutions in the form Z h(u) du = k ln t + η(x). (2.5.4.35) Eliminating t from (2.5.4.2), with ω = k ln t, and (2.5.4.35), we obtain the functional differential equation  ′ Z f (aηx′ )′x f +a(ηx′ )2 +bηx′ g−kceη/k e−H/k = 0, H = h(u) du. (2.5.4.36) h u

Remark 2.30. Equation (2.5.4.36) can be derived from other considerations. Indeed, on rewriting (2.5.2.2) with ξ(x) = 1 in the form

e(H−η−ω)/k = 1,

where k is some constant, we multiply the right-hand side of (2.5.4.2) by e(H−η−ω)/k . On rearranging, we get   ′  e(H−η)/k f eω/k ωt′ = (aηx′ )′x f + a(ηx′ )2 + bηx′ g . (2.5.4.37) c h u The variables in equation (2.5.4.37) are separated: the left-hand side depends on t alone, while the right-hand side depends on x and u. Equating both sides of (2.5.4.37) with a constant λ, we obtain two equations. The left-hand side of (2.5.4.37) gives the equation eω/k ωt′ = λ, whose solution is ω = k ln(t + t0 ) + k ln(λ/k). The right-hand side of (2.5.4.37) with λ = k leads to equation (2.5.4.36).

Solution 7. Let us first look at the degenerate case where the differential form (f /h)′u vanishes. In this case, equation (2.5.4.36) admits solutions if (aηx′ )′x +Abηx′ = 0, Bbηx′ −kceη/k = 0, (2.5.4.38) R where A and B are arbitrary constants and F = f (u) du. It follows from (2.5.4.38) with B = k that the equation   c(x)eη(x)/k −F (u)/k ux , (2.5.4.39) Af (u) + ke c(x)ut = [a(x)f (u)ux ]x + ηx′ (x) h = f,

g = Af +Be−F/k ,

where a(x), c(x), and f (u) are arbitrary functions, and η = η(x) is a solution to the second-order nonlinear ODE [a(x)ηx′ ]′x + Ac(x)eη/k = 0, has the functional separable solution Z f (u) du = k ln t + η(x).

(2.5.4.40)

(2.5.4.41)

153

2.5. Construction of Functional Separable Solutions in Implicit Form

◮ Example 2.41. For a(x) = xn (n 6= 1, 2), c(x) = 1, and A = −k(n−1)(n−2), equation (2.5.4.40) has the exact solution η = k(n − 2) ln x. Hence, the equation   1 ut = [xn f (u)ux ]x + xn−1 −(n − 1)f (u) + (2.5.4.42) e−F (u)/k ux n−2 R admits an exact solution representable in the implicit form f (u) du = k ln t+k(n− ◭ 2) ln x. ◮ Example 2.42. For a(x) = eλx , c(x) = 1, and A = −kλ2 , equation (2.5.4.40) has the exact solution η = kλx. Hence, the convection–diffusion equation   1 (2.5.4.43) ut = [eλx f (u)ux ]x + eλx −λf (u) + e−F (u)/k ux λ R admits an exact solution that can be represented in the implicit form f (u) du = ◭ k ln t + kλx.

Solution 8. Equation (2.5.4.36) is satisfied if we impose the conditions (aηx′ )′x = Aceη/k , a(ηx′ )2 = Bceη/k , bηx′ = ceη/k ,  ′ (2.5.4.44) f + g − k exp(−H/k) = 0, Af + B h u R where A and B are arbitrary constants and H = h(u) du. On substituting c(x) = 1, A = 1/k, and B = 1 into the first three equations of (2.5.4.44), we get a(x) = b(x) = eλx ,

η(x) = x,

λ=

1 . k

(2.5.4.45)

This results in the equation

where g(u) =

ut = [eλx f (u)ux ]x + eλx g(u)ux ,

(2.5.4.46)

    Z 1 d f (u) . exp −λ h(u) du − λf (u) − λ du h(u)

(2.5.4.47)

It admits the solution in implicit form Z 1 h(u) du = x + ln t. λ

(2.5.4.48)

Note that equation (2.5.4.46) with (2.5.4.47) involves two arbitrary functions, f = f (u) and h = h(u). Remark 2.31. The invariant solution (2.5.4.48) of equation (2.5.4.46) can be sought in the explicit form u = U (z) with z = x + (1/λ) ln t, in which case relation (2.5.4.47) is not required. The function U (z) is determined from the ODE

1 ′ Uz = [eλz f (U )Uz′ ]′z + eλz g(U )Uz′ . λ

154

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 9. On setting c(x) = 1, A = (1 + k)/k, and B = 1 in the first three equations of (2.5.4.44), we get a(x) = xn ,

b(x) = xn−1 ,

η(x) = ln x,

n = 2+

1 . k

(2.5.4.49)

As a result, we arrive at the convection–diffusion equation ut = [xn f (u)ux ]x + xn−1 g(u)ux ,

n 6= 2,

(2.5.4.50)

where     Z 1 d f (u) g(u) = , (2.5.4.51) exp −(n−2) h(u) du −(n−1)f (u)− (n − 2) du h(u) which admits the exact solution in implicit form Z 1 h(u) du = ln x + ln t. n−2

(2.5.4.52)

Remark 2.32. The self-similar solution (2.5.4.52) of equation (2.5.4.50) can be sought in the usual form u = U (z) with z = xt1/(n−2) , in which case relation (2.5.4.51) is not used. The function U (z) is determined from the ODE

1 zU ′ = [z n f (U )Uz′ ]′z + z n−1 g(U )Uz′ . n−2 z

Solution 10. Equation (2.5.4.36) can be satisfied if we set  ′ f = Af, exp(−H/k) = Bf, g = f, h u (aηx′ )′x

+

Aa(ηx′ )2

+

bηx′

− Bkce

η/k

(2.5.4.53)

= 0,

R

where A and B are arbitrary constants and H = h(u) du. On substituting A = −1/k = λ and B = 1 into the first three equations of (2.5.4.53), we get f (u) = g(u) = eλu , h(u) = 1. (2.5.4.54) This leads to the equation c(x)ut = [a(x)eλu ux ]x + b(x)eλu ux ,

(2.5.4.55)

which involves three arbitrary functions, a(x), b(x), and c(x), and admits the exact solution in explicit form 1 u = − ln t + η(x). (2.5.4.56) λ The function η = η(x) satisfies the ODE [a(x)eλη ηx′ ]′x + b(x)eλη ηx′ +

1 c(x) = 0. λ

(2.5.4.57)

Equation (2.5.4.57) is reduced to a linear ODE with the change of variable ζ = eλη .

2.5. Construction of Functional Separable Solutions in Implicit Form

155

Solution 11. Setting A = n+ 1, B = 1, and k = −1/n in the first three equations of (2.5.4.53), we get f (u) = un ,

g(u) = un ,

h(u) = 1/u.

(2.5.4.58)

As a result, we arrive at the convection–diffusion equation c(x)ut = [a(x)un ux ]x + b(x)un ux

(2.5.4.59)

where a(x), b(x), and c(x) are arbitrary functions, which has the exact solution ln u = − n1 ln t + η(x). This solution can be rewritten in explicit form as u = t−1/n ζ(x),

ζ(x) = eη(x) .

The function η satisfies the ODE [a(x)ζ n ζx′ ]′x + b(x)ζ n ζx′ +

1 c(x)ζ = 0. n

2.5.5. Nonlinear Klein–Gordon Type Equations with Variable Coefficients A brief overview of exact solutions to nonlinear Klein–Gordon type equations. Nonlinear wave equations and Klein–Gordon type equations arise in gas dynamics, acoustics, relativistic quantum mechanics, field theory, nonlinear optics, plasma physics, and particle physics [46, 170]. Such equations are used to model various physical phenomena including the propagation of dislocations in crystals, ultrashort optical pulses, ferroelectric phase transitions, behavior of particles in condensed matter, growth of crystals, and others [1, 46, 56, 80, 170]. Transformations, symmetries, and exact solutions for various classes of nonlinear Klein–Gordon type equations utt = [f (u)ux ]x + g(u)

(2.5.5.1)

were dealt with in numerous studies (e.g., see [10, 12, 32, 76, 96, 127, 150, 228, 272, 274, 275, 278, 311, 336, 376, 377, 385] and the literature cited therein). Most frequently, the classical and nonclassical methods of symmetry reductions [10, 32, 150, 228, 275, 311, 336] were used to construct exact solutions, as well as the methods of generalized and functional separation of variables [12, 97, 127, 274, 275, 278, 385]. In general, equation (2.5.5.1) admits a traveling wave solution u = U (kx − λt). For g(u) = 0, nonlinear wave equations have self-similar solutions u = U (x/t) [10], as well as more complicated exact solutions, which can be represented in implicit form [275]: p x − t f (u) = ϕ1 (u), p x + t f (u) = ϕ2 (u),

156

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where ϕ1 (u) and ϕ2 (u) are arbitrary functions (the degenerate cases ϕ1 = 0 and ϕ2 = 0 correspond to self-similar solutions of special form). It is noteworthy that for g(u) = 0, equation (2.5.5.1) can be linearized [272, 274, 377] (see also [32]), while some exact solutions for arbitrary f (u) admit a representation in parametric form (see [272, 274, 275]). Furthermore, there are exact solutions to equations of the form (2.5.5.1) where the functions f (u) and g(u) are expressed in terms of a single function h(u) [275]. The studies [142, 144, 146, 253, 272, 274, 275, 376] dealt with nonlinear Klein– Gordon type equations with variable coefficients c(x)utt = [a(x)f (u)ux ]x + b(x)g(u).

(2.5.5.2)

Table 2.6 lists a number of equations of this type with one or two arbitrary functions along with the forms of their exact solutions. (The function cosh( 12 λt) in solution 1 can be replaced with sinh( 12 λt).) Table 2.6. Nonlinear Klein–Gordon type equations with variable coefficients and their exact solutions. No. Equation

Form of solution or remark

1

utt = uxx +eλxg(u)

2

−4/3

3 4

utt = (u

1 λx 2

u = U (z), z = e −1/3

ux)x +b(x)u

k

Reduces to vtt = (v

utt = (x ux)x +g(u), k 6= 2

u = U (z), z = 4x

utt = [x f (u)ux]x +x

u = U (z), z = x

k

k−2

g(u)

2

Literature

cosh( 12 λt)

−4/3

2−k

vz )z

[272, 274] [272, 274, 376]

2 2

−(2−k) t

[272, 274, 376]

t, k 6= 2

[142, 253]

(k−2)/2

5

utt = [x f (u)ux]x +g(u)

u = U (z), z = λt+ln x

[253, 274, 376]

6

utt = (eλxux)x +g(u)

u = U (z), z = 4e−λx −λ2t2

[272, 274, 376]

7

λx

λx

utt = [e f (u)ux]x +e g(u)

8

k

k+1

utt = [a(x)u ux]x +b(x)u

9

λu

λu

utt = [a(x)e ux]x +b(x)e

u = U (z), z = e

λx/2

t

u = ϕ(x)ψ(t) u = ϕ(x)+ψ(t)

10 utt = [a(x)ux]x +b(x)u+cu ln u u = ϕ(x)ψ(t)

[253] [253, 274, 376] [253, 274] [272, 274]

Notation: a(x), b(x), f (u), and g(u) are arbitrary functions; c, k, and λ are free parameters.

The studies [36, 143, 145, 272, 274, 376] describe symmetries and some exact solutions to the nonlinear telegraph type equation c(x)utt = [a(x)f (u)ux ]x + b(x)g(u)ux . In [36, 143], the special case a(x) = b(x) = 1 was considered. Other related and more complicated nonlinear hyperbolic type equations were investigated, for example, in [11, 22, 117, 151, 152, 186, 307].

2.5. Construction of Functional Separable Solutions in Implicit Form

157

Remark 2.33. The studies [193, 287] (see also [190]) obtained some exact solutions to nonlinear Klein–Gordon type equations with delay

utt = auxx + F (u, w),

w = u(x, t − τ ),

where τ > 0 is the delay time. For exact solutions of more complicated related PDEs with delay, see [265, 266, 268].

In what follows, the main focus will be on constructing exact solutions (in an implicit form) to nonlinear Klein–Gordon type equations of a reasonably general form (2.5.5.2), that depend on one or two arbitrary functions. (Exact solutions to PDEs involving arbitrary functions and, hence, possessing significant arbitrariness are of considerable practical interest for testing numerical and approximate analytical methods for solving nonlinear equations.) Remark 2.34. Equation (2.5.5.2) is invariant under the transformations t = −t¯ and t = t˜+ t0 , where t0 is an arbitrary constant. Therefore, the variable t can be replaced with ±t + t0 in all the exact solutions presented below.

Derivation of the functional differential equation. We seek exact solutions to the Klein–Gordon type equation (2.5.5.2) in the implicit form (2.5.2.2). Differentiating (2.5.2.2) with respect to t and x yields ′′ ξωtt ξ ′ ω + ηx′ h′ − ξ 2 (ωt′ )2 u3 , ux = x ; h h h  ′ 1 f f (af ux )x = [(aξx′ )′x ω + (aηx′ )′x ] + a(ξx′ ω + ηx′ )2 . h h h u

ut =

ξωt′ , h

utt =

On substituting these expressions into (2.5.5.2), we arrive at the functional differential equation ′′ ωtt −ξ

h′u ′ 2 (ω ) = Q1 (x, u)ω 2 + Q2 (x, u)ω + Q3 (x, u), h2 t

where the functions Qn are explicitly independent of t and expressed as  ′ a(ξx′ )2 f Q1 (x, u) = , cξ h u  ′   f 1 ′ ′ ′ ′ Q2 (x, u) = (aξx )x f + 2aξx ηx , cξ h u  ′   f 1 Q3 (x, u) = (aηx′ )′x f + a(ηx′ )2 + bgh . cξ h u

(2.5.5.3)

(2.5.5.4)

Equation (2.5.5.3)–(2.5.5.4) depends on three variables, t, x, and u, and involves unknown functions (and their derivatives) with different arguments, which are connected by the additional relation (2.5.2.2). This equation is more complicated than equations of the form (2.5.2.3). The functional differential equation (2.5.5.3)–(2.5.5.4) simplifies significantly in the following two cases: (i) ξx′ = 0 and (ii) (f /h)′u = 0. Let us look at these cases in order (as in [253]).

158

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Case ξ(x) = 1. Traveling wave solutions for ω(t) = kt. For ξx′ = 0, we can set ξ = 1 without loss of generality. On substituting ξ = 1 into (2.5.5.4), we get Q1 (x, u) = Q2 (x, u) = 0. As a result, equation (2.5.5.3) reduces to  ′   h′ 1 f ′′ ωtt − u2 (ωt′ )2 = (aηx′ )′x f + a(ηx′ )2 + bgh . (2.5.5.5) h c h u The variables in (2.5.5.5) separate for ω(t) = kt, where k = const. This corresponds to a traveling wave solution defined in the implicit form Z Z h(u) du = kt + θ(x) dx. (2.5.5.6)

The integrands h(u) and θ(x) = ηx′ (x) are to be determined in the subsequent analysis from the functional differential equation  ′ h′ f 1 f k 2 c u3 + (aθ)′x + aθ2 + bg = 0, (2.5.5.7) h h h h u which is a special case of the bilinear equation (2.5.2.3) with N = 4. Solution 1. First, let us look at the degenerate case where the differential form (f /h)′u in (2.5.5.7) is zero. In this case, according to the splitting principle (we will not refer to it in what follows), equation (2.5.5.7) has solutions under the following conditions: h = f,

g = A + Bf −3 fu′ ,

(aθ)′x + Ab = 0,

Bb + k 2 c = 0,

(2.5.5.8)

where A and B are arbitrary constants. It follows from relation (2.5.5.8) with c(x) = 1 and B = −k 2 that the equation utt = [a(x)f (u)ux ]x + A − k 2

fu′ (u) , f 3 (u)

(2.5.5.9)

which involves two arbitrary functions, a(x) and f (u), has the generalized traveling wave solution Z Z Z x dx dx f (u) du = kt + A + C1 + C2 , (2.5.5.10) a(x) a(x) where C1 , C2 , and k are arbitrary constants. In the special case a(x) = 1, equation (2.5.5.9) and its solution (2.5.5.10) convert into the equation and solution obtained in [274]. Remark 2.35. It is not difficult to verify that the equation

utt = [a(x)f (u)ux ]x + b(x) − k2

fu′ (u) , f 3 (u)

which involves three arbitrary functions, a(x), b(x), and f (u), and generalizes equation (2.5.5.9), has two exact solutions Z  Z Z Z 1 dx f (u) du = ±kt − b(x) dx dx + C1 + C2 . a(x) a(x)

2.5. Construction of Functional Separable Solutions in Implicit Form

159

Solution 2. Equation (2.5.5.7) holds identically if we set  ′ h′u f +A = 0, h2 h u

g = −B

f , h

aθ2 = Ak 2 c,

b=

(aθ)′x , B

(2.5.5.11)

√ where A and B are arbitrary constants. On substituting c(x) = 1 and B = − 21 k A into (2.5.5.11), we arrive at the nonlinear Klein–Gordon type equations a′ (x) utt = [a(x)f (u)ux ]x ∓ px g(u). a(x)

(2.5.5.12)

Here, the functional coefficients f (u) and g(u) are expressed in terms of the arbitrary function h = h(u) as   1 1 √ 1 , (2.5.5.13) f (u) = C1 h + , g(u) = k A C1 + A 2 Ah where C1 is an arbitrary constant. Equations (2.5.5.12) with (2.5.5.13) have the exact solutions Z √ Z dx h(u) du = kt ± k A p + C2 . (2.5.5.14) a(x)

Formulas (2.5.5.13) and (2.5.5.14) involve two arbitrary functions, a(x) and h(u), and four arbitrary constants, A, C1 , C2 , and k. On setting A = 1, C1 = 0, k = 2 in (2.5.5.12)–(2.5.5.14), we find that f = 1 and g = 1/h. As a result, we arrive at the equations a′ (x) g(u), utt = [a(x)ux ]x ∓ px a(x)

(2.5.5.15)

which involve two arbitrary functions, a(x) and g(u), and admit the generalized traveling wave solutions Z Z dx du = 2t ± 2 p + C2 . (2.5.5.16) g(u) a(x) ◮ Example 2.43. On substituting a(x) = x2β into (2.5.5.15) and (2.5.5.16) and

on renaming ∓2βg(u) to g(u), we arrive at the equation utt = (x2β ux )x + xβ−1 g(u),

β 6= 0,

which involves one arbitrary function, g(u), and admits the exact solutions in implicit form ( Z 1 ± β1 t − β(1−β) x1−β + C if β 6= 1, du = g(u) ±t − ln |x| + C if β = 1, where C = ∓C2 /(2β) is an arbitrary constant.

◭

160

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

◮ Example 2.44. On substituting a(x) = e2βx into (2.5.5.15) and (2.5.5.16) and renaming ∓2βg(u) to g(u), we arrive at the equation

utt = (e2βx ux )x + eβx g(u),

β 6= 0,

which involves one arbitrary function, g(u), and admits the exact solutions in implicit form Z du 1 1 = ± t + 2 e−βx + C. g(u) β β ◭ Solution 3. Equation (2.5.5.7) holds if  ′ h′u B f f =A 2, g=− , A(aθ)′x + k 2 c = 0, h h h u

b=

aθ2 , B

(2.5.5.17)

where A and B are arbitrary constants. Using relations (2.5.5.17) with c(x) = 1 and B = −k 4/A2 , we obtain the nonlinear Klein–Gordon type equation utt = [a(x)f (u)ux ]x −

x2 g(u), a(x)

(2.5.5.18)

where a(x) is an arbitrary function. The functions f (u) and g(u) are expressed in terms of the arbitrary function h = h(u) as  ′ h′ k 4 h′u f (u) = A u2 , g(u) = . (2.5.5.19) h Ah h3 u Under condition (2.5.5.19), equation (2.5.5.18) admits the exact solution Z Z x dx k2 + C. (2.5.5.20) h(u) du = kt − A a(x) ◮ Example 2.45. On setting h = u−n−1 and A = −1/(n + 1) in (2.5.5.18)–

(2.5.5.20), we obtain the nonlinear Klein–Gordon type equation with a power-law nonlinearity utt = [a(x)un ux ]x −

x2 3n+1 u , a(x)

n 6= −1, −1/2,

(2.5.5.21)

which involves an arbitrary function, a = a(x). Exact solutions to this equation are defined by formula (2.5.5.20) where Z 1 h(u) du = − n , k = ±[(n + 1)2 (2n + 1)]−1/4 . nu ◭ Solution 4. Equation (2.5.5.7) also has exact solutions if (aθ)′x = Ac,

aθ2 = Bc,

b = c,

where A and B are arbitrary constants.

k2

f 1 h′u +A +B 3 h h h



f h

′

u

+ g = 0, (2.5.5.22)

2.5. Construction of Functional Separable Solutions in Implicit Form

161

1◦ . On substituting a(x) = b(x) = c(x) = 1, θ(x) = κ, A = 0, B = κ2 , and k = λ into (2.5.5.22), we obtain a traveling wave solution (2.5.1.2), which is omitted here. 2◦ . Setting c(x) = 1 and A = B = 1 in the first three equations of (2.5.5.22), we get a(x) = x2 , b(x) = 1, θ(x) = 1/x. (2.5.5.23) As a result, we arrive at the equation utt = [x2 f (u)ux ]x + g(u), where g(u) = −k 2

  h′u (u) f (u) 1 d f (u) , − − h3 (u) h(u) h(u) du h(u)

which has the exact solution in implicit form Z h(u) du = kt + ln x.

(2.5.5.24)

(2.5.5.25)

(2.5.5.26)

Note that equation (2.5.5.24) with (2.5.5.25) involves two arbitrary functions, f = f (u) and h = h(u). Remark 2.36. The invariant solution (2.5.5.26) of equation (2.5.5.24) can be sought in the usual form u = U (z) with z = kt + ln x, in which case relation (2.5.5.25) is not required. The function U (z) satisfies the autonomous ODE ′′ k2 Uzz = [f (U )Uz′ ]′z + f (U )Uz′ + g(U ).

Case ξ(x) = 1. Functional separable solutions for ω(t) = kt2 . On substituting ξ = 1 and ω(t) = kt2 into (2.5.2.2), we seek exact solutions in the implicit form Z h(u) du = kt2 + η(x). (2.5.5.27) Eliminating t from relations (2.5.5.5), with ω = kt2 , and (2.5.5.27), we obtain the functional differential equation (aηx′ )′x

1 f + a(ηx′ )2 h h



f h

′

u

+g−

h′u 4kh′u H 2k − 4kη = 0. + h h3 h3

(2.5.5.28)

For simplicity, it has been assumed that b(x) = c(x) = 1. Solution 5. First, let us look at the degenerate case where the differential form (aηx′ )′x vanishes. In this case, equation (2.5.5.28) has solutions under the following conditions:

C1



(aηx′ )′x = 0,

f h

′

u

− 4k

a(ηx′ )2 = C1 η,

h′u = 0, h2

g=

2k 4kh′u H , − h h3

(2.5.5.29)

162

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where C1 is an arbitrary constant. Integrating the first two equations of (2.5.5.29), we obtain     C22 C1 C1 a= exp − x , η = C3 exp x , (2.5.5.30) C1 C3 C2 C2 where C2 and C3 are arbitrary constants. Integrating further the third equation of (2.5.5.29), we obtain the following representations of f and g in terms of the arbitrary function h: Z 4k 2k 4kh′u f = C4 h − , g= − h(u) du. (2.5.5.31) C1 h h3 ◮ Example 2.46. On setting C1 = C3 = 1, C2 = −1, C4 = 0, and k = − 14

in (2.5.5.30) and (2.5.5.31), we get a = ex , η = e−x , f = 1, and g = − 21 h−1 + R −3 ′ h hu h du. As a result, we arrive at the equation Z 1 h′ utt = (ex ux )x − + u3 h du, 2h h which, for arbitrary h = h(u), has the exact solution in implicit form Z 1 h du = e−x − t2 . 4

◭

Solution 6. Equation (2.5.5.28) holds if we put

C2



f h

′

u

(aηx′ )′x = C1 , − 4k

h′u = 0, h2

a(ηx′ )2 = C2 η, g=

f 2k 4kh′u H − C1 , − h h3 h

(2.5.5.32)

where C1 and C2 are arbitrary constants. The general solution to the system consisting of the first two equations in (2.5.5.32) is a=

1 (C1 x + C3 )2−(C2 /C1 ) , C2 C4

η = C4 (C1 x + C3 )C2 /C1 ,

(2.5.5.33)

where C3 and C4 are arbitrary constants. Integrating further the third equation of (2.5.5.32), we find the functions f and g:   Z 4k 2C1 1 4kh′u f = C5 h − , g = 2k 1 + − h du − C1 C5 , (2.5.5.34) C2 C2 h h3 where h = h(u) is an arbitrary function. ◮ Example 2.47. On setting C1 = 1, C2 = 2−m, C3 = C5 = 0, C4 = 1/(2−m), and k = 14 (m − 2) in (2.5.5.33) and (2.5.5.34), we get a = xm , η = x2−m/(2 − m), R 1 −1 −3 ′ f = 1, and g = 2 mh + (2 − m)h hu h du. As a result, we obtain the equation Z h′u m−4 m + (2 − m) 3 h du, m 6= 2, utt = (x ux )x + 2h h

2.5. Construction of Functional Separable Solutions in Implicit Form

which, for arbitrary h = h(u), admits the exact solution in implicit form Z 1 1 h du = x2−m + (m − 2)t2 . 2−m 4

163

◭

Case ξ(x) = 1. Functional separable solutions for ω(t) = k ln t. On substituting ξ = 1 and ω(t) = k ln t into (2.5.2.2), we seek solutions in the form Z h(u) du = k ln t + η(x). (2.5.5.35) Eliminating t from (2.5.5.5), with ω = k ln t, and (2.5.5.35), we obtain the functional differential equation    ′ h′ f + bgh + kce2η/k 1 + k u2 e−2H/k = 0, (aηx′ )′x f + a(ηx′ )2 h u h (2.5.5.36) Z H = h(u) du. Solution 7. Let us first look at the degenerate case where the differential form (f /h)′u vanishes. In this case, equation (2.5.5.36) has exact solutions if   B fu′ h = f, g = A + 1 + k 2 e−2F/k , f f (2.5.5.37) ′ ′ 2η/k (aηx )x + Ab = 0, Bb + kce = 0, R where A and B are arbitrary constants and F = f (u) du. It follows from relations (2.5.5.37) with B = k that the equation     fu′ (u) −2F (u)/k k 2η(x)/k , 1+k 2 e A+ c(x)utt = [a(x)f (u)ux ]x − c(x)e f (u) f (u) (2.5.5.38) where a(x), c(x), and f (u) are arbitrary functions and η = η(x) is a solution to the nonlinear second-order ODE [a(x)ηx′ ]′x − Ac(x)e2η/k = 0, has the functional separable solution Z f (u) du = k ln t + η(x).

(2.5.5.39)

(2.5.5.40)

◮ Example 2.48. For a(x) = c(x) = 1 and A = −k, equation (2.5.5.39) has the solution η = −k ln cosh x. Hence, the equation     1 f ′ (u) −2F (u)/k utt = [f (u)ux ]x + k cosh−2 x 1 − 1 + k u2 e f (u) f (u) R admits the exact solution f (u) du = k ln t − k ln cosh x. ◭

164

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 8. Equation (2.5.5.36) holds if we set (aηx′ )′x = Ace2η/k , a(ηx′ )2 = Bce2η/k , b = ce2η/k ,  ′   (2.5.5.41) f h′ Af + B + gh + k 1 + k u2 e−2H/k = 0, h u h R where A and B are arbitrary constants and H = h(u) du. Substituting c(x) = 1, A = 2/k, and B = 1 into the first three equations in (2.5.5.41), we find that a(x) = b(x) = eλx ,

η(x) = x,

λ=

2 . k

(2.5.5.42)

As a result, we obtain the equation utt = [eλx f (u)ux ]x + eλx g(u),

(2.5.5.43)

where

      Z 2 h′u f (u) 1 d f (u) 2 1+ , exp −λ h(u) du − λ − g(u) = − λh(u) λ h2 h(u) h(u) du h(u) (2.5.5.44) which has the exact solution in implicit form Z 2 h(u) du = x + ln t. (2.5.5.45) λ Note that equation (2.5.5.43) with (2.5.5.44) involves two arbitrary functions, f = f (u) and h = h(u). Remark 2.37. The invariant solution (2.5.5.45) of equation (2.5.5.43) can be found in the usual form u = U (z) with z = x + (2/λ) ln t, in which case relation (2.5.5.44) between g and h is not used. The function U (z) is determined by the ODE

2 4 ′′ Uzz − Uz′ = [eλz f (U )Uz′ ]′z + eλz g(U ). λ2 λ

Solution 9. On setting c(x) = 1, A = (k + 2)/k, and B = 1 in the first three equations of (2.5.5.41), we get a(x) = xn ,

b(x) = xn−2 ,

η(x) = ln x,

n = 2+

2 . k

(2.5.5.46)

This leads to the Klein–Gordon type equation utt = [xn f (u)ux ]x + xn−2 g(u), where

n 6= 2,

(2.5.5.47)

    Z 2 h′u (u) 2 1+ exp −(n − 2) h(u) du (n − 2)h(u) n − 2 h2 (u)   f (u) 1 d f (u) − (n − 1) , (2.5.5.48) − h(u) h(u) du h(u)

g(u) = −

2.5. Construction of Functional Separable Solutions in Implicit Form

which has the exact solution in implicit form Z h(u) du = ln x +

2 ln t. n−2

165

(2.5.5.49)

Remark 2.38. The self-similar solution (2.5.5.49) of equation (2.5.5.47) can be found in the usual form u = U (z) with z = xt2/(n−2) , in which case relation (2.5.5.48) between g and h is not used. The function U (z) is determined by the ODE

4 2 z(zUz′ )′z − zU ′ = [z n f (U )Uz′ ]′z + z n−2 g(U ). (n − 2)2 n−2 z

Solution 10. Equation (2.5.5.36) can be satisfied if we put    ′ h′ f = Af, 1 + k u2 e−2H/k = Bf, gh = f, h u h (aηx′ )′x

+

Aa(ηx′ )2

+ b + Bkce

(2.5.5.50)

2η/k

= 0, R where A and B are arbitrary constants and H = h(u) du. For A = −2/k = λ and B = 1, the first three equations of (2.5.5.50) have the solution f (u) = g(u) = eλu , h(u) = 1. (2.5.5.51) This results in the equation c(x)utt = [a(x)eλu ux ]x + b(x)eλu ,

(2.5.5.52)

which admits the exact solution in explicit form u=−

2 ln t + η(x), λ

(2.5.5.53)

where the function η = η(x) is determined by the ODE (aηx′ )′x + λa(ηx′ )2 + b −

2 −λη ce = 0. λ

(2.5.5.54)

Equations (2.5.5.52) and (2.5.5.54) involve three arbitrary functions, a = a(x), b = b(x), and c = c(x). Solution 11. By setting A = n + 1, B = 1 − k, and k = −2/n in the first three equations of (2.5.5.50), we find f (u) = un ,

g(u) = un+1 ,

h(u) = 1/u.

(2.5.5.55)

As a result, we obtain the Klein–Gordon type equation c(x)utt = [a(x)un ux ]x + b(x)un+1 ,

(2.5.5.56)

where a(x), b(x), and c(x) are arbitrary functions, which admits an exact solution of the form ln u = −(2/n) ln t + η(x). This solution can be represented explicitly as u = t−2/n ζ(x),

ζ(x) = eη(x) .

(2.5.5.57)

166

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The function ζ is described by the ODE [a(x)ζ n ζx′ ]′x + b(x)ζ n+1 −

2(n + 2) c(x)ζ = 0. n2

(2.5.5.58)

Case h(u) = f (u). Generalized traveling wave solutions for ω(t) = t. If (f /h)′u = 0, we can set h = f without loss of generality. On substituting h = f into (2.5.5.3)–(2.5.5.4), we get the equation ′′ ωtt −ξ

 (aξx′ )′x 1  ′ ′ fu′ ′ 2 (aηx )x f + bf g . (ω ) = fω + t 2 f cξ cξ

(2.5.5.59)

It is apparent that if ω(t) = t, the variables in (2.5.5.59) are separated. This situation corresponds to generalized traveling wave solutions given in implicit form: Z f (u) du = ξ(x)t + η(x), (2.5.5.60) where f (u) is the function appearing in equation (2.5.5.2), while ξ(x) and η(x) are unknown functions to be determined. On eliminating ω = t from (2.5.5.59) using (2.5.5.60) and on rearranging, we obtain the equation Z ξ(aηx′ )′x −η(aξx′ )′x +bξg +cξ 3f −3 fu′ +(aξx′ )′x F = 0, F = f (u) du, (2.5.5.61) which is a special case of equation (2.5.2.3) with N = 4. Solution 12. Let us first consider the degenerate case where two functional coefficients in (2.5.5.61) vanish at once, so that (aξx′ )′x = 0.

(2.5.5.62)

Integrating (2.5.5.62) yields the relation between a = a(x) and ξ = ξ(x): Z dx + C2 , (2.5.5.63) ξ = C1 a(x) where C1 and C2 are arbitrary constants. In the degenerate case (2.5.5.62), equation (2.5.5.61) has solutions under the following conditions: g = k1 + k2 f −3 fu′ ,

(aηx′ )′x + k1 b = 0,

k2 b + cξ 2 = 0,

(2.5.5.64)

where k1 and k2 are arbitrary constants. From relations (2.5.5.63) and (2.5.5.64) it follows that the equation   k1 f ′ (u) , k= c(x)utt = [a(x)f (u)ux ]x − c(x)ξ 2 (x) k + u3 , (2.5.5.65) f (u) k2

167

2.5. Construction of Functional Separable Solutions in Implicit Form

where a(x), c(x), and f (u) are arbitrary functions and the function ξ = ξ(x) is defined by formula (2.5.5.63), admits an exact solution of the form (2.5.5.60), in which η(x) is expressed as Z  Z Z 1 dx 2 η(x) = k + C4 ; (2.5.5.66) c(x)ξ (x) dx dx + C3 a(x) a(x) C3 and C4 are arbitrary constants. ◮ Example 2.49. We set a(x) = c(x) = 1 in equation (2.5.5.65) and C1 = 1, C2 = C3 = C4 = 0 in formulas (2.5.5.63) and (2.5.5.66). As a result, we obtain the equation   fu′ (u) 2 utt = [f (u)ux ]x − x k + 3 , (2.5.5.67) f (u)

which involves an arbitrary function, f (u), and an arbitrary constant, k, and admits the generalized traveling wave solution Z 1 kx4 . f (u) du = xt + 12 In the special case f (u) = eu , equation (2.5.5.67) becomes utt = (eu ux )x − x2 (k + e−2u ). Its exact solution can be represented in the explicit form u = ln(xt +

1 4 12 kx ).

◭

Solution 13. In the degenerate case (2.5.5.62), equation (2.5.5.61) also has other exact solutions under the following conditions: g = k1 ,

f −3 fu′ = −k2 ,

(aηx′ )′x + k1 b − k2 cξ 2 = 0,

(2.5.5.68)

where k1 and k2 are arbitrary constants. From relations (2.5.5.63) and (2.5.5.68) with k1 = 1 and k2 = 12 it follows that the equation c(x)utt = [a(x)u−1/2 ux ]x + b(x),

(2.5.5.69)

involving three arbitrary functions, a(x), b(x), and c(x), admits an exact solution of the form u = 14 [ξ(x)t + η(x)]2 . (2.5.5.70) The function ξ = ξ(x) is defined by (2.5.5.63) and η(x) satisfies the ODE [a(x)ηx′ ]′x =

2 1 2 c(x)ξ

− b(x).

(2.5.5.71)

Since the right-hand side of equation (2.5.5.71) is known, the function η is found by simple integration. Remark 2.39. The equation

c(x)utt = [a(x)u−1/2 ux ]x + b(x)u1/2 + p(x),

(2.5.5.72)

which involves four arbitrary functions, a(x), b(x), c(x), and p(x), and is more general than equation (2.5.5.69), also has exact solutions of the form (2.5.5.70). If p(x) = 0 and p(x)/b(x) = const, equation (2.5.5.72) belongs to the class of equations in question, (2.5.5.2).

168

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 14. In the nondegenerate case, the functional differential equation (2.5.5.61) can be satisfied if we set Z f −3 fu′ = k1 F, g = k2 , F = f (u) du; (2.5.5.73) (aξx′ )′x + k1 cξ 3 = 0, ξ(aηx′ )′x − η(aξx′ )′x + k2 bξ = 0, where k1 and k2 are arbitrary constants. The first equation in (2.5.5.73) with k1 = − 29 admits the exact solution f (u) = u−2/3 (F = 3u1/3 ). Setting further k2 = 1, we arrive at the equation c(x)utt = [a(x)u−2/3 ux ]x + b(x),

(2.5.5.74)

the exact solution of which can be represented in explicit form as u=

1 27 [ξ(x)t

+ η(x)]3 .

(2.5.5.75)

The functions ξ = ξ(x) and η = η(x) satisfy the last two ODEs of (2.5.5.73) with k1 = − 29 and k2 = 1. Remark 2.40. The equation

c(x)utt = [a(x)u−2/3 ux ]x + b(x)u1/3 + p(x),

(2.5.5.76)

which involves four arbitrary functions, a(x), b(x), c(x), and p(x), and is more general than (2.5.5.74), has exact solutions of the form u = [ϕ(x)t + ψ(x)]3 .

(2.5.5.77)

The functions ϕ = ϕ(x) and ψ = ψ(x) are described by the ODEs 3(aϕ′x )′x + bϕ − 6cϕ3 = 0,

3(aψx′ )′x + bψ − 6cϕ2 ψ + p = 0.

If p(x) = 0 and p(x)/b(x) = const, equation (2.5.5.76) belongs to the class of equations in question, (2.5.5.2).

Case h(u) = f (u). Functional separable solutions for ω(t) = eλt . Solution 15. We substitute ω(t) = eλt and η = η0 = const into (2.5.2.2) and (2.5.5.59) and eliminate t to obtain the functional differential equation Z ′ ¯ (aξx′ )′x ¯ b 2 F 2 fu ¯ 2 ¯ −λ F + g = 0, F = f du − η0 . +λ 3F + (2.5.5.78) f f cξ c It is a special case of equation (2.5.2.3) with N = 3. Equation (2.5.5.78) holds identically if (aξx′ )′x = Acξ,

b = c,

where A is an arbitrary constant.

g = λ2

f′ F¯ − λ2 u3 F¯ 2 − AF¯ . f f

(2.5.5.79)

2.5. Construction of Functional Separable Solutions in Implicit Form

169

On substituting b(x) = c(x) = 1 and η0 = 0 into (2.5.5.79), we arrive at the equation Z ′ 2 fu (u) 2 2 F (u) −λ 3 F (u) − AF (u), F = f (u) du, utt = [a(x)f (u)ux ]x + λ f (u) f (u) (2.5.5.80) R which has the exact solution in implicit form f (u) du = eλt ξ(x), with the function ξ = ξ(x) satisfying the linear second-order ODE (aξx′ )′x = Aξ.

(2.5.5.81)

Remark 2.41. The constant A in equations (2.5.5.80) and (2.5.5.81) can be replaced with an arbitrary function A(x).

◮ Example 2.50. On substituting f (u) = uk into (2.5.5.80), we get the equation

utt = [a(x)uk ux ]x +

λ2 A u− uk+1 , (k + 1)2 k+1

(2.5.5.82)

where a(x) is an arbitrary function and A, k, and λ are arbitrary constants (k 6= −1). This PDE admits an exact solution that can be written in the explicit form u = [(k + 1)eλt ξ(x)]1/(k+1) . The function ξ = ξ(x) satisfies the linear ODE (2.5.5.81).

◭

Remark 2.42. Let us look at the more general equation

c(x)utt = [a(x)uk ux ]x + b(x)uk+1 + mc(x)u

(2.5.5.83)

than (2.5.5.82), in which a(x), b(x), and c(x) are arbitrary functions and m 6= 0 is an arbitrary constant. For m = β 2 > 0, equation (2.5.5.83) admits an exact solution as the product of functions with different arguments: u = [C1 exp(−βt) + C2 exp(βt)]θ(x),

where C1 and C2 are arbitrary constants, with the function θ = θ(x) described by the ODE [a(x)θk θx′ ]′x + b(x)θk+1 = 0.

By the change of variable ζ = θk+1 , it is reduced to the linear equation [a(x)ζx′ ]′x + (k + 1)b(x)ζ = 0.

For m = −β 2 < 0, equation (2.5.5.83) admits the exact solution u = [C1 cos(βt) + C2 sin(βt)]θ(x),

with the function θ = θ(x) satisfying ODE (2.5.5.84).

(2.5.5.84)

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Remark 2.43. The equation with an exponential nonlinearity

c(x)utt = [a(x)eλu ux ]x + b(x)eλu + mc(x),

(2.5.5.85)

where a(x), b(x), and c(x) are arbitrary functions and m 6= 0 is an arbitrary constant, has an exact solution as the sum of functions with different arguments u=

1 mt2 + C1 t + θ(x). 2

The function θ = θ(x) is described by the ordinary differential equation [a(x)eλθ θx′ ]′x + b(x)eλθ = 0.

By the change of variable ζ = eλθ , it is reduced to the linear equation [a(x)ζx′ ]′x +λb(x)ζ = 0. Remark 2.44. For a number of other functional separable solutions to equation (2.5.5.2), see the article [253].

2.5.6. Nonlinear Equations with Three or More Independent Variables The method described in Subsection 2.5.2 admits various generalizations and allows one to construct exact solutions to nonlinear PDEs with three or more independent variables. In particular, for unsteady equations with n spatial variables, x1 , . . . , xn , functional separable solutions should be sought in the implicit form Z h(u) du = ξ(x)ω(t) + η(x), x = (x1 , . . . , xn ). (2.5.6.1) Using (2.5.6.1), one finds the partial derivatives and substitutes them into the PDE in question. On eliminating t with (2.5.6.1), one arrives at the functional differential equation (2.5.2.3) in which the functions Φj [x] must be replaced with Φj [x]. Then, one uses a multidimensional analogue of the splitting method to obtains exact solutions. Omitting the details, we will illustrate the above with examples of specific equations. We will only give original nonlinear PDEs and their functional separable solutions. In what follows, we use the notation   n n X X ∂2u ∂ ∂u b(x)f (u) . ∆u = , ∇ · [b(x)f (u)∇u] = ∂x2j ∂xj ∂xj j=1 j=1 Equation 1. The nonlinear reaction–diffusion type equation with n spatial variables k ut = ∇ · [f (u)∇u] + + g(x), f (u) where f (u) and g(x) are arbitrary functions, admits the functional separable solution in implicit form [107, 275]: Z f (u) du = kt + η(x).

2.5. Construction of Functional Separable Solutions in Implicit Form

171

The function η = η(x) satisfies the Poisson equation ∆η + g(x) = 0. For exact solutions to this linear equation, see, for example, [262, 344]. Equation 2. The nonlinear reaction–diffusion type equation with n spatial variables Z aF (u) + b + g(x)[aF (u) + b], F (u) = f (u) du, ut = ∇ · [f (u)∇u] + f (u) where f (u) is an arbitrary function, admits the functional separable solution in implicit form [107, 275]: Z b f (u) du = eat η(x) − . a The function η = η(x) is described by the Helmholtz equation ∆η + ag(x)η = 0. For exact solutions to this linear equation, see, for example, [262, 344]. Equation 3. Consider the nonlinear equation with n spatial variables ut = L[f (u)] +

g(t) + h(x), fu′ (u)

where L is an arbitrary linear differential operator of the second (or any) order in the spatial coordinates with coefficients independent of t, that satisfies the condition L[const] = 0. This equation admits the functional separable solution in implicit form [275]: Z f (u) = g(t) dt + η(x), where the function η = η(x) satisfies the linear PDE L[η] + h(x) = 0. Equation 4. Consider the nonlinear equation with n spatial variables ut = L[f (u)] +

af (u) + b + g(x)[af (u) + b]. fu′ (u)

where L is an arbitrary linear differential operator of the second (or any) order in the spatial coordinates with coefficients independent of t, that satisfies the condition L[const] = 0.

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This equation admits the following functional separable solution in implicit form [275]: b f (u) = eat η(x) − , a where the function η = η(x) is described by the linear PDE L[η] + ag(x)η = 0. Equation 5. Let us look at the nonlinear reaction–diffusion type equation with n spatial variables ut = a(x)∇ · [b(x)f (u)∇u] + c(x) +

k(t) , f (u)

(2.5.6.2)

where a(x), b(x), c(x), k(t), and f (u) are arbitrary functions. It is not difficult to show that equation (2.5.6.2) admits the exact solution in implicit form Z Z f (u) du =

k(t) dt + η(x).

The function η = η(x) satisfies the linear elliptic type equation a(x)∇ · [b(x)∇η] + c(x) = 0.

For b(x) ≡ 1, it coincides with the Poisson equation ∆η = −c(x)/a(x); for exact solutions of this equation, see, for example, [262]. Equation 6. Consider the following nonlinear reaction–diffusion equation with n spatial variables, which is a spatial generalization of the one-dimensional equation (2.5.3.17): ut = ∇ · [a(x)∇u] − a(x)|∇η|2 g(u), (2.5.6.3) where a(x) and g(u) are arbitrary functions and the function η = η(x) is a solution of the linear equation ∇ · [a(x)∇η] = k, (2.5.6.4) with k being some constant. It can be shown that equation (2.5.6.3) admits two functional separable solutions that can be represented in implicit form: −1/2 Z  Z ± 2 g(u) du + C2 du = kt + η(x) + C1 , (2.5.6.5)

where C1 and C2 are arbitrary constants.

◮ Example 2.51. On setting a(x) = 1 in (2.5.6.3)–(2.5.6.5), we find a radik allyPsymmetric particular solution to equation (2.5.6.4): η = 2n |x|2 , where |x|2 = n 2 = j=1 xj . For k = n, the associated equation of the form (2.5.6.3) is

ut = ∆u − |x|2 g(u).

It admits the exact solutions [254] −1/2 Z  Z 1 ± 2 g(u) du + C2 du = nt + |x|2 + C1 . 2

(2.5.6.6)

◭

2.5. Construction of Functional Separable Solutions in Implicit Form

173

Equation 7. Consider the following nonlinear reaction–diffusion equation with n spatial variables, which is a spatial generalization of the one-dimensional equation (2.5.3.25): ut = ∇ · [a(x)f (u)∇u] − b(x)u,

b(x) = ∇ · [a(x)∇η],

(2.5.6.7)

where a(x) and f (u) are arbitrary functions and η = η(x) is a solution to the nonlinear first-order equation |∇η|2 = 4k/a(x). It can be shown that equation (2.5.6.7) admits a functional separable solution that can be represented in the implicit form Z f (u) du = 4k 2 t + η(x) + C, u where C is an arbitrary constant. Equation 8. The nonlinear Klein–Gordon type equation with n spatial variables utt = a(x)∇ · [b(x)f (u)∇u] + c(x) − k 2

fu′ (u) , f 3 (u)

which involves four arbitrary functions, a(x), b(x), c(x), and f (u), admits two exact solutions in implicit form Z f (u) du = ±kt + η(x), where the function η = η(x) satisfies the linear elliptic type equation a(x)∇ · [b(x)∇η] + c(x) = 0. Remark 2.45. See also higher-order equations 5, 6, and 8–10 in Subsection 2.5.7, which admit multidimensional generalizations.

2.5.7. Nonlinear Third- and Higher-Order Equations The studies [105, 114, 274, 275] present some explicit functional separable solutions to nonlinear Korteweg–de Vries type equations as well as a number of other higherorder equations. The method described in Subsection 2.5.2 can successfully be applied to construct exact solutions in implicit form to third- and higher-order nonlinear PDEs. Omitting the details, we illustrate this with specific examples below; we only give original nonlinear equations and their exact solutions. Equation 1. Consider the third-order nonlinear PDE ut = a(x)[b(x)f (u)ux ]xx + c(x) +

k , f (u)

(2.5.7.1)

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where a(x), b(x), c(x), and f (u) are arbitrary functions and k 6= 0 is an arbitrary constant. Using the method described in Subsection 2.5.2, we can construct an exact solution to equation (2.5.7.1) in the implicit form Z Z f (u) du = kt + ζ(x) dx + C1 , (2.5.7.2) where C1 is an arbitrary constant and the function ζ = ζ(x) is described by the linear second-order ODE a(x)[b(x)ζ]′′xx + c(x) = 0. (2.5.7.3) Integrating twice gives the general solution of equation (2.5.7.3):    Z Z 1 c(x) ζ= C2 x + C3 − dx dx , b(x) a(x)

(2.5.7.4)

where C2 and C3 are arbitrary constants. To sum up, formulas (2.5.7.2) and (2.5.7.4) define a functional separable solution to the nonlinear third-order equation (2.5.7.1). Equation 2. Let us look at another nonlinear third-order PDE: ut = [x2 a(x)f (u)ux ]xx − a(x)[k + 2f (u)]ux,

(2.5.7.5)

where a(x) and f (u) are arbitrary functions. It is easy to verify that this equation admits the following functional separable solution in implicit form: Z Z dx + C1 . f (u) du = kt − a(x) Equation 3. The nonlinear nth-order PDE ut = [f (u)ux ](n−1) + x

a +b f (u)

has the exact solution in implicit form [274] Z b n x + Cn−! xn−1 + · · · + C1 x + C0 , f (u) du = at − n! where C0 , C1 , . . . , Cn are arbitrary constants. Equation 4. The nonlinear nth-order PDE ut =

[f (u)ux ](n−1) + x

aF (u) + b + g(x)[aF (u) + b], f (u)

where F (u) =

admits the functional separable solution in implicit form [275] Z b f (u) du = eat η(x) − , a

Z

f (u) du,

2.5. Construction of Functional Separable Solutions in Implicit Form

175

where the function η = η(x) satisfies the linear ODE ηx(n) + ag(x)η = 0. Equation 5. Consider the nonlinear equation √ √ ut = L[ u ] + f (x) + g(x) u, where L is an nth-order linear differential operator with respect to x with coefficients independent of t, and u ≥ 0. This equation admits a generalized separable solution∗ that can be represented in explicit form [114]: u = [ϕ(x) + ψ(x)t]2 , where the functions ϕ = ϕ(x) and ψ = ψ(x) are described by the ordinary differential equations L[ϕ] − 2ϕψ + gϕ + f = 0, L[ψ] − 2ψ 2 + gψ = 0.

Remark 2.46. In equations 5 and 6, the differential operator L and functions f , g can depend on several spatial variables, x = (x1 , . . . , xm ).

Equation 6. Consider the nonlinear equation √ √ ut = L[ u ] + f (x) + g(x) u + cu, where L is an nth-order linear differential operator with respect to x with coefficients independent of t, and u ≥ 0. For c = 0, see the previous equation. This equation admits a generalized separable solution that can be written explicitly [114]:
2  u = ϕ(x) + ψ(x) exp 21 ct ,

where the functions ϕ = ϕ(x) and ψ = ψ(x) are described by the system of ordinary differential equations L[ϕ] + cϕ2 + gϕ + f = 0, L[ψ] + cϕψ + gψ = 0. Equation 7. The nonlinear equation utt = [f (u)ux ](n−1) − a2 x

fu′ (u) + b, f 3 (u)

involving the second derivative with respect to t, admits the following exact solutions in implicit form [274]: Z b n f (u) du = ±at − x + Cn−1 xn−1 + · · · + C1 x + C0 , n! where C0 , C1 , . . . , Cn−1 are arbitrary constants. ∗ The generalized separable solutions to the equations in Examples 5, 6, 8, 9, and 10 can also be treated as functional separable solutions.

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Equation 8. Consider the nonlinear equation √ √ utt = L[ u ] + a(x) + b(x) u, where L is an nth-order linear differential operator with respect to x with coefficients independent of t, and u ≥ 0. This equation admits a generalized separable solution that can be written explicitly as u = [f (x)t2 + g(x)t + h(x)]2 , where the functions f = f (x), g = g(x), h = h(x) are described by the system of ordinary differential equations L[f ] + bf − 12f 2 = 0, L[g] + bg − 12f g = 0,

L[h] + bh + a − 4f h − 2g 2 = 0. Remark 2.47. In equations 8–10, the differential operator L and functions a, b can depend on several spatial variables, x = (x1 , . . . , xm ).

Equation 9. Consider the nonlinear equation √ √ utt = L[ u ] + a(x) + b(x) u + cu. For c = 0, see the preceding equation. 1◦ . Generalized separable solution for c > 0 [114]:  u = f (x) exp

1 2

2 √
√
c t + g(x) exp − 12 c t + h(x) .

The functions f = f (x), g = g(x), and h = h(x) are described by the system of ordinary differential equations L[f ] + L[g] +

3 2 cf h + bf = 0, 3 2 cgh + bg = 0, 2

L[h] + ch + bh + a + 2cf g = 0. 2◦ . Generalized separable solution for c < 0: p 
u = f (x) cos 12 |c| t + g(x) sin

1 2

p
2 |c| t + h(x) ,

The functions f = f (x), g = g(x), and h = h(x) are described by the system of ordinary differential equations L[f ] + L[g] + L[h] +

3 2 cf h + bf = 0, 3 2 cgh + bg = 0, ch2 + bh + a + 12

c(ϕ2 + ψ 2 ) = 0.

2.5. Construction of Functional Separable Solutions in Implicit Form

177

Equation 10. Consider the nonlinear equation utt = L[u1/3 ] + a(x) + b(x)u1/3 , where L is an nth-order linear differential operator with respect to x with coefficients independent of t, and u ≥ 0. This equation admits a generalized separable solution that can be written explicitly [275]: u = [f (x)t + g(x)]3 , where the functions f = f (x) and g = g(x) are described by the system of ordinary differential equations L[f ] + bf − 6f 3 = 0, L[g] + bg + a − 6f 2 g = 0.

¨ 2.5.8. Nonlinear Schrodinger Type Equation Consider the nonlinear Schr¨odinger type equation of general form iwt + wxx + f (|w|)w = 0,

(2.5.8.1)

where w is a complex-valued function of real variables x and t, f (u) is an arbitrary real-valued function of a real variable, and i2 = −1. Nonlinear Schr¨odinger type equations are often used to model different processes in theoretical physics, including nonlinear optics, superconductivity, and plasma physics [100, 167, 188, 339]. Exact solutions of equation (2.5.8.1) are sought in the form w = u(x, t) exp[iv(x, t)],

(2.5.8.2)

where the real-valued functions u = u(x, t) and v = v(x, t) satisfy the following nonlinear system of coupled PDEs: −uvt + uxx − uvx2 + uf (|u|) = 0, ut + ux vx + (uvx )x = 0.

(2.5.8.3) (2.5.8.4)

Some traveling wave solutions (optical solitons) to coupled equations (2.5.8.3) and (2.5.8.4) with f (u) of special form were obtained, for example, in [30, 31, 93]. Below are several functional separable solutions to system (2.5.8.3)–(2.5.8.4) with an arbitrary function f (u) [256, 275]; the intermediate calculations are omitted. Solution 1. There is a traveling wave solution of the form u = u(y),

v = Ax + Bt + C,

y = x − 2At,

where A, B, and C are arbitrary real constants, and the function u = u(y) is determined by the autonomous ODE u′′yy + uf (|u|) − (A2 + B)u = 0.

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Integrating yields the general solution in implicit form Z

du

p = C2 ± y, (A2 + B)u2 − 2F (u) + C1

F (u) =

Z

uf (|u|) du,

where A, B, C, C1 , and C2 are arbitrary real constants. Solution 2. There is a functional separable solution of the form u = u(z),

v = Axt −

2 2 3 3A t

+ Bt + C,

z = x − At2 ,

where A, B, and C are arbitrary real constants; the function u = u(z) is determined by the ODE u′′zz + uf (|u|) − (Az + B)u = 0. Solution 3. There are functional separable solutions of the form 1 √ , u= C1 t

(x + C2 )2 v= + 4t

Z


f |C12 t|−1/2 dt + C3 ,

where C1 , C2 , and C3 are arbitrary real constants (C1 6= 0). Solution 4. There are functional separable solutions of the form u = u(x),

v = C1 t + C2

Z

dx + C3 , u2 (x)

where C1 , C2 , and C3 are arbitrary real constants, and the function u = u(x) is determined by the autonomous ODE u′′xx − C1 u − C22 u−3 + uf (|u|) = 0, whose general solution can be written in implicit form. Solution 5. There is a functional separable solution of the form u = u(ζ),

v = At + φ(ζ),

ζ = kx + λt,

where A, k, and λ are arbitrary real constants; the functions u = u(ζ) and φ = φ(ζ) are determined by the system of coupled ODEs k 2 uφ′′ζζ + 2k 2 u′ζ φ′ζ + λu′ζ = 0, k 2 u′′ζζ − k 2 u(φ′ζ )2 − λuφ′ζ − Au + uf (|u|) = 0. Remark 2.48. The handbook [275] presents a number of exact solutions to more complicated nonlinear Schr¨odinger and Ginzburg–Landau type equations with variable coefficients that can depend on t or x.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

179

2.6. General Functional Separation of Variables. Explicit Representation of Solutions 2.6.1. General Form of Functional Separable Solutions In general, the term a functional separable solution with regard to nonlinear PDEs with two independent variables is used for exact solutions that can be represented as the composition of two functions [251, 257, 300]: u = U (z),

z = ϕ(x, t),

(2.6.1.1)

where the functions U (z) and ϕ(x, t) are unknown and satisfy overdetermined systems of ODEs and PDEs, respectively. In the simplest cases, any of these functions can satisfy a single equation. The representation as a superposition of functions (2.6.1.1) is the most general way of describing solutions to nonlinear PDEs dependent on one or more arbitrary functions of the unknown u. Remark 2.49. If the outer function U = U (z) is described by a single ODE, then the method for seeking functional separable solutions in the form (2.6.1.1) is a special case of the direct method for symmetry reductions [72, 73]. If U = U (z) satisfies an overdetermined system of ODEs, method for seeking functional separable solutions in the form (2.6.1.1) goes beyond the direct method for symmetry reductions (see Chapter 3 for details).

In what follows, we will consider a few classes of nonlinear diffusion and wave type equations that involve an arbitrary function, f (u), and admit exact solutions of the form (2.6.1.1) where U = U (z) is described by a single ODE.

2.6.2. Nonlinear Reaction–Diffusion Type Equations The class of nonlinear PDEs in question. Following [254], we will look at the following class of reaction–convection–diffusion equations with a nonlinear source and variable coefficients: c(x)ut = [a(x)ux ]x + b(x)ux + p(x)f (u),

(2.6.2.1)

where f (u) is an arbitrary function. Some of the four functional coefficients a = a(x) > 0, b = b(x), c = c(x) > 0, and p = p(x) can be free, while the others can be expressed in terms of them as a result of the subsequent analysis; the free coefficients can be chosen differently (see below). Without loss of generality, we assume that p > 0 (for p < 0, the functions p and f must be redefined as −p and −f ). Reduction of nonlinear reaction–convection–diffusion equations to an ODE. Exact solutions to equation (2.6.2.1) will be sought as a composition of functions (2.6.1.1). Substituting (2.6.1.1) into (2.6.2.1) gives  ′′ + [a(x)ϕx ]x + b(x)ϕx − c(x)ϕt Uz′ + p(x)f (U ) = 0. (2.6.2.2) a(x)ϕ2x Uzz In the special case U (z) = z, equation (2.6.2.2) coincides with the original equation (2.6.2.1); hence, no solutions are lost at this stage.

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Suppose that the coefficients of equation (2.6.2.1) satisfy the relations p(x) = a(x)s(ϕ)ϕ2x , c(x)ϕt = [a(x)ϕx ]x + b(x)ϕx +

(2.6.2.3) a(x)k(ϕ)ϕ2x ,

(2.6.2.4)

where s(ϕ) and k(ϕ) are some functions (s 6≡ 0). Then, equation (2.6.2.2) reduces to a single ordinary differential equation: ′′ Uzz − k(z)Uz′ + s(z)f (U ) = 0.

(2.6.2.5)

Exact solutions to the nonlinear ordinary differential equation (2.6.2.5) for some functions k(z), s(z), and f (U ) can be found in [273, 276]. In the special case k(z) ≡ 0, equation (2.6.2.4) is linear, and the general solution to equation (2.6.2.5) with s(z) = 1 and arbitrary f (U ) can be written in implicit form [273]: −1/2 Z  Z C1 − 2 f (U ) dU dU = C2 ± z, (2.6.2.6) where C1 and C2 are arbitrary constants. Equations (2.6.2.3)–(2.6.2.5) allow one to construct exact solutions for a wide class of nonlinear reaction–convection–diffusion equations of the form (2.6.2.1).

Remark 2.50. Without loss of generality, two of four functional coefficients a(x), b(x), c(x), and p(x) in equation (2.6.2.1) can be set equal to unity. In particular, if one divides both Rsides p of the equation by c and changes from t, x to the new independent variables t, y= c/a dx, then one obtains an equation in the canonical form ut = uyy + b1 (y)uy + p1 (y)f (u). It is not difficult to find a transformation t, y¯ = y¯(x) that reduces equation (2.6.2.1) to another canonical form ut = [a2 (¯ y)uy¯]y¯ + p2 (¯ y )f (u). However, dealing with the equation in the general form (2.6.2.1) is more convenient because it includes any canonical and noncanonical forms. Remark 2.51. In equations (2.6.2.1)–(2.6.2.4), the functions of a single variable a(x), b(x), c(x), and p(x) can be replaced with respective functions of two variables, a(x, t), b(x, t), c(x, t), and p(x, t).

Analysis and solutions of the determining system of equations. A direct procedure for constructing exact solutions to nonlinear equations of the form (2.6.2.1) suggests that the functions a(x), b(x), c(x), and f (u) are assumed given, while the functions u = u(x, t) and p = p(x) are unknown. In this case, once the functions k(ϕ) and s(ϕ) are defined somehow, one has first to find particular solutions p(x) and ϕ = ϕ(x, t) of equations (2.6.2.3) and (2.6.2.4); the last equation can be linearized (see below). Then, in view of relation (2.6.2.3), the solution to equation (2.6.2.1) is determined by formula (2.6.1.1), in which U (z) is a solution to the ordinary differential equation (2.6.2.5). In general, for given a = a(x), b = b(x), c = c(x), p = p(x), k(ϕ), and s(ϕ), equations (2.6.2.3) and (2.6.2.4) form an overdetermined nonlinear system of coupled equations for one function ϕ; this system will be called the determining system of equations. We will investigate the properties of equations (2.6.2.3) and (2.6.2.4) step by step.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

181

The nonlinear equation (2.6.2.3)p reduces to two simple first-order separable difp ferential equations: s(ϕ) ϕx = ± p(x)/a(x). Their general solutions are Z p Z p s(ϕ) dϕ = ± p(x)/a(x) dx + ξ(t), (2.6.2.7)

where ξ(t) is an arbitrary function. Therefore, in general, the function ϕ must have the form ϕ = G(y), y = ξ(t) + θ(x). (2.6.2.8) Note that solution (2.6.2.8) also admits another (though equivalent) representation ¯ y ), ϕ = G(¯

¯ θ(x), ¯ y¯ = ξ(t)

(2.6.2.9)

where y¯ = ey , ξ¯ = eξ , and θ¯ = eθ . It is not hard to check that the nonlinear transformations ϕ = F (ψ)

(2.6.2.10)

preserve the form of equations (2.6.2.3) and (2.6.2.4); in this case, the functional coefficients k(ϕ) and s(ϕ) are changed by the rules k(ϕ) =⇒ k(F (ψ))Fψ′ (ψ) +

′′ Fψψ (ψ) , ′ Fψ (ψ)

s(ϕ) =⇒ s(F (ψ))[Fψ′ (ψ)]2 . (2.6.2.11)

The degenerate case k(ϕ) ≡ 0 corresponds to a linear PDE with variable coefficients (2.6.2.4). For k(ϕ) 6≡ 0, the nonlinear PDE (2.6.2.4) can be reduced with the help of the substitution Z  Z ψ = C1 K(ϕ) dϕ + C2 , K(ϕ) = exp k(ϕ) dϕ , (2.6.2.12) which is a special transformation of the form (2.6.2.10) written implicitly, with C1 and C2 being arbitrary constants, to the linear PDE c(x)ψt = [a(x)ψx ]x + b(x)ψx .

(2.6.2.13)

In the special case k(ϕ) = k = const, to linearize equation (2.6.2.4), one can use the substitution ϕ = k −1 ln |ψ|, (2.6.2.14) which follows from (2.6.2.12). Solutions to a linear PDE with autonomous coefficients (2.6.2.13) can be constructed by the method of separation of variables. In particular, this equation has the following additive and multiplicative separable solutions: ψ = λt + η(x), ψ = exp(λt)ζ(x),

[a(x)ηx′ ]′x + b(x)ηx′ − λc(x) = 0; [a(x)ζx′ ]′x + b(x)ζx′ − λc(x)ζ = 0,

(2.6.2.15) (2.6.2.16)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where λ is an arbitrary constant. The equation for η in (2.6.2.15) is easy to integrate with the substitution w(x) = ηx . For solutions to the linear equation for ζ in (2.6.2.16) with various functions a(x), b(x), and c(x), see [273, 276]. Some other exact solutions to equation (2.6.2.13) for certain functions a(x), b(x), and c(x) can be found in [262]. Since transformations of the form (2.6.2.10) only change the functional coefficients k(ϕ) and s(ϕ) in equations (2.6.2.3) and (2.6.2.4), we can, without loss of generality, choose the function F so as to simplify one of these equations. Three possible ways of simplifying these equations are described below. 1◦ . For s(ϕ) = 1 and k = k(ϕ), from formula (2.6.2.7) one finds ϕ = ξ(t) + θ(x),

(2.6.2.17)

which corresponds to G(y) = y in (2.6.2.8). In this case, p(x) = a(x)(θx′ )2 . 2◦ . For s(ϕ) = 1/ϕ and k = k(ϕ), formula (2.6.2.7) gives ¯ θ(x), ¯ ϕ = ξ(t)

(2.6.2.18)

¯ y ) = y¯ in (2.6.2.9). which corresponds to G(¯ 3◦ . For s = s(ϕ) and k(ϕ) = 0, equation (2.6.2.4) is a linear PDE, whose solutions are constructed by the method of separation of variables. In what follows, the simplest representation of solutions of Item 1◦ will be used. Substituting (2.6.2.17) into equation (2.6.2.4) yields the functional differential equation c(x)ξt′ = [a(x)θx′ ]′x + b(x)θx′ + a(x)(θx′ )2 k(ϕ),

ϕ = ξ(t) + θ(x).

(2.6.2.19)

The intention is to find the admissible forms of k(ϕ) for which this equation can have solutions, using the differentiation method (see Section 2.4). To this end, we first divide equation (2.6.2.19) by c to rewrite it as ξt′ = Q(x) + R(x)k(ϕ),

ϕ = ξ(t) + θ(x),

(2.6.2.20)

where Q(x) = [(aθx′ )′x + bθx′ ]/c and R(x) = a(θx′ )2/c, and the arguments of the functions a, b, and c are omitted. Then, differentiating (2.6.2.20) with respect to t, ′′ we rewrite the resulting equation in the form ξtt /ξt′ = R(x)kϕ′ (ϕ). Taking the logarithm of this equation, differentiating with respect to t, and dividing by ξt′ , we ′′ obtain [ln(ξtt /ξt′ )]′t /ξt′ = [ln kϕ′ (ϕ)]′ϕ . On differentiating with respect to x, we get [ln kϕ′ (ϕ)]′′ϕϕ = 0.

(2.6.2.21)

This ODE has the solutions k(ϕ) = k1 ϕ + k2

(degenerate solution),

(2.6.2.22)

k(ϕ) = k1 e−k2 ϕ + k3

(nondegenerate solution),

(2.6.2.23)

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

183

where k1 , k2 , and k3 are arbitrary constants. Formulas (2.6.2.22) and (2.6.2.23) define all admissible functions k(ϕ) for which the functional differential equation (2.6.2.19) can have a solution. Formulas (2.6.2.17), (2.6.2.22), and (2.6.2.23) will be used in the subsequent sections to construct exact solutions of nonlinear reaction–convection–diffusion equations with autonomous coefficients (2.6.2.1). Construction of exact solutions for k(ϕ) = k and s(ϕ) = 1. Direct method of constructing exact solutions. In the simplest case, k(ϕ) = k = const, which corresponds to k1 = 0 and k2 = k in (2.6.2.22), substituting expression (2.6.2.17) into equation (2.6.2.19) gives ξ(t) = t (the constant factor is chosen equal to unity). It follows that in this case, the class of equations (2.6.2.1) admits functional separable solutions of the form (2.6.1.1), where Z ϕ(x, t) = t + g(x) dx. (2.6.2.24) The function g(x) = θx′ (x) can be specified by the researcher or determined in a subsequent analysis, depending on the goal (see below). Substituting (2.6.2.24) into equation (2.6.2.3) with s(ϕ) = 1 and equation (2.6.2.4) with k(ϕ) = k yields p(x) = a(x)g 2 (x), c(x) =

[a(x)g(x)]′x

(2.6.2.25) 2

+ b(x)g(x) + ka(x)g (x).

(2.6.2.26)

Relation (2.6.2.26) links the first three functional coefficients of equation (2.6.2.1) to the function g = g(x) of (2.6.2.24); this relation is differential with respect to a and g and algebraic with respect to b and c. Relation (2.6.2.25) is algebraic; it serves to determine the functional coefficient p(x). If the three functions a(x), b(x), and c(x) are assumed to be given, then relation (2.6.2.26) with k = 6 0 is a Riccati equation for g = g(x). Let us rewrite this ODE in the standard form: a(x)gx′ + ka(x)g 2 + [b(x) + a′x (x)]g − c(x) = 0.

(2.6.2.27)

An extensive list of exact solutions to equation (2.6.2.27) with various a(x), b(x), and c(x) can be found in [273, 276]. We will consider two cases. Degenerate case. For k = 0, the Riccati equation (2.6.2.27) degenerates into a linear ODE whose general solution is Z   Z  c(x) b(x) 1 E(x) dx+ C1 , E(x) = exp − dx , (2.6.2.28) g(x) = a(x) E(x) a(x) where C1 is an arbitrary constant. ◮ Example 2.52. In the case of constant coefficients, a = c = 1 and b = 0, formulas (2.6.2.28) with C1 = 0 give g(x) = x. Substituting this function into (2.6.2.24) and (2.6.2.25), where s(ϕ) = 1, we find that ϕ(x, t) = t+ 12 x2 , p(x) = x2 . It follows that the nonlinear reaction–diffusion equation

ut = uxx + x2 f (u)

(2.6.2.29)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

for arbitrary f (u) admits a functional separable solution u = U (z),

z = t+

x2 .

1 2

(2.6.2.30)

The function U (z) is described by the autonomous ordinary differential equation ′′ Uzz + f (U ) = 0.

(2.6.2.31)

It is obtained by substituting k = 0 and s = 1 into (2.6.2.5). The general solution of ◭ this equation can be represented in the implicit form (2.6.2.6). ◮ Example 2.53. Consider a more complicated situation where one of the equation coefficients depends on the spatial variable, a = a(x), in an arbitrary way and the other two are constants, b(x) = 0 and c(x) = 1. Formulas (2.6.2.28) with C1 = 0 give g(x) = x/a(x). Substituting this and (2.6.2.25), where R xfunction into (2.6.2.24) s(ϕ) = 1, we get ϕ(x, t) = t + a(x) dx and p(x) = x2/a(x). Therefore, the nonlinear reaction–diffusion equation [254]

ut = [a(x)ux ]x +

x2 f (u), a(x)

(2.6.2.32)

dependent on two arbitrary functions, a(x) and f (u), admits the functional separable solution Z x u = U (z), z = t + dx. (2.6.2.33) a(x)

The function U (z) is described by a solvable autonomous ordinary differential equation (2.6.2.31). Substituting a(x) = xn , a(x) = eλx , and a(x) = xeλx into (2.6.2.32) yields the nonlinear PDEs ut = (xn ux )x + x2−n f (u), λx

2 −λx

ut = (e ux )x + x e λx

ut = (xe ux )x + xe

f (u),

−λx

f (u),

(2.6.2.34) (2.6.2.35) (2.6.2.36)

which admit exact solutions for arbitrary f (u). Note that the ODE with variable coefficients ut = (xux )x + xf (u), which is a special case of equation (2.6.2.34) with n = 1, has a noninvariant traveling wave ◭ solution, u = U (x + t). Remark 2.52. Equation (2.6.2.32) and its solution were obtained in Subsection 2.5.3 from other considerations; see equation (2.5.3.17).

Nondegenerate case. For k = const (k 6= 0), the substitution g=

1 yx′ k y

(2.6.2.37)

converts equation (2.6.2.27) to the second-order linear ODE ′′ a(x)yxx + [b(x) + a′x (x)]yx′ − kc(x)y = 0.

(2.6.2.38)

An extensive list of exact solutions to this equation for various a(x), b(x), and c(x) can be found in [273, 276].

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

185

◮ Example 2.54. In the case of constant coefficients, a = c = 1 and b = 0, the general solution of equation (2.6.2.38) is expressed as ( C1 cosh(mx) + C2 sinh(mx) if k = m2 > 0, y= (2.6.2.39) C1 cos(mx) + C2 sin(mx) if k = −m2 < 0,

where C1 and C2 are arbitrary constants. Setting C1 = 1, C2 = 0, and k = 1 in (2.6.2.39) and using formula (2.6.2.37), we get g(x) = tanh x. Substituting this function into (2.6.2.24) and (2.6.2.25) gives ϕ(x, t) = t + ln cosh x,

p(x) = tanh2 x.

It follows that the nonlinear reaction–diffusion equation ut = uxx + tanh2 xf (u)

(2.6.2.40)

for arbitrary f (u) admits the functional separable solution u = U (z),

z = t + ln cosh x,

(2.6.2.41)

with the function U (z) described by the autonomous ordinary differential equation ′′ Uzz − Uz′ + f (U ) = 0.

(2.6.2.42)

The order of equation (2.6.2.42) can be reduced by one using the substitution Uz′ = Φ(U ), which leads to the Abel equation of the second kind in the canonical form. Exact solutions of equation (2.6.2.42) for some specific forms of f (U ) are ◭ available in [273, 276]. Table 2.7 shows nonlinear reaction–diffusion type equations of the form ut = uxx + p(x)f (u), where f (u) is an arbitrary function, that admit functional separable solutions of the form u = U (z), z = ϕ(x, t); the function ϕ is determined up to an additive constant. For equations 1, 2, and 4–7, the function ϕ(x, t) is the sum of functions with different arguments (2.6.2.24). A traveling wave solution (see equation 1) corresponds to a degenerate solution of equation (2.6.2.27) of the form g = α = const. Solutions to some equations of this type with more complicated functions p(x) can be obtained using formulas (2.6.2.39) from Example 2.54. The solution to equation 3 is selfsimilar (see Example 2.56). Other ways of constructing exact solutions. We now consider other possibilities for constructing exact solutions to equations of the form (2.6.2.1) for k(ϕ) = k and s(ϕ) = 1 without integrating the Riccati equation (2.6.2.27). To this end, we assume that g(x) and any two of the three functions a(x), b(x), and c(x) are given, and the remaining function will be found on the basis of (2.6.2.27). Table 2.8 describes the possible situations and provides formulas for determining the required function. The final form of the nonlinear reaction–convection–diffusion equation is determined by substituting the function p(x) = a(x)g 2 (x) into (2.6.2.1).

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Table 2.7. Nonlinear equations ut = uxx + p(x)f (u) that admit exact solutions of the form u = U (z), z = ϕ(x, t) (according to [254]). No.

Function p(x)

Function ϕ(x, t)

Equation for U = U (z)

1

1

t + αx

′′ α2 Uzz − Uz′ + f (U ) = 0

2

x2

3

x−2

xt−1/2

4

tanh2 (αx)

t + α−2 ln cosh(αx)

′′ Uzz − α2 Uz′ + α2 f (U ) = 0

5

coth2 (αx)

t + α−2 ln |sinh(αx)|

′′ Uzz − α2 Uz′ + α2 f (U ) = 0

6

tan2 (αx)

′′ + α2 Uz′ + α2 f (U ) = 0 Uzz

7

cot2 (αx)

t − α−2 ln |cos(αx)|

t+

1 2

x2

′′ Uzz + f (U ) = 0 ′′ Uzz +

t − α−2 ln |sin(αx)|

1 2

zUz′ + z −2 f (U ) = 0

′′ Uzz + α2 Uz′ + α2 f (U ) = 0

Notation: f (u) is an arbitrary function, α is an arbitrary constant (α 6= 0). Table 2.8. Different ways of specifying the functional coefficients of equation (2.6.2.1) with p(x) = a(x)g 2 (x) and equation (2.6.3.1) with p(x) = a(x)g(x). No.

Known (preset) functions

Unknown functions

1

a = a(x), b = b(x), g = g(x)

c(x) = agx′ + kag 2 + (b + a′x )g

b(x) = g −1 (c − agx′ ) − a′x − kag  R 3 b = b(x), c = c(x), g = g(x) a(x) = g −1 E (c − bg)E −1 dx + C1 R
Notation: k and C1 are arbitrary constants, g −1 = 1/g, E = exp −k g dx . 2

a = a(x), c = c(x), g = g(x)

◮ Example 2.55. We use the third way specified in Table 2.8, with b = 0 and c = 1, to obtain an alternative representation of the equations and their exact solutions. There are two possible cases.

1◦ . Degenerate case for k = 0. From row 3 of Table 2.8 with C1 = 0, we get a(x) = x/g(x) and p(x) = x2/a(x), which leads to equation (2.6.2.32). 2◦ . Nondegenerate case for Rk 6= 0. From row 3 of Table 2.8, where k 6= 0 and −1 −1 C1 = 0, we have R a(x) = g E E dx. We introduce a new function, h = h(x), by putting h = E −1 dx. Differentiating this expression and taking into account the
R formula E = exp −k g dx , we express g in terms of h. After simple calculations, we finally get g = k −1 h′′xx /h′x , a = kh/h′′xx , and p = k −1 hh′′xx /(h′x )2 . It follows that the nonlinear equation ut = [a(x)ux ]x + p(x)f (u),

a(x) = k

h , h′′xx

p(x) =

1 hh′′xx , (2.6.2.43) k (h′x )2

where f (u) and h = h(x) are arbitrary functions and k 6= 0 is an arbitrary constant,

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

187

admits the functional separable solution u = U (z),

z = t+

1 ln |h′x |. k

′′ The function U (z) is determined from the ordinary differential equation Uzz −kUz′ + f (U ) = 0. For example, on setting h = sinh(αx) and k = α2 in (2.6.2.43), we obtain equation 4 of Table 2.7. Substituting h = ln(αx) and k = −1 into (2.6.2.43), we arrive at the equation

ut = [x2 ln(αx)ux ]x + ln(αx)f (u), which admits an exact solution of the form u = U (z) with z = t + ln x.

(2.6.2.44) ◭

Direct construction of exact solutions for k(ϕ) 6= const. 1◦ . Case of k(ϕ) = k1 ϕ and s(ϕ) = 1. For k(ϕ) = k1 ϕ, which corresponds to k2 = 0 in (2.6.2.22), substituting expression (2.6.2.17) into equation (2.6.2.19) gives ξ(t) = eλt . It follows that the class of equations (2.6.2.1) admits functional separable solutions of the form (2.6.1.1) with ϕ(x, t) = eλt + θ(x).

(2.6.2.45)

Substituting (2.6.2.45) into relation (2.6.2.3) with s(ϕ) = 1 and equation (2.6.2.19) with k(ϕ) = k1 ϕ, we obtain c(x) =

k1 a(x)(θx′ )2 , λ

p(x) = a(x)(θx′ )2 .

(2.6.2.46)

In this case, the functions a(x) and b(x) remain arbitrary, while the function θ = θ(x) is determined by solving the ODE with a cubic nonlinearity [a(x)θx′ ]′x + b(x)θx′ + k1 a(x)θ(θx′ )2 = 0. (2.6.2.47)
R The substitution η = exp 12 k1 θ2 dθ reduces (2.6.2.47) to the linear equation [a(x)ηx′ ]′x + b(x)ηx′ = 0, whose general solution is defined as  Z  Z 1 b exp − dx dx + C2 . η = C1 a a 2◦ . Case of k(ϕ) = k1 e−k2 ϕ +k3 and s(ϕ) = 1. For k(ϕ) = k1 e−k2 ϕ +k3 , which corresponds to the use of (2.6.2.23), substituting expression (2.6.2.17) into equation (2.6.2.19) gives ξ(t) = k2−1 ln t. In this case, the class of equations (2.6.2.1) admits functional separable solutions of the form (2.6.1.1) with ϕ(x, t) =

1 ln t + θ(x). k2

(2.6.2.48)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting (2.6.2.48) into relation (2.6.2.3) with s(ϕ) = 1 and equation (2.6.2.19) with k(ϕ) = k1 e−k2 ϕ + k3 , we find that p(x) = a(x)(θx′ )2 ,

c(x) = k1 k2 a(x)e−k2 θ (θx′ )2 .

(2.6.2.49)

The functions a(x) and b(x) remain arbitrary, while θ = θ(x) is determined by solving the ODE with a quadratic nonlinearity [a(x)θx′ ]′x + b(x)θx′ + k3 a(x)(θx′ )2 = 0.

(2.6.2.50)

This equation is easy to integrate, since the substitution ζ(x) = θx′ converts it to a Bernoulli equation. In particular, for k3 = 0, the general solution of equation (2.6.2.50) is expressed as  Z  Z 1 b θ(x) = C1 exp − dx dx + C2 . a a ◮ Example 2.56. Let

a(x) = 1,

b(x) = 0,

k1 = − 21 ,

k2 = −2,

k3 = 1.

(2.6.2.51)

Then, equation (2.6.2.50) has a solution θ = ln x. Substituting this function into (2.6.2.48) and (2.6.2.49) and taking into account (2.6.2.51), we obtain ϕ(x, t) = − 12 ln t + ln x,

p(x) = x−2 ,

c(x) = 1.

(2.6.2.52)

Therefore, the equation ut = uxx + x−2 f (u)

(2.6.2.53)

admits the self-similar solution u = U (z),

z = − 21 ln t + ln x ≡ ln(xt−1/2 ),

(2.6.2.54)

with the function U (z) satisfying the ordinary differential equation ′′ Uzz + ( 12 e2z − 1)Uz′ + f (U ) = 0.

(2.6.2.55)

Note that in applications, an alternative representation of such solutions is usually used. It suggests the introduction of the self-similar variable z¯ = ez = xt−1/2 and reduction of equation (2.6.2.53) to the equation Uz¯′′z¯ + 12 z¯Uz¯′ + z¯−2 U = 0 (see row 3 ◭ in Table 2.7). ◮ Example 2.57. Equations (2.6.2.49) and (2.6.2.50) are satisfied if we set

a(x) = 1, b(x) = 0, c(x) = e−x , p(x) = 1, θ(x) = x, k1 = k2 = 1, k3 = 0. Hence, equation e−x ut = uxx + f (u) admits an exact solution u = U (z) with ◭ z = x + ln t. The results obtained for the single equation (2.6.2.1) can be extended to systems of equations [255]. Omitting the details, we will illustrate this with a specific system.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

189

◮ Example 2.58. The nonlinear system consisting of two reaction–diffusion type

PDEs ut = [a(x)ux ]x + [x2/a(x)]f (u, v), vt = [a(x)ux ]x + [x2/a(x)]g(u, v), which involves three arbitrary functions, a(x), f (u, v), and g(u, v), has the generalized traveling wave solution Z x dx u = U (z), v = V (z), z = t + . a(x) The functions U = U (z) and V = V (z) are described by the autonomous nonlinear system of ODEs ′′ Uzz + f (U, V ) = 0, ′′ Vzz + g(U, V ) = 0.

◭

Reaction–convection–diffusion equations with several spatial variables. The results obtained for PDE (2.6.2.1) can be generalized to the case of a multidimensional reaction–convection–diffusion equation with a nonlinear source   X N N X ∂ ∂u ∂u an + bn + pf (u), cut = ∂xn ∂xn ∂xn n=1 n=1

(2.6.2.56)

whose coefficients can depend on the spatial coordinates and time: an = an (x, t), bn = bn (x, t), c = c(x, t), p = p(x, t), x = (x1 , . . . , xN ), n = 1, . . . , N . We seek exact solutions to equation (2.6.2.56) in the form of a composition of functions u = U (z), z = ϕ(x, t). (2.6.2.57) We require that the coefficients of equation (2.6.2.56) and function ϕ are connected by two relations p = s(ϕ)

N X

n=1

an



∂ϕ ∂xn

2

,

(2.6.2.58)

   X 2 N N N X X ∂ϕ ∂ϕ ∂ϕ ∂ an + bn + k(ϕ) an , (2.6.2.59) cϕt = ∂xn ∂xn ∂xn ∂xn n=1 n=1 n=1 where s(ϕ) and k(ϕ) are some functions (s 6≡ 0). As a result, we arrive at an ordinary differential equation for U (z): ′′ Uzz − k(z)Uz′ + s(z)f (U ) = 0.

(2.6.2.60)

190

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Equations (2.6.2.58) and (2.6.2.59) preserve their form under any nonlinear transformation ϕ = F (ψ); in this case, the functional coefficients k(ϕ) and s(ϕ) change according to the rules (2.6.2.11). The transformation (2.6.2.12) converts the nonlinear PDE (2.6.2.59) to the linear PDE cψt =

  X N N X ∂ψ ∂ψ ∂ an + bn . ∂x ∂x ∂x n n n n=1 n=1

(2.6.2.61)

For equation (2.6.2.56) with autonomous coefficients an = an (x), bn = bn (x), c = c(x), and p = p(x), the solution of equation (2.6.2.58) has the form ϕ = G(y) with y = ξ(t) + θ(x). Without loss of generality, we set s(ϕ) = 1. Then, exact solutions to equations (2.6.2.58) and (2.6.2.59) can be found as functions with additive separation of variables ϕ = ξ(t) + θ(x). (2.6.2.62) On substituting (2.6.2.62) into equation (2.6.2.59), we obtain a functional differential equation that admits exact solutions for s(ϕ) in the forms (2.6.2.22) and (2.6.2.23). ◮ Example 2.59. It is easy to verify that for an = 1, bn = 0 (n = 1, . . . , N ), c = 1, k(ϕ) = 0, and s(ϕ) = 1, equations (2.6.2.58) and (2.6.2.59) can be satisfied if we put p = |x|2 , ϕ = N t + 12 |x|2 , where |x|2 = x21 + · · · + x2n . Therefore, the N -dimensional nonlinear reaction–diffusion equation

ut = ∆u + |x|2 f (u), where ∆ is the Laplace operator and f (u) is an arbitrary function, admits a functional separable solution of the form u = U (z),

z = Nt +

2 1 2 |x| .

′′ The function U (z) is described by autonomous ordinary differential equation Uzz + f (U ) = 0, whose general solution can be written in the implicit form (2.6.2.6). ◭

Remark 2.53. The functional coefficients an in equation (2.6.2.56) can have unlike signs. In particular, for a1 = −1, an > 0 for n = 2, . . . , N , and c = 0, equation (2.6.2.56) is a hyperbolic type equation.

2.6.3. Nonlinear Convection–Diffusion Type Equations The class of equations in question. We will look at nonlinear convection–diffusion equations with variable coefficients of the form c(x)ut = [a(x)ux ]x + [b(x) + p(x)f (u)]ux ,

(2.6.3.1)

where f (u) is an arbitrary function. Some of the four functional coefficients a = a(x) > 0, b = b(x), c = c(x) > 0, and p = p(x) can be treated as free, while the others can be expressed in terms of them, which can be done in several ways (see below).

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

191

Reduction of nonlinear convection–diffusion type equations to ODEs. Following [255], we seek exact solutions to equation (2.6.3.1) in the form of the composition of functions (2.6.1.1). Substituting (2.6.1.1) into (2.6.3.1) gives  ′′ a(x)ϕ2x Uzz + [a(x)ϕx ]x + b(x)ϕx − c(x)ϕt Uz′ + p(x)ϕx f (U )Uz′ = 0. (2.6.3.2) In the special case U (z) = z, equation (2.6.3.2) coincides with the original equation (2.6.3.1). At this stage, no solutions are lost. Let the coefficients of the equation satisfy the relations p(x) = a(x)s(ϕ)ϕx , c(x)ϕt = [a(x)ϕx ]x + b(x)ϕx +

(2.6.3.3) a(x)k(ϕ)ϕ2x ,

(2.6.3.4)

where s(ϕ) and k(ϕ) are some functions (s 6≡ 0). Then, equation (2.6.3.2) reduces to the ordinary differential equation ′′ Uzz + [s(z)f (U ) − k(z)]Uz′ = 0.

(2.6.3.5)

Exact solutions to the nonlinear ordinary differential equation (2.6.3.5) for some specific functions k(z), s(z), and f (U ) can be found in [273, 276]. In the special case of k(z) = k = const and s(z) = s = const, the general solution to equation (2.6.3.5) for any f (U ) can be written in implicit form: −1 Z  Z kU − s f (U ) dU + C1 dU = z + C2 ,

(2.6.3.6)

where C1 and C2 are arbitrary constants. Equations (2.6.3.3)–(2.6.3.5) allow one to construct exact solutions for a wide class of nonlinear convection–diffusion equations of the form (2.6.3.1). Analysis and solutions of the determining system of equations. The direct procedure for constructing exact solutions to nonlinear equations of the form (2.6.3.1) suggests that the functions a(x), b(x), c(x), and f (u) are assumed given, while the functions u = u(x, t) and p = p(x) are unknown and to be determined. In this case, with k(ϕ) and s(ϕ) given in some way, first one has to find particular solutions p(x) and ϕ = ϕ(x, t) of equations (2.6.3.3) and (2.6.3.4); the last equation can be linearized (see below). After this, in view of relation (2.6.3.3), a solution to equation (2.6.3.1) is determined by formula (2.6.1.1), where the function U (z) is a solution to the ordinary differential equation (2.6.3.5). In the general case, for given a = a(x), b = b(x), c = c(x), p = p(x), k(ϕ), and s(ϕ), equations (2.6.3.3) and (2.6.3.4) form an overdetermined nonlinear system of coupled equations for the single function ϕ; this system will be called the determining system of equations. The properties of equations (2.6.3.3) and (2.6.3.4) will be investigated below step by step. The general solution of equation (2.6.3.3) is given by the formula Z Z p(x) dx + ξ(t), (2.6.3.7) s(ϕ) dϕ = a(x)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where ξ(t) is an arbitrary function. Therefore, in general, the function ϕ must have the form ϕ = G(y), y = ξ(t) + θ(x). (2.6.3.8) Note that solution (2.6.3.8) also admits another (though equivalent) representation: ¯ y ), ϕ = G(¯

¯ θ(x), ¯ y¯ = ξ(t)

(2.6.3.9)

where y¯ = ey , ξ¯ = eξ , and θ¯ = eθ . It is not hard to check that the nonlinear transformations ϕ = F (ψ)

(2.6.3.10)

preserve the form of equations (2.6.3.3) and (2.6.3.4); in this case, the functional coefficients k(ϕ) and s(ϕ) are changed by the rule: k(ϕ) =⇒ k(F (ψ))Fψ′ (ψ) +

′′ Fψψ (ψ) , ′ Fψ (ψ)

s(ϕ) =⇒ s(F (ψ))Fψ′ (ψ).

(2.6.3.11)

For the subsequent presentation, it is important to note that equation (2.6.3.4) coincides with equation (2.6.2.4). The degenerate case k(ϕ) ≡ 0 corresponds to the linear PDE with variable coefficients (2.6.3.4). For k(ϕ) 6≡ 0, the nonlinear PDE (2.6.3.4) can be reduced with the help of the substitution (2.6.2.12) to the linear PDE (2.6.2.13). In the special case k(ϕ) = k = const, one can use the simple substitution (2.6.2.14), which follows from (2.6.2.12). Solutions to the linear PDE with autonomous coefficients (2.6.2.13) with k(ϕ) = 0 can be constructed by the method of separation of variables. In particular, this equation has the additive and multiplicative separable solutions (2.6.2.15) and (2.6.2.16). Since transformations of the form (2.6.3.10) only change the functional coefficients k(ϕ) and s(ϕ) in equations (2.6.3.3) and (2.6.3.4), one can choose the function F , without loss of generality, so as to simplify one of these equations. Three possible ways of simplifying these equations are described below. 1◦ . For s(ϕ) = 1 and k = k(ϕ), from formula (2.6.3.7) one finds ϕ = ξ(t) + θ(x),

(2.6.3.12)

which corresponds to G(y) = y in (2.6.3.8). In this case, p(x) = a(x)θx′ . 2◦ . For s(ϕ) = 1/ϕ and k = k(ϕ), formula (2.6.3.7) leads to ¯ θ(x), ¯ ϕ = ξ(t)

(2.6.3.13)

¯ y ) = y¯ in (2.6.3.9). which corresponds to G(¯ ◦ 3 . For s = s(ϕ) and k(ϕ) = 0, equation (2.6.3.4) is a linear PDE with autonomous coefficients, solutions of which are constructed by the method of separation of variables. In what follows, the simplest representation of the solution of Item 1◦ will be used. Substituting (2.6.3.12) into equation (2.6.3.4) yields the functional differential equation c(x)ξt′ = [a(x)θx′ ]′x + b(x)θx′ + a(x)(θx′ )2 k(ϕ),

ϕ = ξ(t) + θ(x),

(2.6.3.14)

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

193

which coincides with equation (2.6.2.19). By reasoning as in Subsection 2.6.2, we obtain the ordinary differential equation (2.6.2.21) for k = k(ϕ), which has the following solutions: k(ϕ) = k1 ϕ + k2 k(ϕ) = k1 e

−k2 ϕ

+ k3

(degenerate solution),

(2.6.3.15)

(nondegenerate solution),

(2.6.3.16)

where k1 , k2 , and k3 are arbitrary constants. Formulas (2.6.3.15) and (2.6.3.16) define all admissible functions k(ϕ) for which the functional differential equation (2.6.3.14) can have a solution. Below we will use formulas (2.6.3.12), (2.6.3.15), and (2.6.3.16) to construct exact solutions of nonlinear convection–diffusion equations with autonomous coefficients (2.6.3.1). Construction of exact solutions for k(ϕ) = k and s(ϕ) = 1. Direct method of constructing exact solutions. In the simplest case, k(ϕ) = k = const, which corresponds to k1 = 0 and k2 = k in (2.6.3.15), substituting expression (2.6.3.12) into equation (2.6.3.14) gives ξ(t) = t (the constant factor is chosen equal to unity). Therefore, the class of equations (2.6.3.1) admits functional separable solutions of the form (2.6.1.1), where Z ϕ(x, t) = t + g(x) dx. (2.6.3.17) The function g(x) = θx′ (x) can be prescribed by the researcher or determined in the subsequent analysis, depending on the goal (see below). Substituting (2.6.3.17) into equation (2.6.3.3) with s(ϕ) = 1 and equation (2.6.3.4) with k(ϕ) = k yields p(x) = a(x)g(x), c(x) =

[a(x)g(x)]′x

(2.6.3.18) 2

+ b(x)g(x) + ka(x)g (x).

(2.6.3.19)

Relation (2.6.3.19) links the first three functional coefficients of equation (2.6.3.1) to the function g = g(x) in (2.6.3.17) (this relation is differential with respect to a and g and algebraic with respect to b and c). Relation (2.6.3.18) is algebraic and serves to determine the functional coefficient p(x). If the functions a(x), b(x), and c(x) are given and g = g(x) is unknown, then (2.6.3.19) with k = 6 0 is a Riccati equation, which coincides with (2.6.2.27). A large number of exact solutions to this ODE for various a(x), b(x), and c(x) can be found in [273, 276]. Let us look at two cases. Degenerate case. For k = 0, the Riccati equation (2.6.3.19) degenerates into a linear equation whose general solution is Z   Z  c(x) b(x) 1 E(x) dx+C1 , E(x) = exp − dx , (2.6.3.20) g(x) = a(x) E(x) a(x) where C1 is an arbitrary constant.

194

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

◮ Example 2.60. In the case of constant coefficients, a = c = 1 and b = 0, using formulas (2.6.3.20) with C1 = 0, we get g(x) = x. Substituting this function into (2.6.3.17) and (2.6.3.18) gives ϕ(x, t) = t + 21 x2 and p(x) = x. It follows that the nonlinear convection–diffusion equation

ut = uxx + xf (u)ux

(2.6.3.21)

for arbitrary f (u) admits a functional separable solution u = U (z),

z = t+

1 2

x2 .

(2.6.3.22)

Here, the function U (z) is described by the autonomous ordinary differential equation ′′ Uzz + f (U )Uz′ = 0 (2.6.3.23) It is obtained by substituting k = 0 and s = 1 into (2.6.3.5) and its general solution ◭ can be represented implicitly by (2.6.3.6). ◮ Example 2.61. Consider a more complicated situation where one of the coefficients of the equation depends in an arbitrary way on the spatial variable, a = a(x), and the other two are constants, b(x) = 0 and c(x) = 1. Using formulas (2.6.3.20) with C1 = 0, we find that g(x) = x/a(x).R Substituting this into (2.6.3.17) and x dx and p(x) = x. Therefore, the (2.6.3.18) with s(ϕ) = 1 gives ϕ(x, t) = t + a(x) nonlinear convection–diffusion equation

ut = [a(x)ux ]x + xf (u)ux ,

(2.6.3.24)

dependent on two arbitrary functions, a(x) and f (u), admits the functional separable solution Z x u = U (z), z = t + dx, (2.6.3.25) a(x) where the function U (z) is described by the solvable autonomous ordinary differential equation (2.6.3.23). Substituting a(x) = xn and a(x) = eλx into (2.6.3.24) yields the nonlinear equations ut = (xn ux )x + xf (u)ux , λx

ut = (e ux )x + xf (u)ux ,

(2.6.3.26) (2.6.3.27)

which admit the exact solutions described above for arbitrary f (u). Interestingly, the equation ut = (xux )x + xf (u)ux , which is a special case of equation (2.6.3.26) with n = 1, admits a noninvariant traveling wave solution ◭ u = U (x + t). Nondegenerate case. For k = const (k 6= 0), the substitution g=

1 yx′ k y

(2.6.3.28)

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

195

converts the first-order nonlinear ODE (2.6.3.19) to the second-order linear ODE ′′ a(x)yxx + [b(x) + a′x (x)]yx′ − kc(x)y = 0.

(2.6.3.29)

An extensive list of exact solutions to this equation for various forms of the functions a(x), b(x), and c(x) can be found in [273, 276]. ◮ Example 2.62. In the case of constant coefficients, a = c = 1 and b = 0, the general solution of equation (2.6.3.29) is ( C1 cosh(mx) + C2 sinh(mx) if k = m2 > 0, y= (2.6.3.30) C1 cos(mx) + C2 sin(mx) if k = −m2 < 0,

where C1 and C2 are arbitrary constants. By setting C1 = 1, C2 = 0, and k = 1 in (2.6.3.30) and using formula (2.6.3.28), we find that g(x) = tanh x. Substituting this function into (2.6.3.17) and (2.6.3.18), we get ϕ(x, t) = t + ln cosh x,

p(x) = tanh x.

It follows that the nonlinear convection–diffusion equation ut = uxx + tanh xf (u)ux , with arbitrary f (u) admits the functional separable solution u = U (z),

z = t + ln cosh x.

where the function U (z) is described by the autonomous ordinary differential equation ′′ Uzz + [f (U ) − 1]Uz′ = 0. Its general solution is given by formula (2.6.3.6) with k = s = 1.

◭

Table 2.9 lists nonlinear convection–diffusion equations ut = uxx + p(x)f (u)ux , where f (u) is an arbitrary function, that admit functional separable solutions of the form u = U (z), z = ϕ(x, t); the function ϕ is determined to within an additive constant. For equations 1, 2, and 4–7, ϕ(x, t) is the sum of functions with different arguments (2.6.3.17). A traveling wave solution (see equation 1) corresponds to a degenerate solution g = α = const of equation (2.6.3.19) with a = c = 1 and b = 0. Solutions to some equations of this type with more complicated functions p(x) can be obtained using formulas (2.6.3.30) from Example 2.62. The solution to equation 3 is self-similar (see Example 2.64). Other ways of constructing exact solutions. We now look at other possibilities for constructing exact solutions of equations of the form (2.6.3.1), where k(ϕ) = k

196

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Table 2.9. Nonlinear of equations ut = uxx + p(x)f (u)ux that admit exact solutions of the form u = U (z), z = ϕ(x, t). No.

Function p(x)

Function ϕ(x, t) t + αx

Equation for U = U (z) 2

′′ α Uzz + [αf (U ) − 1]Uz′ = 0

1

1

2

x

3

x−1

xt−1/2

′′ Uzz + [ 12 z + z −1 f (U )]Uz′ = 0

4

tanh(αx)

t + α−2 ln cosh(αx)

′′ Uzz + [αf (U ) − α2 ]Uz′ = 0

5

coth(αx)

t + α−2 ln |sinh(αx)|

′′ Uzz + [αf (U ) − α2 ]Uz′ = 0

6

tan(αx)

t − α−2 ln |cos(αx)|

′′ Uzz + [αf (U ) + α2 ]Uz′ = 0

7

cot(αx)

t − α−2 ln |sin(αx)|

t+

1 2

x2

′′ Uzz + f (U )Uz′ = 0

′′ Uzz + [αf (U ) + α2 ]Uz′ = 0

Notation: f (u) is an arbitrary function, α is an arbitrary constant (α 6= 0).

and s(ϕ) = 1, without integrating the Riccati equation (2.6.3.19). To this end, we assume that g(x) and any two of the three functions a(x), b(x), and c(x) are given, and the remaining function will be found from (2.6.3.19). Table 2.8 lists the possible situations and provides formulas for determining the desired functions. The final form of the nonlinear convection–diffusion equation is determined by substituting the function p(x) = a(x)g(x) in (2.6.3.1). ◮ Example 2.63. Let us use the third way specified in Table 2.8 with b = 0 and c = 1, to get alternative representations of equations and their exact solutions. Two cases are possible.

1◦ . Degenerate case with k = 0. Using row 3 of Table 2.8 with C1 = 0, we find that a(x) = xg −1 (x) and p(x) = x, which leads to equation (2.6.3.24). 2◦ . Degenerate caseR with k 6= 0. For row 3 of Table 2.8 with k 6= 0 and C1 = 0, we have a(x)R = g −1 E E −1 dx. Let us introduce a new function, h = h(x), by setting h = RE −1 dx.
Differentiating this expression and taking into account that E = exp −k g dx , we express g through h. After simple calculations, we finally obtain g = k −1 h′′xx /h′x , a = kh/h′′xx, and p = h/h′x . It follows that the nonlinear convection–diffusion equation ut = [a(x)ux ]x + p(x)f (u)ux ,

a(x) = k

h , h′′xx

p(x) =

h , h′x

(2.6.3.31)

where f (u) and h = h(x) are arbitrary functions and k 6= 0 is an arbitrary constant, admits the functional separable solution u = U (z),

z = t+

1 ln |h′x |, k

′′ with the function U (z) determined by the autonomous ODE Uzz +[f (U )−k]Uz′ = 0, which is easy to integrate.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

197

Setting, for example, h = sinh(αx) and k = α2 in (2.6.3.31) and renaming f (U ) ◭ to αf (U ), we obtain equation 4 from Table 2.9. Direct construction of exact solutions for k(ϕ) 6= const. 1◦ . Case of k(ϕ) = k1 ϕ and s(ϕ) = 1. For k(ϕ) = k1 ϕ, which corresponds to k2 = 0 in (2.6.3.15), substituting expression (2.6.3.12) in equation (2.6.3.14) gives ξ(t) = eλt . Therefore, in this case, the class of equations (2.6.3.1) admits functional separable solutions of the form (2.6.1.1) with ϕ(x, t) = eλt + θ(x).

(2.6.3.32)

Substituting (2.6.3.32) into relation (2.6.3.3) with s(ϕ) = 1 and equation (2.6.3.14) with k(ϕ) = k1 ϕ, we obtain c(x) =

k1 a(x)(θx′ )2 , λ

p(x) = a(x)θx′ .

(2.6.3.33)

In this case, a(x) and b(x) remain arbitrary, and θ = θ(x) is determined by solving the ordinary differential equation [a(x)θx′ ]′x + b(x)θx′ + k1 a(x)θ(θx′ )2 = 0. (2.6.3.34)
R The substitution η = exp 21 k1 θ2 dθ reduces (2.6.3.34) to the simpler linear ODE [a(x)ηx′ ]′x + b(x)ηx′ = 0, the general solution of which is expressed as  Z  Z 1 b η = C1 exp − dx dx + C2 . a a 2◦ . Case of k(ϕ) = k1 e−k2 ϕ +k3 and s(ϕ) = 1. For k(ϕ) = k1 e−k2 ϕ +k3 , which corresponds to using the nondegenerate solution (2.6.3.16), substituting expression (2.6.3.12) into equation (2.6.3.14) gives ξ(t) = k2−1 ln t. In this case, the class of equations (2.6.3.1) admits functional separable solutions of the form (2.6.1.1) where ϕ(x, t) =

1 ln t + θ(x). k2

(2.6.3.35)

Substituting (2.6.3.35) into relation (2.6.3.3) with s(ϕ) = 1 and equation (2.6.3.14) with k(ϕ) = k1 e−k2 ϕ + k3 , we get c(x) = k1 k2 a(x)e−k2 θ (θx′ )2 ,

p(x) = a(x)θx′ .

(2.6.3.36)

The functions a(x) and b(x) remain arbitrary, while θ = θ(x) is determined by solving the nonlinear ODE [a(x)θx′ ]′x + b(x)θx′ + k3 a(x)(θx′ )2 = 0.

(2.6.3.37)

This equation is easy to integrate, since the substitution ζ(x) = θx′ converts it to a Bernoulli equation. In particular, for k3 = 0, the general solution of equation (2.6.3.37) is  Z  Z 1 b θ(x) = C1 exp − dx dx + C2 . a a

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

◮ Example 2.64. Let

a(x) = 1,

b(x) = 0,

k1 = − 21 ,

k2 = −2,

k3 = 1.

(2.6.3.38)

In this case, equation (2.6.3.37) has a solution θ = ln x. Substituting this function into formulas (2.6.3.35) and (2.6.3.36), and taking into account (2.6.3.38), we obtain ϕ(x, t) = − 12 ln t + ln x,

p(x) = x−1 ,

c(x) = 1.

(2.6.3.39)

Therefore, the equation ut = uxx + x−1 f (u)ux

(2.6.3.40)

admits the self-similar solution u = U (z),

z = − 21 ln t + ln x ≡ ln(xt−1/2 ),

where the function U (z) satisfies the ODE   ′′ Uzz + 12 e2z − 1 + f (U ) Uz′ = 0.

(2.6.3.41)

(2.6.3.42)

Note that an alternative representation of similar solutions is often used in applications. It is based on introducing the self-similar variable z¯ = ez = xt−1/2 and ′′ reducing PDE (2.6.3.40) to the ODE Uzz + [ 12 z + z −1 f (U )]Uz′ = 0 (see equation 3 ◭ in Table 2.9). ◮ Example 2.65. Relations (2.6.3.36) and equation (2.6.3.37) hold identically if

we set a(x) = 1, b(x) = 0, c(x) = e−x , p(x) = 1, θ(x) = x, k1 = k2 = 1, k3 = 0. Therefore, the equation e−x ut = uxx + f (u)ux admits an exact solution of the form ◭ u = U (z) with z = x + ln t. More complicated one-dimensional diffusion type equations. The proposition stated below allows one to construct exact solutions to more complex second-order PDEs. Proposition 1. Suppose ϕ = ϕ(x, t) is a solution to the parabolic equation with quadratic nonlinearity c(x, t)ϕt = [a(x, t)ϕx ]x + b(x, t)ϕx + ka(x, t)ϕ2x ,

(2.6.3.43)

where k is an arbitrary constant. Then the PDE with a more complex nonlinearity c(x, t)ut = [a(x, t)ux ]x + b(x, t)ux + a(x, t)ϕ2x F (ϕ, u, ux /ϕx ),

(2.6.3.44)

where F (ϕ, u, w) is an arbitrary function of three arguments, admits a functional separable solution of the form u = U (z),

z = ϕ(x, t).

(2.6.3.45)

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

199

The function U (z) is determined by solving the ODE ′′ Uzz − kUz′ + F (z, U, Uz′ ) = 0.

(2.6.3.46)

Thus, exact solutions to equation (2.6.3.43) generate related exact solutions to the nonlinear equation (2.6.3.44). Note that equation (2.6.3.43) is invariant under translation: ϕ ⇒ ϕ + const. Proposition 1 is proved by direct verification by substituting (2.6.3.45) into equation (2.6.3.44) while taking into account relation (2.6.3.43). Remark 2.54. For k = 0, we get the linear PDE (2.6.3.43). For k 6= 0, the substitution ϕ = k−1 ln |ψ| reduces the nonlinear equation (2.6.3.43) to the linear equation

c(x, t)ψt = [a(x, t)ψx ]x + b(x, t)ψx .

(2.6.3.47)

Thus, exact solutions to the linear equation (2.6.3.47) generate related exact solutions to the nonlinear equation (2.6.3.44). Exact solutions to equation (2.6.3.47) for certain functions a(x, t), b(x, t), and c(x, t) can be found in [262]. ◮ Example 2.66. Equation (2.6.3.43) holds identically if we put

Z

x dx . a(x) (2.6.3.48) On substituting (2.6.3.48) and F (ϕ, u, w) = f (u)w2 + g(u)w + h(u) into (2.6.3.44), we get the nonlinear PDE a(x, t) = a(x),

b(x, t) = 0,

c(x, t) = 1,

k = 0,

ut = [a(x)ux ]x + a(x)f (u)u2x + xg(u)ux +

ϕ(x, t) = t +

x2 h(u), a(x)

(2.6.3.49)

which involves four arbitrary functions, a(x), f (u), g(u), and h(u). From Proposition 1 it follows that equation (2.6.3.49) has the exact solution Z x dx u = U (z), z = t + , a(x) with the function U = U (z) described by autonomous ODE ′′ + f (U )(Uz′ )2 + g(U )Uz′ + h(U ) = 0. Uzz

◭

Proposition 1 admits a generalization to systems of coupled equations [255]. Omitting the details, we will illustrate this with a specific system. ◮ Example 2.67. The nonlinear system of coupled PDEs

ut = uxx + xf (u, v)vx , vt = vxx + xg(u, v)ux involving two arbitrary functions, f (u, v) and g(u, v), has an exact solution of the form u = U (z), v = V (z), z = t + 12 x2 .

200

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The functions U = U (z) and V = V (z) are described by the autonomous nonlinear system of ODEs ′′ Uzz + f (U, V )Vz′ = 0, ′′ Vzz + g(U, V )Uz′ = 0.

◭

Diffusion type equations with several spatial variables. Proposition 1 admits a multidimensional generalization. Proposition 2. Suppose ϕ = ϕ(x, t) is a solution to the parabolic equation with a quadratic nonlinearity c(x, t)ϕt = ∆ϕ + b(x, t) · ∇ϕ + k|∇ϕ|2 ,

(2.6.3.50)

where x = (x1 , . . . , xn ), ∆ is the Laplace operator, ∇ is the gradient operator, and k is an arbitrary constant. Then the PDE with a more complex nonlinearity c(x, t)ut = ∆u + b(x, t) · ∇u + |∇ϕ|2 F (ϕ, u, |∇u|/|∇ϕ|),

(2.6.3.51)

where F (ϕ, u, w) is an arbitrary function of three arguments, admits a functional separable solution of the form u = U (z),

z = ϕ(x, t).

(2.6.3.52)

The function U (z) is described by ODE (2.6.3.46). Proposition 2 is proved by direct verification by substituting function (2.6.3.52) into equation (2.6.3.51), while taking into account relation (2.6.3.50). Remark 2.55. For k 6= 0, the substitution ϕ = k −1 ln |ψ| reduces the nonlinear generating

equation (2.6.3.50) to the linear equation c(x, t)ψt = ∆ψ + b(x, t) · ∇ψ .

2.6.4. Nonlinear Klein–Gordon Type Equations and Nonlinear Telegraph Equations Preliminary remarks. The class of nonlinear PDEs in question. As already noted in Subsection 2.5.5, nonlinear Klein–Gordon type equations play an important role in relativistic quantum mechanics, field theory, nonlinear optics, plasma physics, and particle physics. Nonlinear telegraph type equations arise, for example, in studying unsteady processes in electrical transmission lines, migration of biological populations, and branching random walks [5, 89]. We look at nonlinear telegraph equations with variable coefficients of the form c(x)utt + d(x)ut = [a(x)ux ]x + b(x)ux + p(x)f (u),

(2.6.4.1)

where f (u) is an arbitrary function. Some of the five functional coefficients a = a(x), b = b(x), c = c(x), d = d(x), and p = p(x) can be treated as free, while the others will be expressed in terms of them; this can be done in different ways (see below). The nonlinear Klein–Gordon equation is a special case of equation (2.6.4.1) with b(x) ≡ d(x) ≡ 0.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

201

For c(x) ≡ 0, equation (2.6.4.1) degenerates into a nonlinear convection–diffusion equation with volume reaction. Reduction of nonlinear telegraph type equations to an ODE. Following [387], we seek exact solutions to equation (2.6.4.1) in the form of a composition of functions (2.6.1.1). Substituting (2.6.1.1) into (2.6.4.1) yields the equation  ′′   + [a(x)ϕx ]x − c(x)ϕtt a(x)ϕ2x − c(x)ϕ2t Uzz + b(x)ϕx − d(x)ϕt Uz′ + p(x)f (U ) = 0. (2.6.4.2) In the special case U (z) = z, equation (2.6.4.2) coincides with the original equation (2.6.4.1), which means that no solution is lost at this stage. Now let us require that the relations  p(x) = s(ϕ)[a(x)ϕ2x − c(x)ϕ2t , (2.6.4.3)  2 2 c(x)ϕtt + d(x)ϕt = [a(x)ϕx ]x + b(x)ϕx + k(ϕ)[a(x)ϕx − c(x)ϕt (2.6.4.4) hold, where s(ϕ) and k(ϕ) are some functions (s 6≡ 0). Then the variables separate and equation (2.6.4.2) reduces to the ordinary differential equation ′′ Uzz − k(z)Uz′ + s(z)f (U ) = 0.

(2.6.4.5)

For exact solutions to the nonlinear ordinary differential equation (2.6.4.5) with some specific functions k(z), s(z), and f (U ), see [273]. In the special case k(z) ≡ 0, in which equation (2.6.4.4) becomes linear, the general solution to equation (2.6.4.5) with s(z) = 1 and any f (U ) can be represented in implicit form: −1/2 Z  Z C1 − 2 f (U ) dU dU = C2 ± z, (2.6.4.6) where C1 and C2 are arbitrary constants. Equations (2.6.4.3)–(2.6.4.5) allow one to find exact solutions to telegraph equations of the form (2.6.4.1).

Remark 2.56. In equations (2.6.4.1) and (2.6.4.2)–(2.6.4.4), the functions a(x), b(x), c(x), d(x), and p(x) can be replaced with functions of two arguments: a(x, t), b(x, t), c(x, t), d(x, t), and p(x, t). All reasoning will remain the same.

Direct procedure for seeking exact solutions. In the generic case, equations (2.6.4.3) and (2.6.4.4) with prescribed functions a = a(x), b = b(x), c = c(x), d = d(x), p = p(x), k(ϕ), and s(ϕ) make up an overdetermined nonlinear system of equations for ϕ. It will be referred to as the determining system of equations. The nonlinear transformations ϕ = F (ψ)

(2.6.4.7)

preserve the form of equations (2.6.4.3) and (2.6.4.4); in this case, the functional coefficients k(ϕ) and s(ϕ) change by the rule k(ϕ) =⇒ k(F (ψ))Fψ′ (ψ) +

′′ Fψψ (ψ) , ′ Fψ (ψ)

s(ϕ) =⇒ s(F (ψ))[Fψ′ (ψ)]2 . (2.6.4.8)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The degenerate case k(ϕ) ≡ 0 corresponds to the linear hyperbolic equation with variable coefficients (2.6.4.4). If k(ϕ) 6≡ 0, the substitution Z  Z ψ = C1 K(ϕ) dϕ + C2 , K(ϕ) = exp k(ϕ) dϕ , (2.6.4.9) where C1 and C2 are arbitrary constants, linearizes equation (2.6.4.4): c(x)ψtt + d(x)ψt = [a(x)ψx ]x + b(x)ψx .

(2.6.4.10)

In the special case k(ϕ) = k = const, to linearize equation (2.6.4.4), one can use the substitution ϕ = k −1 ln |ψ|, (2.6.4.11) which follows from (2.6.4.9). Since transformations of the form (2.6.4.7) only change the functional coefficients k(ϕ) and s(ϕ) in equations (2.6.4.3) and (2.6.4.4), the function F can be chosen, without loss of generality, so as to simplify one of these equations. The direct procedure for constructing exact solutions to nonlinear equations of the form (2.6.4.1) assumes the functions a(x), b(x), c(x), d(x), and f (u) to be given and the functions u = u(x) and p = p(x) to be unknown. Then, by setting k(ϕ) and s(ϕ) in one way of another, one needs first to find particular solutions p(x) and ϕ = ϕ(x, t) to equations (2.6.4.3) and (2.6.4.4); recall that the latter equation can be linearized. After that, one obtains the corresponding solution to equation (2.6.4.1) by formula (2.6.1.1), with the function U (z) being a solution to the ordinary differential equation (2.6.4.5). Solutions to the linear equations (2.6.4.4), with k(ϕ) ≡ 0, and (2.6.4.10) can be obtained using the method of separation of variables. In particular, equation (2.6.4.10) with d(x) ≡ 0 admits the exact solutions ψ = αt2 + βt + ζ(x), ψ = (αe−λt + βeλt )ζ(x), ψ = [α cos(λt) + β sin(λt)]ζ(x),

[a(x)ζx′ ]′x + b(x)ζx′ − 2αc(x) = 0; (2.6.4.12) [a(x)ζx′ ]′x + b(x)ζx′ − λ2 c(x)ζ = 0; [a(x)ζx′ ]′x

+

b(x)ζx′

(2.6.4.13)

2

+ λ c(x)ζ = 0, (2.6.4.14)

where α, β, and λ are arbitrary constants. The equation in (2.6.4.12) is easy to integrate with the substitution w(x) = ζx′ . Solutions to the linear ODEs in (2.6.4.13) and (2.6.4.14) with various specific a(x), b(x), and c(x) can be found in [273]. For other exact solutions to linear PDEs of the form (2.6.4.10) with specific functional coefficients, see [262]. Without performing an exhaustive analysis of the overdetermined system of equations (2.6.4.3) and (2.6.4.4), we will show below how this system allows one to construct exact solutions to equations of the form (2.6.4.1), once suitable functional coefficients k(ϕ) and s(ϕ) are selected.

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

203

Case of k(ϕ) = k and s(ϕ) = 1. Generalized traveling wave solutions. The nonlinear telegraph equation (2.6.4.1) admits generalized traveling wave solutions (2.6.1.1) with Z ϕ(x, t) = t +

g(x) dx.

(2.6.4.15)

The function g(x) can be prescribed or determined in the subsequent analysis, depending on the goal (see below). On substituting (2.6.4.15) into (2.6.4.3) and (2.6.4.4) and on setting s(ϕ) = 1 and k(ϕ) = k = const, we get p(x) = a(x)g 2 (x) − c(x),

d(x) =

[a(x)g(x)]′x

(2.6.4.16) 2

+ b(x)g(x) + k[a(x)g (x) − c(x)].

(2.6.4.17)

Equation (2.6.4.17) relates the first four functional coefficients of equations (2.6.4.1) and the function g = g(x) appearing in (2.6.4.15); it is a differential relation with respect to a and g and an algebraic relation with respect to b, c, and d. Formula (2.6.4.16) serves to determine the functional coefficient p(x). If the four functions a(x), b(x), c(x), and d(x) are assumed to be given, relation (2.6.4.17) with k = 6 0 represents a Riccati equation for g = g(x), which can be written as a(x)gx′ + ka(x)g 2 + [b(x) + a′x (x)]g − kc(x) − d(x) = 0. (2.6.4.18) An extensive list of exact solutions to this equation for various specific a(x), b(x), c(x), and d(x) can be found in [273, 276]. Below we will look at two cases. Degenerate case. If k = 0, the Riccati equation (2.6.4.18) degenerates into a linear equation, whose general solution is Z   Z  d(x) b(x) E(x) dx + m , E(x) = exp − dx , (2.6.4.19) g(x) = a(x) E(x) a(x)

where m is an arbitrary constant. ◮ Example 2.68. Suppose that one of the equation coefficients is arbitrarily dependent on the space variable, a = a(x), while the other three are constant, b(x) = d(x) = 0 and c(x) = 1. By formula (2.6.4.19) we get g(x) = m/a(x). Inserting this R dx m2 and p(x) = a(x) − 1. into (2.6.4.15) and (2.6.4.16), we find that ϕ(x) = t + m a(x) Hence, the nonlinear Klein–Gordon type equation [387]  2  m utt = [a(x)ux ]x + − 1 f (u), (2.6.4.20) a(x)

which involves two arbitrary functions, a(x) and f (u), admits the two generalized traveling wave solutions Z dx u = U (z), z = t ± m , (2.6.4.21) a(x) with the function U (z) determined by the autonomous ODE ′′ Uzz + f (U ) = 0.

(2.6.4.22)

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This equation is obtained by substituting k = 0 and s = 1 into (2.6.4.5); its general solution can be represented in the implicit form (2.6.4.6). For instance, by setting a(x) = eλx in (2.6.4.20), we get the nonlinear equation utt = (eλx ux )x + (m2 e−λx − 1)f (u),

(2.6.4.23)

which admits two generalized traveling wave solutions for arbitrary f (u).

◭

◮ Example 2.69. Let us look at the case of a = a(x), b(x) = 0, and c(x) = d(x) = 1. By formulas (2.6.4.19) with m = 0, we get g(x) = x/a(x). Substituting R x dx x2 this into (2.6.4.15) and (2.6.4.16) gives ϕ(x, t) = t + a(x) and p(x) = a(x) − 1. It follows that the nonlinear telegraph equation [387]  2  x utt + ut = [a(x)ux ]x + − 1 f (u), (2.6.4.24) a(x)

involving two arbitrary functions, a(x) and f (u), admits the functional separable solution Z x dx u = U (z), z = t + , (2.6.4.25) a(x) where U (z) is a function determined by the solvable ODE (2.6.4.22).

◭

◮ Example 2.70. Now we set a = a(x), b = −ax (x), c(x) = 1, and d(x) = 0 and use formulas (2.6.4.19) to get g(x) = m. Substituting into (2.6.4.15) and (2.6.4.16) gives ϕ(x) = t + mx and p(x) = m2 a(x) − 1. It follows that the Klein–Gordon type equation utt = a(x)uxx + [m2 a(x) − 1]f (u), (2.6.4.26)

which involves two arbitrary functions, a(x) and f (u), admits two exact solutions u = U (z),

z = t ± mx,

(2.6.4.27)

with the function U (z) described by the solvable autonomous ODE (2.6.4.22).

◭

Remark 2.57. Solutions (2.6.4.27) are noninvariant traveling wave solutions to equation (2.6.4.26); they cannot be obtained using the classical group analysis. The function a(x) in equation (2.6.4.26) can be replaced with a(x, t).

Nondegenerate case. If k = const (k 6= 0), the change of variable g=

1 yx′ k y

(2.6.4.28)

converts the first-order nonlinear ODE (2.6.4.18) to the second-order linear ODE ′′ a(x)yxx + [b(x) + a′x (x)]yx′ − k[kc(x) + d(x)]y = 0.

(2.6.4.29)

An extensive list of exact solutions to this equation for various specific a(x), b(x), c(x), and d(x) can be found in [273].

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

205

◮ Example 2.71. In the case of a = c = 1 and b = d = 0, the general solution of equation (2.6.4.29) is

y = C1 cosh(kx) + C2 sinh(kx),

(2.6.4.30)

where C1 and C2 are arbitrary constants. By setting C1 = 1, C2 = 0, and k = 1 in (2.6.4.30) and using formula (2.6.4.28), we find that g(x) = tanh x. Inserting this into (2.6.4.15) and (2.6.4.16), we get ϕ(x) = t + ln cosh x,

p(x) = −1/cosh2 x.

It follows that the nonlinear Klein–Gordon type equation [387] utt = uxx −

1 f (u) cosh2 x

(2.6.4.31)

with arbitrary f (u) admits the functional separable solution u = U (z),

z = t + ln cosh x,

(2.6.4.32)

where U (z) is a function satisfying the autonomous ODE ′′ Uzz − Uz′ + f (U ) = 0.

(2.6.4.33)

The order of equation (2.6.4.33) can be reduced by one with the substitution Uz′ = Φ(U ), which leads to an Abel equation of the second kind. For functions f (U ) such that the general solution to equation (2.6.4.33) can be written in a closed form, ◭ see [273, 276]. Other ways of constructing exact solutions. We will now discuss other possibilities to construct exact solutions to equations of the form (2.6.4.1) with k(ϕ) = k and s(ϕ) = 1 without integrating the Riccati equation (2.6.4.18). To this end, we preset the function g(x) as well as any three out of the four functions a(x), b(x), c(x), and d(x) (treat these functions to be free), while the remaining function will be expressed in terms of them using (2.6.4.18). Table 2.10 lists possible situations and gives formulas that express the unknown functions in terms of the free ones. One should set p(x) = a(x)g 2 (x) − c(x) in the telegraph equation (2.6.4.1). ◮ Example 2.72. To obtain exact solutions to the equation in question, we use the fourth way of setting the functional coefficients (see Table 2.10) with b = d = 0 and c = 1. Two cases are possible.

1◦ . Degenerate case, k = 0. From row 4 of Table 2.10, we get a(x) = C1 g −1 (x) and p(x) = C1 g(x) − 1. Up to notation, this leads to equation (2.6.4.20) and its solution (2.6.4.21).

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Table 2.10. Different ways of setting the functional coefficients in (2.6.4.1) with p(x) = a(x)g 2(x) − c(x). No.

Preset free functions

Function expressed through preset ones

1

a = a(x), b = b(x), d = d(x), g = g(x)

c(x) = k−1[agx′ + kag 2 + (b + a′x)g − d]

2

a = a(x), c = c(x), d = d(x), g = g(x)

b(x) = g −1(kc + d − agx′ ) − a′x − kag

3

a = a(x), b = b(x), c = c(x), g = g(x)

4

d(x) = agx′ + kag 2 + (b + a′x)g − kc R  b = b(x), c = c(x), d = d(x), g = g(x) a(x) = g −1E (kc + d − bg)E −1dx + C1 R
Here, k and C1 are arbitrary constants, g −1 = 1/g, and E = exp −k g dx .

2◦ . Nondegenerate case, Rk 6= 0. From row 4 of Table 2.10 with kR 6= 0 and −1 C1 = 0, we get a(x) = kg −1 E E −1 dx. Introducing the function h(x) = R E
dx, computing its derivative, and taking into account that E = exp −k g dx , we express g in terms of h. On rearranging, we find that g = k −1 h′′xx /h′x , a = k 2 h/h′′xx , and p = h(h′x )−2 h′′xx − 1. It follows that the equation utt = [a(x)ux ]x + p(x)f (u),

a(x) = k 2

h , h′′xx

p(x) =

hh′′xx − 1, (2.6.4.34) (h′x )2

where f (u) and h = h(x) are arbitrary functions and k 6= 0 is an arbitrary constant, admits the functional separable solution u = U (z),

z = t+

1 ln |h′x |. k

′′ The function U (z) is determined by the autonomous ODE Uzz − kUz′ + f (U ) = 0. By setting, for instance, h = sinh x and k = 1 in (2.6.4.34), we obtain equation ◭ (2.6.4.31) and its exact solution of the form (2.6.4.32).

Case of d(x) = 0, k(ϕ) = k0 /ϕ, and s(ϕ) = s0 /ϕ. Functional separable solutions. If d(x) = 0, k(ϕ) = k0 /ϕ, and s(ϕ) = s0 /ϕ, the overdetermined system (2.6.4.3)–(2.6.4.4) admits solutions of the form ϕ(x, t) = θ(x) − (t + t0 )2 ,

(2.6.4.35)

where t0 is an arbitrary constant. The function θ = θ(x) is determined by the firstorder ODE a(x)(θx′ )2 = 4c(x)θ. (2.6.4.36) Equations (2.6.4.3) and (2.6.4.4) become p(x) = 4s0 c(x),

(2.6.4.37)

[a(x)θx′ ]′x + b(x)θx′ + (4k0 + 2)c(x) = 0.

(2.6.4.38)

Let us look at the special case b(x) = 0 in more detail. We assume the function c = c(x) to be given. Then the function p(x) is determined by formula (2.6.4.37),

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

207

while a = a(x) and ξ = ξ(x) are found from equations (2.6.4.36) and (2.6.4.38). Omitting the intermediate calculations, we obtain: if k1 = 4k0 + 2 6= 0, Z 4 1 +2 − 4 a(x) = I k1 , θ = C1 I k1 , I = C2 − k1 c(x) dx; (2.6.4.39) 4C1 c(x) if k0 = − 12 , a(x) =

C12 , 4C2 c(x)E(x)

  Z 4 E(x) = exp c(x) dx , (2.6.4.40) C1

θ = C2 E(x),

where C1 and C2 are arbitrary constants. and k0 = − 12 , relation (2.6.4.37) holds identically, while formulas (2.6.4.40) become     C12 4 4 a(x) = exp − x , θ(x) = C2 exp x . (2.6.4.41) 4C2 C1 C1 ◮ Example 2.73. With b(x) = 0, c(x) = p(x) = 1, s0 =

1 4,

On setting C1 = −4/λ and C2 = 4/λ2 in (2.6.4.41), we arrive at the nonlinear Klein– Gordon type equation utt = (eλx ux )x + f (u). (2.6.4.42) For arbitrary f (u), it admits a functional separable solution of the form u = U (z),

z = 4λ−2 e−λx − t2 ,

(2.6.4.43)

where U (z) is a function satisfying the nonautonomous ODE ′′ 4zUzz + 2Uz′ + f (U ) = 0.

◭

Remark 2.58. Solution (2.6.4.43) to equation (2.6.4.42) was obtained in [272].

◮ Example 2.74. Now we set a(x) = c(x) = p(x) = 1 and s0 =

1 4,

in which case equation (2.6.4.37) holds identically, while solutions to equations (2.6.4.36) and (2.6.4.38) are given by b(x) = −

2k0 , x + C1

θ(x) = (x + C1 )2 ,

where C1 is an arbitrary constant. To C1 = 0 and k0 = − 12 (n − 1) there corresponds an n-dimensional nonlinear Klein–Gordon equation with radial symmetry [272]: utt = uxx +

n−1 ux + f (u), x

where x is a radial coordinate. This PDE admits a solution of the form u = U (z) with z = x2 − (t + t0 )2 . ◭

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Case d(x) = 0. Solutions to the determining system in the form ϕ = ξ(x)t. We look for compatible solutions to the determining system (2.6.4.3)–(2.6.4.4) in the form ϕ = ξ(x)t. (2.6.4.44) A simple analysis shows that solution (2.6.4.44) satisfies both equation (2.6.4.3) and (2.6.4.4) if and only if the following relations hold: p = s0 cξ 2 ,

k(ϕ) =

k0 ϕ , A2 ϕ2 − 1

s(ϕ) =

s0 , A2 ϕ2 − 1

(2.6.4.45)

where A, k0 , and s0 are some constants. The functions a = a(x), b = b(x), c = c(x), and ξ = ξ(x) are connected by two differential-algebraic relations a(ξx′ )2 = A2 cξ 4 ,

(aξx′ )′x + bξx′ + k0 cξ 3 = 0.

(2.6.4.46)

Any two of the four functions in (2.6.4.46) can be treated as given (arbitrarily) and the other two as unknown. Let us look at the special case of b(x) = 0 and c(x) = 1. Eliminating a from (2.6.4.46), we arrive at the following ODE for ξ: ′′ ξξxx = (k0 A−2 + 4)(ξx′ )2 .

(2.6.4.47)

Equation (2.6.4.47) is autonomous and generalized homogeneous. Its general solution is  2  − A C1 (x + C2 ) k0 +3A2 if k0 6= −3A2 , (2.6.4.48) ξ=  C1 eλx if k0 = −3A2 ,

where C1 , C2 , and λ are arbitrary constants. The function a is expressed via ξ as a = A2 ξ 4 (ξx′ )−2 , which follows from the first equation of (2.6.4.46).

◮ Example 2.75. By setting A = C1 = s0 = 1, k0 = −3, λ = 1/2, b(x) = 0, and c(x) = 1 in (2.6.4.45) and (2.6.4.48), we get ξ(x) = ex/2 and a(x) = p(x) = ex . It follows that the nonlinear Klein–Gordon type equation

utt = (ex ux )x + ex f (u) with arbitrary f (u) admits an exact invariant solution of the form u = U (z) with ◭ z = ex/2 t.

2.6.5. Anisotropic Heat and Wave Equations with Three or More Independent Variables The equations and solutions described in Example 2.21 admit a number of generalizations. Omitting the details, we will present some results obtained in this direction [279, 301, 302, 376] (see also [275]).

2.6. General Functional Separation of Variables. Explicit Representation of Solutions

209

Nonlinear steady-state heat and diffusion equations in anisotropic media. 1◦ . The three-dimensional steady-state heat equation with power-law anisotropy and a nonlinear source (axk ux )x + (by m uy )y + (cz n uz )z = f (u) admits functional separable solutions of the form   x2−k y 2−m z 2−n 2 u = U (ζ), ζ = A + + , a(2 − k)2 b(2 − m)2 c(2 − n)2

(2.6.5.1)

(2.6.5.2)

where A is a free parameter and U (ζ) is a function satisfying the ODE   D ′ 1 1 1 4 ′′ Uζζ + Uζ = f (U ), D = 2 + + − 1. (2.6.5.3) ζ A 2−k 2−m 2−n For D = 0 and arbitrary kinetic function f (U ), the general solution to ODE (2.6.5.3) can be written in implicit form. The general solution to this equation can also be obtained for D = 1 and f (U ) = σeβU , where α and β are arbitrary constants (e.g., see [273, 276]). 2◦ . The three-dimensional steady-state heat equation with a nonlinear source and exponential anisotropy (aeβx ux )x + (beµy uy )y + (ceλz uz )z = f (u) has functional separable solutions of the form  −βx  e e−µy e−λz u = U (ζ), ζ 2 = A + + , aβ 2 bµ2 cλ2

(2.6.5.4)

(2.6.5.5)

where A is a free parameter and U (ζ) is a function satisfying the ODE ′′ Uζζ −

1 ′ 4 U = f (U ). ζ ζ A

(2.6.5.6)

3◦ . The three-dimensional steady-state heat equation with mixed anisotropy and a nonlinear source (axk ux )x + (by m uy )y + (ceλz uz )z = f (u) admits functional separable solutions of the form   x2−k y 2−m e−λz 2 u = U (ζ), ζ = A + + , a(2 − k)2 b(2 − m)2 cλ2 where A is a free parameter and U (ζ) is a function satisfying the ODE   D ′ 4 1 1 ′′ Uζζ + Uζ = f (U ), D = 2 + − 1. ζ A 2−k 2−m

(2.6.5.7)

(2.6.5.8)

(2.6.5.9)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

4◦ . A generalization of Items 1◦ to 3◦ . The nonlinear equation with q independent variables     p q X X ∂ ∂ ki ∂u βi xi ∂u ai xi + bi e = f (u) ∂xi ∂xi ∂xi ∂xi i=1 i=p+1

(2.6.5.10)

admits functional separable solutions of the form u = U (ζ),

X p ζ2 = A i=1

 q i X e−βi xi x2−k i + , ai (2 − ki )2 bi βi2 i=p+1

(2.6.5.11)

where A is a free parameter and U (ζ) is a function satisfying the ordinary differential equation ′′ Uζζ +

4 D ′ U = f (U ), ζ ζ A

D=

p X i=1

2 − 1. 2 − ki

(2.6.5.12)

Nonlinear wave equations in anisotropic media. 1◦ . The above results remain valid if the coefficients a, b, and c appearing in equations (2.6.5.1), (2.6.5.4), (2.6.5.7) and their solutions (2.6.5.2), (2.6.5.5), (2.6.5.8) have unlike signs. For abc < 0, equations (2.6.5.1), (2.6.5.4), and (2.6.5.7) are hyperbolic, in which case the sign of the free parameter A is chosen so as to ensure the condition ζ 2 > 0 in solutions (2.6.5.2), (2.6.5.5), and (2.6.5.8). Likewise, the coefficients ai and bi in equation (2.6.5.10) can have unlike signs. ◮ Example 2.76. To illustrate the above, we put c = −1, n = 0, and z = t in (2.6.5.1). As a result, we arrive at the nonlinear Klein–Gordon type equation in an anisotropic medium with two spatial variables

utt = (axk ux )x + (by m uy )y − f (u). It admits functional separable solutions of the form   y 2−m t2 x2−k + − , u = U (ζ), ζ 2 = A a(2 − k)2 b(2 − m)2 4

(2.6.5.13)

(2.6.5.14)

where A is a free parameter and U (ζ) is a function satisfying the ODE ′′ Uζζ +

D ′ 4 Uζ = f (U ), ζ A

D=

2 2 + . 2−k 2−m

(2.6.5.15)

One can set A = sign Ω in solution (2.6.5.14) and equation (2.6.5.15), where y 2−m t2 x2−k Ω = Ω(t, x, y) ≡ a(2−k) 2 + b(2−m)2 − 4 . It follows that A must have different signs in the spatio-temporal domains Ω(t, x, y) > 0 and Ω(t, x, y) < 0. ◭

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

211

2◦ . So far, we have looked at nonlinear PDEs with anisotropy dependent on spatial variables. Below we describe a few exact solutions to nonlinear wave equations in an inhomogeneous medium with anisotropic properties dependent arbitrarily on the unknown function u. The nonlinear wave type equation utt = [f (u)ux ]x + [g(u)uy ]y + [h(u)uz ]z , involving three arbitrary functions f (u), g(u), and h(u), has two exact solutions that can be written in implicit form [275]: xϕ1 (u) + yϕ2 (u) + zϕ3 (u) = ψ(u) + t, xϕ1 (u) + yϕ2 (u) + zϕ3 (u) = ψ(u) − t, where ϕ1 (u), ϕ2 (u), and ψ(u) are arbitrary functions, while the function ϕ3 (u) is determined from the normalization type condition f (u)ϕ21 (u) + g(u)ϕ22 (u) + h(u)ϕ23 (u) = 1. 3◦ . A generalization of Item 2◦ . The nonlinear wave type equation   n X ∂2u ∂ ∂u = fi (u) ∂t2 ∂xi ∂xi i=1 admits two exact solutions representable in the implicit form n X i=1

xi ϕi (u) = ψ(u) ± t,

(2.6.5.16)

where the functions ϕ1 (u), . . . , ϕn (u) and ψ(u) are related by one normalization type condition: n X fi (u)ϕ2i (u) = 1; i=1

this means that n functions in (2.6.5.16) can be chosen arbitrarily.

Remark 2.59. For many other exact solutions to nonlinear diffusion and wave type equations for an anisotropic medium, see [275].

2.7. General Functional Separation of Variables. Implicit Representation of Solutions 2.7.1. Description of the Method. The Generalized Splitting Principle Nonlinear transformations of special type. The generalized splitting principle. To be specific, we will look at nonlinear partial differential equations of mathematical

212

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

physics with two independent variables F (x, t, ux , ut , uxx , uxt , utt , . . . ) = 0,

(2.7.1.1)

where u = u(x, t) is the unknown function. To construct exact solutions of equation (2.7.1.1), we first introduce a new dependent variable, ϑ, related to the original unknown function u by a nonlinear transformation of the form [257, 299, 300] Z ϑ = ζ(u) du. (2.7.1.2) The functions ϑ = ϑ(x, t) and ζ = ζ(u) will be sought in the subsequent analysis. Once these functions are determined, the integral relation (2.7.1.2) will define an exact solution to the equation in question in implicit form. In rare case, the solution can be written out explicitly. Remark 2.60. The use of the nonlinear transformation (2.7.1.2) in the initial stage of the analysis is a significant generalization of the approach based on representing solutions in the implicit form (2.5.2.2). Indeed, on setting

ϑ = ξ(x)ω(t) + η(x),

ζ(u) = h(u),

in (2.7.1.2), we arrive at relation (2.5.2.2).

Differentiating (2.7.1.2) with respect to the independent variables, we find the partial derivatives ux =

ϑx ϑt ϑxx ϑ2x ζu′ ϑxt ϑx ϑt ζu′ , ut = , uxx = − 3 , uxt = − , . . . (2.7.1.3) ζ ζ ζ ζ ζ ζ3

We assume that after substituting expressions (2.7.1.3) into (2.7.1.1), the resulting equation can be converted to the form N X

Φn Ψn = 0,

(2.7.1.4)

n=1

where

Φn = Φn (x, t, ϑx , ϑt , ϑxx , . . . ), ′′ Ψn = Ψn (u, ζ, ζu′ , ζuu , . . . ).

(2.7.1.5)

To construct exact solutions of equation (2.7.1.4)–(2.7.1.5), we take advantage of the generalized splitting principle described below. Generalized splitting principle [257, 300]. Solutions to the equation in question are sought in the form various linear combinations of elements of two sets, {Φj } and {Ψj }, appearing in (2.7.1.4): N X

αni Φn = 0,

i = 1, . . . , l;

n=1 N X

n=1

(2.7.1.6) βnj Ψn = 0,

j = 1, . . . , m,

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

213

where 1 ≤ l ≤ N − 1 and 1 ≤ m ≤ N − 1. The constants αni and βnj in (2.7.1.6) are chosen so that the bilinear relation (2.7.1.4) holds identically (this can always be done as shown below). Importantly, relations (2.7.1.6) are purely algebraic in nature and are independent of any particular expressions of the differential forms (2.7.1.5). Once relations (2.7.1.6) are obtained, we substitute the differential forms (2.7.1.5) into them to arrive at systems of differential equations (often overdetermined) for the unknown functions ϑ = ϑ(x, t) and ζ = ζ(u) that appear in (2.7.1.2). Remark 2.61. The degenerate cases where, apart from linear relations (2.7.1.6), one or more differential forms Φn and/or Ψn vanish must be treated separately. Remark 2.62. Bilinear functional differential equations that are similar in appearance to (2.7.1.4)–(2.7.1.5) arise when one looks for exact solutions to nonlinear equations of mathematical physics using the methods of generalized and functional separation of variables with a preset solution structure (see Sections 1.5.1 and 2.5.2). However, there is a fundamental difference in this case: the differential forms Φn and Ψn in (2.7.1.5) depend, in view of transformation (2.7.1.2), on the same independent variables, x and t, whereas in the methods described previously, the differential forms depend on different independent variables. This circumstance significantly expands the possibilities for constructing exact solutions by switching to equivalent equations (see Subsection 2.7.2 for details). Remark 2.63. Instead of transformation (2.7.1.2), we can also use the transformation ϑ = Z(u), which leads to slightly more complex equations. The method for constructing functional separable solutions described above, based on transformation (2.7.1.2), is more convenient and often leads to lower-order differential equations than when exact solutions are sought in the explicit form (2.6.1.1). Remark 2.64. This approach, based on transformation (2.7.1.2) with ϑ = ϑ(x), can also be used to construct exact solutions of nonlinear ordinary differential equations with variable coefficients [260].

Some formulas ensuring that relation (2.7.1.4) holds identically. 1◦ . For any N , relation (2.7.1.4) can be satisfied if all Φi but one are put proportional to a selected element Φj (j 6= i). As a result, we get Φi = −Ai Φj ,

i = 1, . . . , j − 1, j + 1, . . . , N ;

Ψj = A1 Ψ1 + · · · + Aj−1 Ψj + Aj+1 Ψj+1 + · · · + AN ΨN ,

(2.7.1.7)

where Ai are arbitrary constants. In formula (2.7.1.7), the symbols can be swapped, Φn ⇄ Ψ n . 2◦ . For even N , relation (2.7.1.4) is satisfied if N/2 individual pairwise sums Φi Ψi + Φj Ψj vanish. In this case, we have the relations Φi − Aij Φj = 0,

Aij Ψi + Ψj = 0

(i 6= j),

(2.7.1.8)

where Aij are arbitrary constants, while the subscripts i and j together run through all values from 1 to N .

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

3◦ . For N ≥ 3, relation (2.7.1.4) also holds identically if we put Φm − Am ΦN −1 − Bm ΦN = 0,

m = 1, . . . , N − 2;

ΨN −1 + A1 Ψ1 + · · · + AN −2 ΨN −2 = 0,

(2.7.1.9)

ΨN + B1 Ψ1 + · · · + BN −2 ΨN −2 = 0,

where Ai and Bi are arbitrary constants. In formulas (2.7.1.9), the symbols can be swapped, Φn ⇄ Ψn , or simultaneous pairwise permutations Φi ⇄ Φj and Ψi ⇄ Ψj can be made. To construct more complex linear combinations of the form (2.7.1.6) that would identically satisfy the bilinear relation (2.7.1.4) for any N , one can use the formulas for the coefficients αni and βnj given in Subsection 1.5.2 (see formulas (1.5.2.9), where adequate renaming should be made).

2.7.2. Usage of Equivalent Equations. Simplification of Equations Generalizations based on using equivalent equations. One can obtain a number of other exact solutions to equation (2.7.1.1) if, instead of (2.7.1.4)–(2.7.1.5), one takes advantage of equivalent differential equations [251, 257]; these reduce to (2.7.1.4)– (2.7.1.5) on the set of functions satisfying relation (2.7.1.2). Below we indicate two classes of such equations, which will be used in Subsection 2.7.3. 1◦ . One can use the equations N X

n=1

e nΨ e n = 0, Φ

e n = Φn ηn (ϑ), Φ

en = Ψ

Ψn , ηn (Z)

Z=

Z

ζ(u) du, (2.7.2.1)

which preserve the bilinear structure and, by virtue of (2.7.1.2) implying that ϑ = Z, are equivalent to equation (2.7.1.4) for arbitrary ηn (ϑ). Equations (2.7.2.1) must be treated in conjunction with (2.7.1.5). 2◦ . One can use equations of a different form, G(x, t, u, ϑ) − G(x, t, u, Z) +

N X

Φn Ψn = 0,

(2.7.2.2)

n=1

which are equivalent to equation (2.7.1.4) for any G(x, t, u, ϑ). The function G can depend explicitly on ϑ and ζ (and their derivatives) as well as the functional coefficients of the original PDE, which means that G depends on the original variables x, t, and u explicitly. Equations (2.7.2.2) must be treated in conjunction with (2.7.1.5). The application of the generalized splitting principle to equation (2.7.1.4) and equivalent equations of the form (2.7.2.1) or (2.7.2.2) results generally in different exact solutions of the original equation (2.7.1.1).

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

215

PN Further generalizations are also possible. In particular, the sum n=1 Φn Ψn PN e nΨ e n , where the tilde quantities are dein (2.7.2.2) can be replaced with n=1 Φ fined in (2.7.2.1). The functions G(x, t, u, ϑ) and G(x, t, u, Z) can be multiplied by ηN +1 (ϑ)/ηN +1 (Z) and ηN +2 (ϑ)/ηN +2 (Z), respectively. For greater clarity, Fig. 2.2 displays the main steps of finding functional separable solutions using the generalized splitting principle.

Figure 2.2. The schematic of constructing functional P e e separable solutions using the generalized Φn Ψn = 0, can also be used instead of the splitting principle. Equivalent P equations, functional differential equation Φn Ψn = 0 (see Section 2.7.2).

Using transformation (2.7.1.2) to simplify equations. Transformation (2.7.1.2) can also be used to simplify nonlinear PDEs. To illustrate this, we consider the equation ut = auxx + f (u)u2x + b(x)g(u)ux + c(x)h(u),

(2.7.2.3)

where a is a constant. Transformation (2.7.1.2) takes equation (2.7.2.3) to the form ϑt = aϑxx + ϑ2x

  1 ζ′ f (u) − a u + b(x)g(u)ϑx + c(x)h(u)ζ. ζ ζ

(2.7.2.4)

216

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

To simplify (2.7.2.4), we set f (u) − a Whence

ζu′ = 0, ζ

 Z  1 ζ = exp f (u) du , a

g(u) = 1,

h(u)ζ = 1.

  Z 1 h(u) = exp − f (u) du . a

As a result, we obtain the nonlinear equation   Z 1 f (u) du , ut = auxx + f (u)u2x + b(x)ux + c(x) exp − a

(2.7.2.5)

where b(x), c(x), and f (u) are arbitrary functions, which can be reduced with the transformation  Z  Z 1 ϑ = exp f (u) du du (2.7.2.6) a

to the linear PDE

ϑt = aϑxx + b(x)ϑx + c(x).

(2.7.2.7)

Some exact solutions of this equation can be found in [262]. Remark 2.65. In equations (2.7.2.5) and (2.7.2.7), the functional coefficients a(x) and b(x) can be replaced with a(x, t) and b(x, t).

◮ Example 2.77. In the special case of a = 1 and f (u) = 1, equation (2.7.2.5)

becomes ut = uxx + u2x + b(x)ux + c(x)e−u , and transformation (2.7.2.6) can be rewritten in explicit form as u = ln ϑ. As a result, ◭ we obtain equation (2.7.2.7) with a = 1.

2.7.3. Nonlinear Reaction–Convection–Diffusion Equations The class of equations in question. Reduction to the bilinear form. We look at a wide class of nonlinear diffusion type equations with variable coefficients, ut = [a(x)f (u)ux ]x + b(x)g(u)ux + c(x)h(u),

(2.7.3.1)

which contain reaction and convective terms. Note that exact solutions to some simpler equations belonging to the class (2.7.3.1) can be found in numerous publications (e.g., see [42, 44, 45, 58, 62, 63, 65, 85, 86, 97, 107, 114–116, 119, 132, 135, 154, 155, 157, 159, 165, 176, 216, 223, 239, 250, 251, 253, 275, 278, 304, 314, 350, 351, 353, 362]). Remark 2.66. The reaction–convection–diffusion equation (2.7.3.1) with a(x) = 1 and b(x) = c(x) = −1 is also known as Richards’ equation; it is used to model the seepage of water through unsaturated soils. In R this equation, u is the volumetric water content, f = f (u) is the soil-water diffusivity, K = g(u) du is the hydraulic conductivity, and h = h(u) is the root water uptake rate [45, 217].

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

217

Using the method described in Subsection 2.7.1, we further obtain a lot of exact solutions to equations of the reasonably general form (2.7.3.1), in which at least two functional coefficients, a(x) and f (u), are given arbitrarily (and the others are expressed through them). In what follows, for brevity, we often omit the arguments of the functions appearing in transformation (2.7.1.2) and equation (2.7.3.1). Having made transformation (2.7.1.2), we substitute the derivatives (2.7.1.3) into (2.7.3.1). On rearranging, we get  ′ f −ϑt + (aϑx )x f + aϑ2x + bϑx g + chζ = 0. (2.7.3.2) ζ u For ζ = 1, equation (2.7.3.2) coincides with the original equation (2.7.3.1), where u = ϑ. Therefore, no solutions are lost at this stage. We introduce the notation Φ1 = −ϑt , Φ2 = (aϑx )x , Φ3 = aϑ2x ,

Ψ1 = 1,

Ψ2 = f,

Ψ3 =

(f /ζ)′u ,

Φ4 = bϑx , Φ5 = c; Ψ4 = g,

Ψ5 = hζ.

(2.7.3.3)

As a result, equation (2.7.3.2) can be represented in the bilinear form (2.7.1.4) with N = 5: 5 X Φn Ψn = 0. (2.7.3.4) n=1

Following [257], we will now proceed to construct exact solutions to nonlinear equations of the form (2.7.3.1) by using relations (2.7.3.3), (2.7.3.4), and (2.7.1.7)– (2.7.1.9) and applying the approach described in Subsection 2.7.1. Exact solutions obtained by analyzing equation (2.7.3.2). Solution 1. Equation (2.7.3.4) can be satisfied identically if we use the linear relations Φ1 = −Φ5 , Φ2 = 0, kΦ3 = −Φ4 ; (2.7.3.5) Ψ1 = Ψ5 , Ψ3 = kΨ4 , where k is an arbitrary constant. Substituting (2.7.3.3) into (2.7.3.5), we arrive at the equations ϑt = c, (aϑx )x = 0, kaϑ2x = −bϑx ; (2.7.3.6) hζ = 1, (f /ζ)′u = kg.

The solution of the overdetermined system consisting of the first three equations of (2.7.3.6) is Z dx b0 + C1 , b(x) = b0 , c(x) = c0 , (2.7.3.7) ϑ(x, t) = c0 t − k a(x) where a(x) is an arbitrary function, while b0 , c0 , and C1 are arbitrary constants. The solution to the system consisting of the last two equations in (2.7.3.6) can be written as Z kG(u) + C2 f h= , ζ= , G(u) = g(u) du, (2.7.3.8) f kG(u) + C2

218

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where f (u) and g(u) are arbitrary functions. From formulas (2.7.3.7) and (2.7.3.8) with b0 = c0 = 1 we obtain the equation ut = [a(x)f (u)ux ]x + g(u)ux +

kG(u) + C2 , f (u)

which admits the generalized traveling wave solution in implicit form Z Z 1 dx f (u) du =t− + C1 . kG(u) + C2 k a(x)

(2.7.3.9)

(2.7.3.10)

Note that equation (2.7.3.9) contains three arbitrary functions, a(x), f (u), and g(u), as well as two arbitrary constants, C2 and k. Solution 2. Equation (2.7.3.4) can be satisfied if we put Φ1 = −Φ4 , Φ2 = 0, kΦ3 = −Φ5 ; Ψ1 = Ψ4 , Ψ3 = kΨ5 ,

(2.7.3.11)

where k is an arbitrary constant. Substituting (2.7.3.3) into (2.7.3.11), we arrive at the equations ϑt = bϑx , (aϑx )x = 0, kaϑ2x = −c; (2.7.3.12) g = 1, (f /ζ)′u = khζ. The solutions of the first three equations (2.7.3.12) are Z kC12 λ dx a(x), c(x) = − + C2 , b(x) = , (2.7.3.13) ϑ(x, t) = λt + C1 a(x) C1 a(x) where a(x) is an arbitrary function and C1 , C2 , and λ are arbitrary constants. The last two equations in (2.7.3.12) give two functions g(u) = 1,

−1/2  Z , ζ(u) = ±f (u) 2k f (u)h(u) du + C3

(2.7.3.14)

where f = f (u) and h = h(u) are arbitrary functions, while C3 is an arbitrary constant. Setting C1 = 1 in (2.7.3.13) and (2.7.3.14), we get the equation ut = [a(x)f (u)ux ]x + λa(x)ux −

k h(u), a(x)

(2.7.3.15)

where a(x), f (u), and h(u) are arbitrary functions, while k and λ are arbitrary constants. This equation admits two exact solutions ±

Z

−1/2  Z Z du = λt + f (u) 2k f (u)h(u) du + C3

where C2 and C3 are arbitrary constants.

dx + C2 , a(x)

(2.7.3.16)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

219

Solution 3. Equation (2.7.3.4) holds identically if we set Φ1 = −k1 Φ5 , Ψ3 = 0,

Φ2 = −k2 Φ5 ,

Φ4 = −k3 Φ5 ;

Ψ5 = k1 Ψ1 + k2 Ψ2 + k3 Ψ4 ,

(2.7.3.17)

where k1 , k2 , and k3 are arbitrary constants. Substituting (2.7.3.3) into (2.7.3.17), we get ϑt = k1 c, (aϑx )x = −k2 c, bϑx = −k3 c; (2.7.3.18) (f /ζ)′u = 0, hζ = k1 + k2 f + k3 g. The solution to the overdetermined system consisting of the first three equations of (2.7.3.18) can be represented as Z Z x dx dx ϑ(x, t) = c0 k1 t − c0 k2 − C1 + C2 , a(x) a(x) (2.7.3.19) c0 k3 a(x) , c(x) = c0 , b(x) = c0 k2 x + C1 where a(x) is an arbitrary function and c0 , C1 , and C2 are arbitrary constants. From the last two equations of (2.7.3.18) we obtain h=

k1 g + k2 + k3 , f f

ζ = f,

(2.7.3.20)

where f = f (u) and g = g(u) are arbitrary functions. For c0 = k3 = 1, formulas (2.7.3.19) and (2.7.3.20) lead to the equation ut = [a(x)f (u)ux ]x +

k1 + k2 f (u) + g(u) a(x) g(u)ux + , k2 x + C1 f (u)

which has the generalized traveling wave solution Z Z Z x dx dx f (u) du = k1 t − k2 − C1 + C2 . a(x) a(x) Solution 4. Equation (2.7.3.4) holds if we set

Φ1 = −Φ5 , Φ2 = −kΦ4 ; Ψ1 = Ψ5 , kΨ2 = Ψ4 , Ψ3 = 0,

(2.7.3.21)

where k is an arbitrary constant. Substituting (2.7.3.3) into (2.7.3.21) yields ϑt = c, (aϑx )x = −kbϑx; hζ = 1, kf = g, (f /ζ)′u = 0.

(2.7.3.22)

The general solution to the overdetermined system consisting of the first two equations of (2.7.3.22) is given by ϑ(x, t) = c(x)t + s(x),   Z Z dx b dx + C2 , c(x) = C1 exp −k a a   Z Z dx b dx + C4 , s(x) = C3 exp −k a a

(2.7.3.23)

220

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where a = a(x) and b = b(x) are arbitrary functions, while C1 , . . . , C4 are arbitrary constants. The solution to the system consisting of the last three equations of (2.7.3.22) is 1 (2.7.3.24) g = kf, h = , ζ = f. f With relations (2.7.3.23) and (2.7.3.24), we obtain the equation ut = [a(x)f (u)ux ]x + kb(x)f (u)ux +

c(x) , f (u)

(2.7.3.25)

which admits the exact solution in implicit form Z f (u) du = c(x)t + s(x).

(2.7.3.26)

Here, a(x), b(x), and f (u) are arbitrary functions, while c(x) and s(x) are the functions defined in (2.7.3.23). ◮ Example 2.78. Setting C2 = λ, C1 = 0, and k = 1 in (2.7.3.23), (2.7.3.25), and (2.7.3.26), we get the equation

ut = [a(x)f (u)ux ]x + b(x)f (u)ux +

λ , f (u)

whose solution is  Z  Z Z b(x) dx f (u) du = λt + C3 exp − dx + C4 . a(x) a(x)

◭

Solution 5. Equation (2.7.3.4) can be satisfied by setting Φ1 + Φ2 + Φ4 = 0, Φ3 = −kΦ5 ; Ψ2 = Ψ1 , Ψ4 = Ψ1 , kΨ3 = Ψ5 ,

(2.7.3.27)

where k is an arbitrary constant. Substituting (2.7.3.3) into (2.7.3.27), we get −ϑt + (aϑx )x + bϑx = 0, f = g = 1,

k(f /ζ)u′

= hζ.

aϑ2x = −kc;

The first two equations of (2.7.3.28) admit the solution Z λ (ar)′x ϑ(x, t) = λt + r(x) dx + C1 , b = − , r r

(2.7.3.28)

c=−

ar2 , k

where a = a(x) and r = r(x) are arbitrary functions, while λ and C1 are arbitrary constants. From the last equation of (2.7.3.28) we get kζ −3 ζu′ = −h, which gives two solutions −1/2  Z 2 , h du + C2 ζ=± k where h = h(u) is an arbitrary function and C2 is an arbitrary constant.

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

221

Solution 6. Equation (2.7.3.4) holds if we set Φ1 = λΦ5 ,

Φ2 = k1 Φ5 ,

λΨ1 + k1 Ψ2 + Ψ5 = 0,

Φ4 = k2 Φ3 ; Ψ3 = −k2 Ψ4 ,

(2.7.3.29)

where k1 , k2 , and λ are arbitrary constants. Substituting (2.7.3.3) into (2.7.3.29), we obtain ϑt = −λc, (aϑx )x = k1 c, bϑx = k2 aϑ2x ; (2.7.3.30) λ + k1 f + hζ = 0, (f /ζ)′u = −k2 g. The solutions to the first three equations of (2.7.3.30) are expressed as Z Z x dx dx ϑ(x, t) = −λt + k1 + C1 + C2 , a(x) a(x) (2.7.3.31) b(x) = k2 (k1 x + C1 ), c(x) = 1, where a(x) is an arbitrary function, while C1 and C2 are arbitrary constants. The solutions to the last two equations of (2.7.3.30) are given by  Z   Z −1 k1 f + λ h= k2 g du + C3 , ζ = −f k2 g du + C3 , (2.7.3.32) f where f = f (u) and g = g(u) are arbitrary functions and C3 is an arbitrary constant. Setting k1 = k and k2 = 1 in (2.7.3.31) and (2.7.3.32), we arrive at the equation Z kf (u) + λ G(u), G(u) = g(u) du+C3 , ut = [a(x)f (u)ux ]x +(kx+C1 )g(u)ux + f (u) where a(x), f (u), and g(u) are arbitrary functions, while C1 , C3 , k, and λ are arbitrary constants. This equation admits the exact solution in implicit form Z Z Z dx f (u) x dx du = λt − k − C1 − C2 . (2.7.3.33) G(u) a(x) a(x) Solution 7. Equation (2.7.3.4) can be satisfied by setting Φ2 = k1 Φ5 , Ψ5 = −k1 Ψ2 ,

Φ3 = −k22 Φ1 , Ψ1 −

k22 Ψ3

Φ4 = −k3 Φ1 ;

− k3 Ψ4 = 0,

(2.7.3.34)

where k1 , k2 , and k3 are arbitrary constants. Substituting (2.7.3.3) in (2.7.3.34) yields (aϑx )x = k1 c, aϑ2x = k22 ϑt , bϑx = k3 ϑt ; (2.7.3.35) hζ = −k1 f, 1 − k22 (f /ζ)′u − k3 g = 0. The solutions to the first three equations of (2.7.3.35) can be represented as √ √ Z dx k3 √ k2 λ a′x √ , λa, c(x) = ϑ(x, t) = λt + k2 λ √ + C1 , b(x) = a k2 2k1 a (2.7.3.36)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where a = a(x) is an arbitrary constant, while C1 and λ are arbitrary constants. The solutions to the last two equations of (2.7.3.35) are given by   f k2 1 1 + 2 h′u , ζ = −k1 , (2.7.3.37) g= k3 k1 h where f = f (u) and h = h(u) are√arbitrary functions. Setting k1 = k3 = 1, k2 = 1/ λ, and C2 = −C in (2.7.3.36) and (2.7.3.37), we arrive at the equation ut = [a(x)f (u)ux ]x +

p 1 a′ (x) h(u), a(x) [λ + h′u (u)]ux + px 2 a(x)

(2.7.3.38)

which contains three arbitrary functions, a(x), f (u), and h(u), and has the exact solution Z Z dx f (u) du = −λt − p + C. (2.7.3.39) h(u) a(x) Solution 8. Equation (2.7.3.4) holds if we put Φ1 = −k1 Φ4 ,

Φ2 = −k2 Φ4 ,

Ψ4 = k1 Ψ1 + k2 Ψ2 ,

Ψ3 = Ψ5 ,

Φ3 = −Φ5 ;

(2.7.3.40)

where k1 and k2 are arbitrary constants. Substituting (2.7.3.3) into (2.7.3.40), we arrive at the equations ϑt = k1 bϑx , (aϑx )x = −k2 bϑx , g = k1 + k2 f, (f /ζ)′u = hζ.

aϑ2x = −c;

Integrating the first three equations of (2.7.3.41) gives Z k2 λ x + C1 ϑ(x, t) = λt − dx + C2 , k1 a(x) k 2 λ2 (x + C1 )2 a(x) , c(x) = − 2 2 , b(x) = − k2 (x + C1 ) k1 a(x)

(2.7.3.41)

(2.7.3.42)

where a(x) is an arbitrary function, while C1 , C2 , and λ are arbitrary constants. The last two equations of (2.7.3.41) give two solutions g(u) = k1 + k2 f (u),

−1/2  Z , (2.7.3.43) ζ(u) = ±f (u) 2 f (u)h(u) du + C3

where f = f (u) and h = h(u) are arbitrary functions and C3 is an arbitrary constant. Setting C1 = s, k1 = −1, k2 = k, and λ = k in (2.7.3.42) and (2.7.3.43), we obtain the equation ut = [a(x)f (u)ux ]x −

a(x) (x + s)2 [k + f (u)]ux − h(u), x+s a(x)

(2.7.3.44)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

223

where a(x), f (u), and h(u) are arbitrary functions, while k and s are arbitrary constants. This equation admits the exact solutions ±

Z

−1/2  Z Z x+s du = kt − f (u) 2 f (u)h(u) du + C3 dx + C2 , a(x)

where C2 and C3 are arbitrary constants. ◮ Example 2.79. In the special case of k = −1, f (u) = 1, and s = 0, equation (2.7.3.44) simplifies to become

ut = [a(x)ux ]x −

x2 h(u). a(x)

Up to renaming the function h, this equation coincides with (2.5.3.17).

◭

◮ Example 2.80. Setting h(u) = 0, C3 = 0, and s = 0 in (2.7.3.44) and renaming a(x) to xa(x), we obtain the equation

ut = [xa(x)f (u)ux ]x − a(x)[k + f (u)]ux , which has the solutions ±

Z

f (u) du = kt −

Z

dx + C2 . a(x)

◭

Solution 9. Equation (2.7.3.4) holds if we set Φ1 + Φ3 + k1 Φ4 + Φ5 = 0, Φ2 + k2 Φ4 = 0; Ψ3 = Ψ1 , Ψ4 = k1 Ψ1 + k2 Ψ2 , Ψ5 = Ψ1 ,

(2.7.3.45)

where k1 and k2 are arbitrary constants. Substituting (2.7.3.3) into (2.7.3.45), we obtain the equations −ϑt + aϑ2x + k1 bϑx + c = 0, (f /ζ)′u = 1,

g = k1 + k2 f,

(aϑx )x + k2 bϑx = 0; hζ = 1.

(2.7.3.46)

In the special case k1 = k2 = 0, the solution of system (2.7.3.46) leads to the equation   u β2 , (2.7.3.47) ut = [a(x)f (u)ux ]x + λ − a(x) f (u) where a(x) and f (u) are arbitrary functions, while β and λ are arbitrary constants. This equation admits two exact solutions Z Z dx f (u) du = λt ± β + C1 . (2.7.3.48) u a(x)

224

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Solution 10. Equation (2.7.3.4) can be satisfied if we put Φ1 = −k1 Φ5 ,

Φ2 = −k2 Φ3 ,

Ψ5 = k1 Ψ1 + k3 Ψ4 ,

Φ4 = −k3 Φ5 ;

Ψ3 = k2 Ψ2 ,

(2.7.3.49)

where k1 , k2 , and k3 are arbitrary constants. Substituting (2.7.3.3) into (2.7.3.49), we arrive at the equations ϑt = k1 c, (aϑx )x = −k2 aϑ2x , bϑx = −k3 c; hζ = k1 + k3 g, (f /ζ)′u = k2 f. The solutions to the first three equations of (2.7.3.50) are  Z  dx 1 ln k2 + C1 + C2 , ϑ(x, t) = k1 t + k2 a(x)  Z  dx b(x) = −k3 a(x) k2 + C1 , c(x) = 1, a(x)

(2.7.3.50)

(2.7.3.51)

where a(x) is an arbitrary function, while C1 and C2 are arbitrary constants. The solutions to the last two equations in (2.7.3.50) are given by   Z k1 + k3 g(u) h(u) = k2 f (u) du + C3 , f (u) (2.7.3.52) −1  Z ζ(u) = f (u) k2 f (u) du + C3 , where f (u) and g(u) are arbitrary functions and C3 is an arbitrary constant. In particular, setting a(x) = xn , C1 = C2 = 0, C3 = m, k1 = k, k2 = 1 − n, and k3 = 1 in (2.7.3.51) and (2.7.3.52), we obtain the equation   Z k + g(u) n (1 − n) f (u) du + m . ut = [x f (u)ux ]x − xg(u)ux + f (u) Solution 11. Equation (2.7.3.4) can be satisfied if we use the relations Φ3 = Φ1 , Φ4 = k1 Φ1 + k2 Φ2 , Φ5 = Φ1 ; Ψ1 + Ψ3 + k1 Ψ4 + Ψ5 = 0, Ψ2 + k2 Ψ4 = 0,

(2.7.3.53)

where k1 and k2 are arbitrary constants. Substituting (2.7.3.3) in (2.7.3.53) yields aϑ2x = −ϑt , 1+

(f /ζ)′u

bϑx = −k1 ϑt + k2 (aϑx )x ,

+ k1 g + hζ = 0,

c = −ϑt ;

f + k2 g = 0.

(2.7.3.54)

The first three equations (2.7.3.54) admit two solutions, which are given by Z √ 1 dx ϑ(x, t) = −t ± √ + C1 , b(x) = ±k1 a + k2 a′x , c(x) = 1, (2.7.3.55) 2 a

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

225

where a = a(x) is an arbitrary function and C1 is an arbitrary constant; in both formulas, the upper or lower signs are taken simultaneously. From the last equation of (2.7.3.54) we get g = −f /k2 . Then the penultimate equation, which serves to determine the function ζ, converts into the Abel equation of the second kind   k1 ξξu′ + 1 − f ξ + f h = 0, ζ = f /ξ. (2.7.3.56) k2 Setting k1 = ±k and k2 = 1 in (2.7.3.55) and (2.7.3.56), we obtain the equation   p ut = [a(x)f (u)ux ]x − k a(x) + 12 a′x (x) f (u)ux + h(u).

It has two generalized traveling wave solutions that can be represented in the implicit form Z Z f (u) dx + C1 , (2.7.3.57) du = −t ± p ξ(u) a(x) where the function ξ = ξ(u) is described by the Abel equation ξξu′ + [1 ∓ kf (u)]ξ + f (u)h(u) = 0. A large number of exact solutions to Abel equations for various functions f (u) and h(u) can be found in [273, 276]. Remark 2.67. Sometimes, it is quite helpful to make a transformation of the unknown function ϑ = ϑ(x, t) in equation (2.7.3.2).

Let us illustrate this remark with specific examples. Solution 12. We set a = b = c = 1 in (2.7.3.2) and then make the substitution ϑ = ϑ¯ + αx + βt, where α and β are free parameters, to obtain  ′ 2 f ¯ ¯ ¯ −ϑt + ϑxx f + (ϑx + α) + ϑ¯x g − β + αg + hζ = 0. ζ u

(2.7.3.58)

(2.7.3.59)

Below we give three solutions to equation (2.7.3.59), which lead to different solutions of the original PDE (2.7.3.1). 1◦ . A particular solution to equation (2.7.3.59) is sought in the form ϑ¯ = C1 eλt+γx + C2 ,

ζ = f,

(2.7.3.60)

where C1 and C2 are arbitrary constants. On substituting (2.7.3.60) into (2.7.3.59), we get C1 (−λ + γ 2 f + γg)eλt+γx − β + αg + hζ = 0. This relation holds identically if −λ + γ 2 f + γg = 0,

−β + αg + hζ = 0.

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Then, in view of the second relation in (2.7.3.60), we find that   λ αλ 1 g= − γf, h = αγ + β − , ζ = f. γ γ f Thus, we arrive at the equation     1 αλ λ ut = [f (u)ux ]x + − γf (u) ux + αγ + β − , γ γ f (u)

(2.7.3.61)

which depends on an arbitrary function, f = f (u), and admits the exact solution in implicit form Z f (u) du = αx + βt + C1 eλt+γx + C2 .

(2.7.3.62)

Setting λ/γ = σ, β − (αλ/γ) = µ, and αγ = ε in (2.7.3.61) and (2.7.3.62), we obtain the more compact equation ut = [f (u)ux ]x + [σ − γf (u)]ux + ε +

µ , f (u)

which has the exact solution   Z ε εσ t + C1 eγσt+γx + C2 . f (u) du = x + µ + γ γ 2◦ . For g ≡ 0, equation (2.7.3.59) has the steady-state particular solution ϑ¯ = −kx2 + C,

h=

β + 2k, f

ζ =f

(C and k are arbitrary constants),

which leads to the PDE ut = [f (u)ux ]x + 2k +

β . f (u)

(2.7.3.63)

NoteRthat equation (2.7.3.63) is a special case of equation (2.5.3.12) and has the solution f (u) du = −kx2 + αx + βt + C.

3◦ . For g ≡ 0 and α = 0, equation (2.7.3.59) has another steady-state particular solution Z f F ¯ , F = f (u) du, ϑ = ln(C1 x + C2 ), h = β , ζ = f F

which leads to a special case of equation (2.5.3.91). Solution 13. In (2.7.3.2), we set ζ = f and then make the transformation Z dx ϑ = ϑ¯ + βt + k , (2.7.3.64) a(x)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

227

where β and k are free parameters, to obtain −ϑ¯t + (aϑ¯x )x f + bϑ¯x g − β + k

b g + cf h = 0. a

(2.7.3.65)

¯ We are looking for a steady-state solution ϑ¯ = ϑ(x) to equation (2.7.3.65). In this case, using the splitting principle, we set g = −µf + λ, h = γ + (σ/f ), (aϑ¯′x )′x − µbϑ¯′x + cγ − kµ(b/a) = 0, bλϑ¯′ g + σ − β + kλ(b/a) = 0,

(2.7.3.66)

x

where µ, λ, γ, and σ are arbitrary constants. The system of equations (2.7.3.66) admits the solution β γ = kµ, σ = β, g(u) = −µf (u), h(u) = kµ + , f (u) Z eµx ¯ dx + C2 , b(x) = a(x), c(x) = 1, ϑ(x) = C1 a(x)

λ = 0,

where C1 , C2 , and µ are arbitrary constants. In view of the constraint (2.7.3.64), we get the equation ut = [a(x)f (u)ux ]x − µa(x)f (u)ux + σ +

β , f (u)

which admits the exact solution Z Z Z µx σ dx e f (u) du = βt + + C1 dx + C2 . µ a(x) a(x) Solution 14. We seek a particular solution to equation (2.7.3.65) as the product of functions with different arguments ϑ¯ = eλt ξ(x).

(2.7.3.67)

−λξ + (aξx′ )′x f + bξx′ g = 0, b −β + k g + cf h = 0. a

(2.7.3.68)

We arrive at the equations

For g = const, we find that f = const and h = const, which corresponds to a linear equation. Therefore, we further assume that g 6= const. The first equation of (2.7.3.68) holds if we put (aξx′ )′x − Abξx′ = 0,

Bbξx′ − λξ = 0,

g = B − Af,

(2.7.3.69)

228

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where A and B are arbitrary constants (A 6= 0). The first two equations of (2.7.3.69) involve three functions, a = a(x), b = b(x), and ξ = ξ(x), one of which can be set arbitrarily. Assuming that the function ξ = ξ(x) in (2.7.3.69) is given, we find that   Z 1 Aλ λξ a= ′ . (2.7.3.70) ξ dx + C1 , b = ξx B Bξx′ If we assume that b = b(x) is given, then the first two equations of (2.7.3.69) have the solutions  Z  Z    Z dx λ dx λ A exp dx + C1 , a(x) = b(x) exp − B b(x) B b(x)  Z  (2.7.3.71) λ dx ξ(x) = C2 exp , B b(x) where C1 and C2 are arbitrary constants (C2 6= 0). In particular, by setting B = 1 and b(x) = 1 in (2.7.3.71), we find that a(x) =

A + C1 e−λx , λ

ξ(x) = C2 eλx .

If B = 1 and b(x) = x, we get a(x) =

A x2 + C1 x1−λ , λ+1

ξ(x) = C2 xλ .

The last equation of (2.7.3.68) can be satisfied in two cases, which are considered below. 1◦ . For β = 0, the last equation in (2.7.3.68) has the solution c(x) = k

b(x) , a(x)

h(u) = A −

B . f (u)

These formulas are derived taking into account the last relation of (2.7.3.69). Thus, the equation   b(x) B ut = [a(x)f (u)ux ]x + b(x)[B − Af (u)]ux + k A− , a(x) f (u) where b(x) and f (u) are arbitrary functions and a = a(x) is expressed via b = b(x) by the first formula of (2.7.3.71), admits the solution   Z Z Z dx λ dx λt . + C2 e exp f (u) du = k a(x) B b(x) 2◦ . For k = 0, the last equation of (2.7.3.68) has the solution c(x) = 1,

h(u) =

β . f (u)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

229

As a result, we obtain the equation ut = [a(x)f (u)ux ]x + b(x)[B − Af (u)]ux +

β , f (u)

where b(x) and f (u) are arbitrary functions and a = a(x) is expressed via b = b(x) by the first formula of (2.7.3.71). This equation admits the solution  Z  Z λ dx f (u) du = βt + C2 eλt exp . B b(x) Solution 15. Equation (2.7.3.4) can be satisfied if we take all Φi (i = 1, 2, 3, 4) proportional to Φ5 . So we have Φ1 = k1 Φ5 ,

Φ2 = k2 Φ5 ,

Φ3 = k3 Φ5 ,

Φ4 = k4 Φ5 ,

k1 Ψ1 + k2 Ψ2 + k3 Ψ3 + k4 Ψ4 + Ψ5 = 0.

(2.7.3.72)

Substituting (2.7.3.3) into (2.7.3.72) yields ϑt = −k1 c, (aϑx )x = k2 c, aϑ2x = k3 c, k1 + k2 f + k3 (f /ζ)′u + k4 g + hζ = 0.

bϑx = k4 c;

(2.7.3.73)

Consider two cases. 1◦ . The simplest solution of the first four equations of (2.7.3.73) is given by a(x) = b(x) = c(x) = 1,

θ(x, t) = −k1 t + k4 x + C1 ,

k2 = 0,

k3 = k42 .

It leads to a traveling wave solution of the original equation (2.7.3.1); this solution will not be discussed here. 2◦ . The first four equations in (2.7.3.73) also admit a different solution a(x) = x2 ,

b(x) = x,

c(x) = 1,

ϑ(x, t) = −k1 t + k2 ln x + C1 ,

k3 = k22 ,

k4 = k2 .

(2.7.3.74)

Setting k = k1 and k2 = 1 in (2.7.3.74) and using the last equation in (2.7.3.73), we arrive at the reaction–diffusion type equation ut = [x2 f (u)ux ]x + xg(u)ux + h(u),

(2.7.3.75)

where h(u) = −

 ξ(u)  k + f (u) + g(u) + ξu′ (u) , f (u)

ξ(u) =

f (u) , ζ(u)

(2.7.3.76)

while f = f (u), g = g(u), and ξ = ξ(u) are arbitrary functions. This equation admits the exact solution Z f (u) du = −kt + ln x + C1 . (2.7.3.77) ξ(u)

230

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Remark 2.68. The invariant solution (2.7.3.77) to equation (2.7.3.75) can be sought in the explicit form u = U (z) with z = −kt + ln x, in which case relation (2.7.3.76) is not required. The function U (z) is described by the ODE

[f (U )Uz′ ]′z + [f (U ) + g(U ) + k]Uz′ + h(U ) = 0.

Solution 16. Equation (2.7.3.4) holds if we use the linear relations Φ1 = k1 Φ4 + k2 Φ5 , Ψ3 = k3 Ψ2 ,

Φ2 = −k3 Φ3 ;

Ψ4 = −k1 Ψ1 ,

Ψ5 = −k2 Ψ1 ,

(2.7.3.78)

where k1 , k2 , and k3 are arbitrary constants. This leads to the equations ϑt = −k1 bϑx − k2 c,

(f /ζ)′u = k3 f,

(aϑx )x = −k3 aϑ2x ;

g = −k1 ,

hζ = −k2 .

The solutions to the first two equations of (2.7.3.79) are  Z  1 dx ϑ(x, t) = λt + ln k3 + C1 + C2 , k3 a(x)  Z  k2 c(x) + λ dx b(x) = − a(x) k3 + C1 , k1 a(x)

(2.7.3.79)

(2.7.3.80)

where a(x) and c(x) are arbitrary functions, while C1 and C2 are arbitrary constants. From the last three equations of (2.7.3.79) we get  Z   Z −1 k2 h(u) = − k3 f (u) du + C3 , ζ(u) = f (u) k3 f (u) du + C3 . f (u) (2.7.3.81) Substituting C1 = C3 = 0 and k3 = 1 into (2.7.3.80) and (2.7.3.81), we obtain the equation Z  Z dx c(x) ut = [a(x)f (u)ux ]x + a(x)[c(x) + λ] ux − f (u), (2.7.3.82) a(x) f (u) which has the solution

Z

Z

dx , (2.7.3.83) a(x) where C4 is an arbitrary constant. Formula (2.7.3.83) was obtained using the relation −1 Z  Z Z f f (u) du + C3 du = ln f (u) du + C3 + const. f (u) du = C4 eλt

It is noteworthy that the diffusion term of equation (2.7.3.82) vanishes by virtue of solution (2.7.3.83), [a(x)f (u)ux ]x = 0, although uxx 6≡ 0. Seeking exact solutions with the help of equivalent equations. Using the considerations outlined in Subsection 2.7.2, we will now obtain a few other exact solutions to equation (2.7.1.1). To this end, instead of (2.7.1.4)–(2.7.1.5), we will consider equivalent differential equations, which reduce to (2.7.1.4)–(2.7.1.5) on the set of functions satisfying relation (2.7.1.2).

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

231

Solution 17. Let us revisit the class of reaction–diffusion equations of the form (2.7.3.1). Having made substitution (2.7.1.2), we consider, instead of (2.7.3.2), the more complex equation  ′ f λϑ −λZ 2 −e e ϑt + (aϑx )x f + aϑx + bϑx g + chζ = 0, (2.7.3.84) ζ u R where Z = ζ du and λ is an arbitrary constant. Equations (2.7.3.2) and (2.7.3.84) are equivalent, since, by virtue of transformation (2.7.1.2), the relation ϑ = Z holds. Equation (2.7.3.84) can be represented in the bilinear form (2.7.3.4) where Φ1 = −eλϑ ϑt , Φ2 = (aϑx )x , Φ3 = aϑ2x ,

Ψ1 = e

−λZ

,

Ψ2 = f,

Φ5 = c;

Ψ5 = hζ. (2.7.3.85) As previously, equation (2.7.3.4) can be satisfied using relations (2.7.3.5). Substituting (2.7.3.85) into (2.7.3.5), we arrive at the equations eλϑ ϑt = c, hζ = e−λZ ,

Ψ3 =

Φ4 = bϑx ,

(f /ζ)′u ,

(aϑx )x = 0, (f /ζ)′u = kg,

Ψ4 = g,

kaϑ2x = −bϑx ;

(2.7.3.86)

which coincide with (2.7.3.6) at λ = 0. The solution to the overdetermined system consisting of the first three equations of (2.7.3.86) is Z dx b0 1 + C1 , ϑ(x, t) = ln(t + C2 ) − λ k a(x)   (2.7.3.87) Z 1 dx b0 λ b(x) = b0 , c(x) = exp − + C1 λ , λ k a(x) where a(x) is an arbitrary function and b0 , C1 , C2 , k, and λ are arbitrary constants. The solution to the system consisting of the last two equations of (2.7.3.86) is written as   Z Z f (u) 1 ζ(u) = , h(u) = exp −λ ζ(u) du , G(u) = g(u) du, kG(u) + C2 ζ(u) (2.7.3.88) where f (u) and g(u) are arbitrary functions. Solution 18. Equation (2.7.3.4) can also be satisfied with relations (2.7.3.17). Substituting (2.7.3.85) into (2.7.3.17) yields eλϑ ϑt = k1 c, (f /ζ)′u

= 0,

(aϑx )x = −k2 c,

hζ = k1 e

−λZ

bϑx = −k3 c;

(2.7.3.89)

+ k2 f + k3 g.

The solution of the overdetermined system consisting of the first three equations (2.7.3.89) is 1 ln[k1 (λt + C1 )c(x)], λ   Z c(x) a(x) = ′ C2 − k2 λ c(x) dx , cx (x)

ϑ(x, t) =

b(x) = −

k3 λc2 (x) , c′x (x)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where c(x) is an arbitrary function (other than a constant), while C1 , C2 , and λ are arbitrary constants. The solutions to the last two equations of (2.7.3.89) are expressed as   Z   1 h(u) = k1 exp −λ f (u) du + k2 f (u) + k3 g(u) , ζ(u) = f (u), f (u) where f = f (u) and g = g(u) are arbitrary functions. Solution 19. As before, equation (2.7.3.4) holds if relations (2.7.3.21) are used. Substituting (2.7.3.85) into (2.7.3.21), we get the equations eλϑ ϑt = c, hζ = e

−λZ

,

(aϑx )x = −kbϑx ; kf = g,

(f /ζ)′u = 0.

(2.7.3.90)

The solution to the overdetermined system consisting of the first two equations of (2.7.3.90) can be written as   Z Z 1 b dx ϑ(x, t) = ln(t + C1 ) + C2 exp −k dx + C3 , λ a a   Z   Z b dx 1 dx + C3 λ , c(x) = exp C2 λ exp −k λ a a where a = a(x) and b = b(x) are arbitrary functions, while C1 , C2 , C3 , k, and λ are arbitrary constants. The solution to the system consisting of the last three equations of (2.7.3.90) is given by   Z 1 g = kf, h = exp −mλ f du , ζ = mf, mf where m 6= 0 is an arbitrary constant. Solution 20. Substituting (2.7.3.85) into (2.7.3.72), we arrive at the equations eλϑ ϑt = −k1 c, k1 e

−λZ

+ k2 f +

(aϑx )x = k2 c, k3 (f /ζ)′u

aϑ2x = k3 c,

bϑx = k4 c;

+ k4 g + hζ = 0.

(2.7.3.91)

The first four equations of system (2.7.3.91) admit a solution for exponential equation coefficients: a(x) = b(x) = c(x) = eλx , k1 = −

1 , λ

k2 = λ,

ϑ(x, t) =

1 ln t + x, λ

k3 = k4 = 1.

Using the last equation in (2.7.3.91), we arrive at the nonlinear PDE ut = [eλx f (u)ux ]x + eλx g(u)ux + eλx h(u),

(2.7.3.92)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

233

where   ′  1 1 −λZ f h(u) = − − e + λf + +g , ζ λ ζ u

Z=

Z

ζ du,

(2.7.3.93)

and f = f (u), g = g(u), and ζ = ζ(u) are arbitrary functions. Equation (2.7.3.92) admits the exact solution Z 1 (2.7.3.94) ζ(u) du = ln t + x. λ Remark 2.69. The invariant solution (2.7.3.94) to equation (2.7.3.92) can be sought in the explicit form u = U (z) with z = λ1 ln t + x, in which case relation (2.7.3.93) is not required. The function U (z) is described by the ODE

1 ′ U = [eλz f (U )Uz′ ]′z + eλz g(U )Uz′ + eλz h(U ). λ z

Solution 21. The first four equations of system (2.7.3.91) also admit an exact solution for power-law equation coefficients: a(x) = xn ,

b(x) = xn−1 ,

λ = n − 2,

k1 = −

c(x) = xn−2 ,

1 , n−2

k2 = n − 1,

ϑ(x, t) =

1 ln t + ln x, n−2

k3 = k4 = 1.

Using the last equation in (2.7.3.91), we arrive at the nonlinear PDE ut = [xn f (u)ux ]x + xn−1 g(u)ux + xn−2 h(u),

(2.7.3.95)

where    ′ 1 f 1 −(n−2)Z +g , e +(n−1)f + h(u) = − − ζ n−2 ζ u

Z=

Z

ζ du, (2.7.3.96)

while f = f (u), g = g(u), and ζ = ζ(u) are arbitrary functions. Equation (2.7.3.95) has the self-similar solution Z 1 ζ(u) du = ln t + ln x. (2.7.3.97) n−2 Remark 2.70. The self-similar solution (2.7.3.97) of equation (2.7.3.95) can be sought in the standard form u = U (z) with z = xt1/(n−2) , in which case relation (2.7.3.96) is not required. The function U (z) is described by the ODE

1 zUz′ = [z n f (U )Uz′ ]′z + z n−1 g(U )Uz′ + z n−2 g(U ). n−2

Solution 22. Equation (2.7.3.4) holds if we take all Ψi (i = 1, 3, 4, 5) to be proportional to Ψ2 . As a result, we get Ψ1 = k1 Ψ2 ,

Ψ3 = k2 Ψ2 ,

Ψ4 = k3 Ψ2 ,

k1 Φ1 + Φ2 + k2 Φ3 + k3 Φ4 + k4 Φ5 = 0.

Ψ5 = k4 Ψ2 ,

(2.7.3.98)

234

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting (2.7.3.85) into (2.7.3.98), we obtain the equations e−λZ = k1 f, −k1 e

λϑ

(f /ζ)′u = k2 f,

ϑt + (aϑx )x +

k2 aϑ2x

g = k3 f,

hζ = k4 f,

+ k3 bϑx + k4 c = 0.

(2.7.3.99)

The first four equations of system (2.7.3.99) admit an exact solution for exponential functional coefficients: f (u) = g(u) = h(u) = e−λu , k1 = k3 = k4 = 1,

ζ = 1,

Z = u;

k2 = −λ.

In this case, we obtain the reaction–diffusion type equation ut = [a(x)eβu ux ]x + b(x)eβu ux + c(x)eβu ,

λ = −β,

(2.7.3.100)

which has the additive separable solution u=−

1 ln t + η(x), β

(2.7.3.101)

with the function η = η(x) described by the ODE −

1 = [a(x)eβη ηx′ ]′x + b(x)eβη ηx′ + c(x)eβη . β

(2.7.3.102)

Equations (2.7.3.100) and (2.7.3.102) contain three arbitrary functions, a(x), b(x), and c(x). It is noteworthy that equation (2.7.3.102) reduces with the substitution ξ = eβη to the linear second-order ODE [a(x)ξx′ ]′x + b(x)ξx′ + βc(x)ξ + 1 = 0. Solution 23. The first four equations of system (2.7.3.99) also admit a solution for power-law functional coefficients: f (u) = un , λ = −n,

g(u) = un ,

h(u) = un+1 ,

k1 = k3 = k4 = 1,

ζ(u) = 1/u,

Z = ln u,

k2 = n + 1.

In this case, the solution to the last equation in (2.7.3.99) is expressed as ϑ = −(1/n) ln t + η(x), with the function η = η(x) satisfying the ODE 1 −nη e + (aηx′ )′x + (n + 1)a(ηx′ )2 + bηx′ + c = 0. n As a result, we get the nonlinear PDE ut = [a(x)un ux ]x + b(x)un ux + c(x)un+1 ,

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

235

the exact solution of which can be represented as the product of functions with different arguments, u = t−1/n ξ(x), with the function ξ(x) = eη described by the ODE 1 [a(x)ξ n ξx′ ]′x + b(x)ξ n ξx′ + c(x)ξ n+1 + ξ = 0. n Solution 24. Let us revisit the class of reaction–diffusion equations of the form (2.7.3.1). Having made the substitution (2.7.1.2), we consider, instead of (2.7.3.2), the more complicated equation −ϑt + (aϑx )x f + aϑ2x



f ζ

′

+ bϑx g + chζ

u

ϑ = 0, Z

(2.7.3.103)

R where Z = ζ du. Equations (2.7.3.2) and (2.7.3.103) are equivalent, since, by virtue of transformation (2.7.1.2), the relation ϑ = Z holds. Equation (2.7.3.103) can be represented in the bilinear form (2.7.3.4) where Φ1 = −ϑt, Φ2 = (aϑx)x, Φ3 = aϑ2x, Φ4 = bϑx, ′ Ψ1 = 1, Ψ2 = f, Ψ3 = (f /ζ)u, Ψ4 = g,

Φ5 = cϑ; (2.7.3.104) Ψ5 = hζ/Z.

Equation (2.7.3.4) holds identically if relations (2.7.3.17) are used. Substituting (2.7.3.104) into (2.7.3.17), we get ϑt = k1 cϑ, (f /ζ)′u

= 0,

(aϑx )x = −k2 cϑ,

bϑx = −k3 cϑ;

hζ/Z = k1 + k2 f + k3 g,

(2.7.3.105)

where k1 , k2 , and k3 are arbitrary constants. Let a = a(x), f = f (u), and g = g(u) be arbitrary functions. Then the solutions of equations (2.7.3.105) are given by b(x) = −

k3 λ ω , k1 ωx′

c(x) =

1 h = (k1 + k2 f + k3 g)F, f

λ = const, k1 ζ = f,

ϑ(x, t) = eλt ω(x), Z F = f du,

(2.7.3.106)

where λ is an arbitrary constant, and the function ω = ω(x) solves the linear secondorder ODE (aωx′ )′x = −(k2 λ/k1 )ω. In the special case of a(x) = const and k3 = 0, formulas (2.7.3.106) lead to a nonlinear reaction–diffusion equation and its solution, which are given in [275]. Solution 25. Consider the special case a = b = c = 1,

ζ = f.

(2.7.3.107)

We look for a solution to equation (2.7.3.103) under conditions (2.7.3.107) in the form ϑ = (γx + δ)eαx+βt . (2.7.3.108)

236

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting (2.7.3.108) into (2.7.3.103) and taking into account (2.7.3.107), we obtain   γxeαx+βt −β + α2 f + αg + (f /F )h   + eαx+βt −βδ + (α2 δ + 2αγ)f + (αδ + γ)g + δ(f /F )h = 0, (2.7.3.109)

R where F = f du. Equating the expressions in square brackets in (2.7.3.109) with zero, we arrive at the equations −β + α2 f + αg + (f /F )h = 0,

−βδ + (α2 δ + 2αγ)f + (αδ + γ)g + δ(f /F )h = 0. Solving these equations for g and h gives   β F. g = −2αf, h = α2 + f As a result, we arrive at the equation 

β ut = [f (u)ux ]x − 2αf (u)ux + α + f (u) which has the exact solution Z

2

Z

f (u) du,

f (u) du = (γx + δ)eαx+βt ,

where γ and δ are arbitrary constants. Solution 26. We look for a solution to equation (2.7.3.103) under conditions (2.7.3.107) in the form ϑ = Aeαx+βt + Beγx+δt . Omitting the intermediate calculations, we arrive at the equation   δ−β ut = [f (u)ux ]x + − (α + γ)f (u) ux γ−α Z  βγ − αδ 1 f (u) du, + αγ + γ − α f (u) which has the solution Z

f (u) du = Aeαx+βt + Beγx+δt ,

where A and B are arbitrary constants.

(2.7.3.110)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

237

◮ Example 2.81. In the special case of γ = −α and δ = β, equation (2.7.3.110) simplifies and takes the form  Z β 2 ut = [f (u)ux ]x + −α + f (u) du; f (u)

its solution is written as

Z

f (u) du = eβt (Aeαx + Be−αx ). ◭

Solution 27. Assuming that conditions (2.7.3.107) hold, we look for a solution to equation (2.7.3.103) in the form ϑ = Aeαt sin(βx + σt + δ), where A and δ are arbitrary constants. On rearranging, we obtain the equation Z  α f (u) du, ut = [f (u)ux ]x + γux + β 2 + f (u) where γ = σ/β, which admits the exact solution Z f (u) du = Aeαt sin(βx + βγt + δ). Remark 2.71. In the case (2.7.3.107), equation (2.7.3.103) also admits a more complicated solution of the form

ϑ = Aeµx+αt sin(βx + σt + δ),

which we do not discuss here.

Solution 28. Instead of (2.7.3.103), we can look at the more complex equation  ′ ϑn f ϑ − n ϑt + (aϑx )x f + aϑ2x = 0, (2.7.3.111) + bϑx g + chζ Z ζ u Z R where Z = ζ du and n is an arbitrary constant. Equations (2.7.3.2) and (2.7.3.111) are equivalent, since, by virtue of transformation (2.7.1.2), the relation ϑ = Z holds. Equation (2.7.3.111) can be represented in the bilinear form (2.7.3.4) where Φ1 = −ϑnϑt, Φ2 = (aϑx)x, Φ3 = aϑ2x,

Ψ1 = Z

−n

,

Ψ2 = f,

Φ4 = bϑx, Φ5 = cϑ;

Ψ3 = (f /ζ)′u,

Ψ5 = hζ/Z. (2.7.3.112) Equation (2.7.3.4) holds identically if relations (2.7.3.17) are used. Substituting (2.7.3.112) into (2.7.3.17) yields ϑn ϑt = k1 cϑ, (f /ζ)′u

= 0,

(aϑx )x = −k2 cϑ,

hζ/Z = k1 Z

−n

Ψ4 = g,

bϑx = −k3 cϑ;

+ k2 f + k3 g,

(2.7.3.113)

238

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where k1 , k2 , and k3 are arbitrary constants. Let a = a(x), f = f (u), and g = g(u) be arbitrary functions. Then the solutions to equations (2.7.3.113) are expressed as b(x) = −

k3 ω n+1 , k1 n ωx′

c(x) =

ωn , k1 n

1 h = (k1 F −n + k2 f + k3 g)F, f

ϑ(x, t) = t1/n ω(x), Z ζ = f, F = f du,

(2.7.3.114)

where the function ω = ω(x) is a solution to the nonlinear Emden–Fowler type ODE (aωx′ )′x = −

k2 n+1 ω . k1 n

(2.7.3.115)

We set k3 = k1 n and k = k2 /(k1 n). From relations (2.7.3.114) it follows that the nonlinear reaction–diffusion type equation   1 −n ω n+1 n F (u) kf (u) + g(u) + ut = [a(x)f (u)ux ]x − g(u)u + ω F (u) , x ωx′ f (u) n (2.7.3.116) where f (u), Rg(u), and a(x) are arbitrary functions, k and n are arbitrary constants, and F (u) = f (u) du, admits the functional separable solution in implicit form Z f (u) du = ω(x)t1/n . (2.7.3.117) The function ω = ω(x) in (2.7.3.116) and (2.7.3.117) is described by the nonlinear ordinary differential equation [a(x)ωx′ ]′x + kω n+1 = 0.

(2.7.3.118)

Note that for n = −1, the general solution to equation (2.7.3.118) is Z Z dx x dx + C1 + C2 , ω = −k a(x) a(x) where C1 and C2 are arbitrary constants. ◮ Example 2.82. Substituting a(x) = 1 and k = 0 into (2.7.3.116)–(2.7.3.118),

we get the equation n+1

ut = [f (u)ux ]x − x

  1 1 1−n g(u)ux + x g(u)F (u) + F (u) . (2.7.3.119) f (u) n n

It has the noninvariant self-similar solution (2.7.3.117), which makes the diffusion ◭ term [f (u)ux ]x vanish. Solution 29. We now consider the equation  ′ f + bϑx g + chζ = 0, −ϑt + λϑ − λZ + (aϑx )x f + aϑ2x ζ u

(2.7.3.120)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

239

where λ = const, which, by virtue of (2.7.1.2) (ϑ = Z), is equivalent to (2.7.3.2). Equation (2.7.3.120) is invariant under the transformation ϑ = ϑ¯ + C1 eλt ,

(2.7.3.121)

where C1 is an arbitrary constant. It is easy to verify that if the constants a, b, and c are all set, without loss of generality, equal to 1, equation (2.7.3.120) has the particular solution Z λ−β λ ϑ¯ = C2 eβt−µx , g = µf + , h= f du, ζ = f, (2.7.3.122) µ f where f = f (u) is an arbitrary function and C2 , β, and µ are arbitrary constants. Given (2.7.3.121), we obtain the equation   Z λ λ−β ux + ut = [f (u)ux ]x + µf (u) + f (u) du, (2.7.3.123) µ f (u) which has the exact solution in implicit form Z f (u) du = C1 eλt + C2 eβt−µx . By setting β = λ−σµ, we rewrite equation (2.7.3.123) in the more compact form Z λ ut = [f (u)ux ]x + [µf (u) + σ]ux + f (u) du. f (u) R In this case, its solution is f (u) du = C1 eλt + C2 e(λ−σµ)t−µx . Solution 30. We look for a steady-state particular solution ϑ = ϑ(x) of equation (2.7.3.120). In this case, we have aϑ2x = k1 ϑ, (aϑx )x = k2 , bϑx = k3 , c = 1; λ + k1 (f /ζ)′u = 0, −λZ + k2 f + k3 g + hζ = 0,

(2.7.3.124)

where k1 , k2 , and k3 are arbitrary constants. The solution to the first three equations of (2.7.3.124) with k1 k2 6= 0 can be represented as 1 (k2 x + C3 )2−(k1 /k2 ) , C2 k1 ϑ(x) = C2 (k2 x + C3 )k1 /k2 , a(x) =

b(x) =

k3 (k2 x + C3 )1−(k1 /k2 ) , C2 k1

(2.7.3.125) where C2 and C3 are arbitrary constants. The solution to the system consisting of the last two equations in (2.7.3.124) is written as   Z k1 f du f λ(u + C4 ) ζ =− k2 f + k3 g + k1 , (2.7.3.126) , h= λ u + C4 k1 f u + C4 where f = f (u) and g = g(u) are arbitrary functions and C4 is an arbitrary constant.

240

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

Substituting C2 = 1/k, C3 = C4 = 0, k1 = k, k2 = k3 = 1, and λ = kσ into (2.7.3.125) and (2.7.3.126), we arrive at the equation   Z f (u) σu f (u) + g(u) + k du . ut = [x2−k f (u)ux ]x + x1−k g(u)ux + f (u) u (2.7.3.127) For k 6= 0, this equation admits the exact solution Z σ f (u) du = Cekσt − xk , C = −C1 σ. (2.7.3.128) u k We have taken into account that equation (2.7.3.120) is invariant under transformation (2.7.3.121). Solution 31. First, we note that equation (2.7.3.120) is equivalent to equation (2.7.3.2) for any λ = λ(x, t, u). Now we set λ = p(x)f (u) and ζ = f (u) in (2.7.3.120) and divide by f = f (u) to obtain −ϑt

1 g + (aϑx )x + pϑ + bϑx + ch − pF = 0, f f

(2.7.3.129)

R where F = f (u) du. Assuming the function f to be given arbitrarily, we look for the functions g and h in the form   1 1 (2.7.3.130) g = f k1 + k2 + k3 F , h = m1 + m2 + m3 F, f f

where ki and mi are some constants (i = 1, 2, 3). Substituting (2.7.3.130) into (2.7.3.129), we arrive at the equations (aϑx )x + pϑ + k1 bϑx + m1 c = 0, −ϑt + k2 bϑx + m2 c = 0,

(2.7.3.131)

−p + k3 bϑx + m3 c = 0.

These admit the exact solution k2 = k3 = 0,

ϑ = m2 c(x)t + η(x),

p = m3 c(x).

(2.7.3.132)

In addition, we find that the three functions a = a(x), b = b(x), and c = c(x) are connected by one equation (ac′x )′x + k1 bc′x + m3 c2 = 0,

(2.7.3.133)

and the function η is described by the linear ODE (aηx′ )′x + k1 bηx′ + m3 cη + m1 c = 0.

(2.7.3.134)

Note that for given a and c, equation (2.7.3.133) is algebraic in b; for given b and c, it is a first-order linear ODE with respect to a (which is easy to integrate); and

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

241

for given a and b, it is a second-order ODE with a quadratic nonlinearity with respect to c. To sum up, we have obtained the nonlinear reaction–diffusion type equation   Z m2 + m3 f (u) du , ut = [a(x)f (u)ux ]x + b(x)f (u)ux + c(x) m1 + f (u) (2.7.3.135) where f (u) is an arbitrary function. Any two out of the three functions a = a(x), b = b(x), and c = c(x) can be set arbitrarily; the remaining function satisfies equation (2.7.3.133) with k1 = 1. Equation (2.7.3.135) has an exact solution that can be represented in the implicit form Z f (u) du = m2 c(x)t + η(x), where η(x) is determined by ODE (2.7.3.134) with k1 = 1. Remark 2.72. The more general nonlinear PDE

ut = [a(x)f (u)ux ]x + b(x)f (u)ux + m(x) +

c(x) + n(x) f (u)

Z

f (u) du

than (2.7.3.135), in which f = f (u) and m = m(x) are arbitrary functions, and the four functions a = a(x), b = b(x), c = c(x), and n = n(x) are connected by one equation (algebraic in b and n, and differential in a and c) (ac′x )′x + bc′x + cn = 0,

admits the exact solution

Z

f (u) du = c(x)t + η(x).

The function η(x) is determined by the ODE (aηx′ )′x + bηx′ + nη + m = 0.

Solution 32. Solutions to equation (2.7.3.129) can be sought in the form g = f (k1 f −1 + k2 ),

h = k3 f −1 + k4 ,

F = k5 f −1 + k6 ,

(2.7.3.136)

where kn are some constants; the last relation in (2.7.3.136) serves to determine the function f . By setting k1 = 0, k2 = 1, k5 = 2, and k6 = 0 in (2.7.3.136), we obtain f = g = u−1/2 , h = k3 u1/2 + k4 , and F = 2u1/2 . The corresponding nonlinear diffusion type equation ut = [a(x)u−1/2 ux ]x + b(x)u−1/2 ux + c(x)(k3 u1/2 + k4 ),

(2.7.3.137)

where a(x), b(x), and c(x) are arbitrary functions, while k3 and k4 are arbitrary constants, has an exact solution, which can be represented in explicit form as u=

1 4 [ξ(x)t

+ η(x)]2 .

(2.7.3.138)

242

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

The functions ξ = ξ(x) and η = η(x) are determined by solving the system of ODEs (aξx′ )′x + bξx′ + (aηx′ )′x

+

bηx′

+

1 2 k3 cη

1 1 2 2 k3 cξ − 2 ξ − 12 ξη + k4 c

= 0, = 0.

(2.7.3.139)

For c(x) = 1, the first equation in (2.7.3.139) is satisfied if we set ξ(x) = k3 . Remark 2.73. The equation

ut = [a(x)u−1/2 ux ]x + b(x)u−1/2 ux + c(x)u1/2 + d(x),

(2.7.3.140)

which is more general than (2.7.3.137), also has an exact solution of the form (2.7.3.138). In the case d(x)/c(x) = const, equation (2.7.3.140) belongs to the class of equations (2.7.3.1) in question.

Solution 33. Now we consider the equation  ′  ′  ′ f f f −ϑt + ϑxx f + ϑ2x −k ϑ+k Z + hζ = 0. ζ u ζ u ζ u

(2.7.3.141)

which, by virtue of (2.7.1.2), is equivalent to equation (2.7.3.2) with a = c = 1 and b = 0. An exact solution to equation (2.7.3.141) will be sought in the form ϑ = Ax2 + Bx + Ce−λt , where A, B, C, and λ are constants to be found. Omitting the intermediate calculations, we ultimately arrive at the equation Z 1 u f (u) u ut = [f (u)ux ]x − λu − γ 2 −λ du, (2.7.3.142) 2 f (u) f (u) u which has two exact solutions Z f (u) 1 du = λx2 ± γx + βe−λt , u 4

(2.7.3.143)

where β, γ, and λ are arbitrary constants. Remark 2.74. The method employed in this section allows one to obtain a number of other exact solutions to equation (2.7.3.1), which we do not discuss here (recall that this section only deals with nonlinear PDEs of a reasonably general form, that depend on a few arbitrary functions).

2.7.4. Generalized Porous Medium Equations with a Nonlinear Source The class of equations in question. Reduction to the bilinear form. We consider the class of nonlinear generalized porous medium equations with variable coefficients of the form [300] ut = [a(x)f (u)um x ]x + b(x)g(u).

(2.7.4.1)

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

243

For m = 1, this equation coincides with the simpler equation (2.5.3.1), which was considered in Subsection 2.5.3. Some solutions with m 6= 1 can be found in [97, 156, 158, 275]. In what follows, we assume that a(x) 6≡ 0, f (u) 6≡ 0, b(x) 6≡ 0, and g(u) 6≡ 0. By employing the approach described in Section 2.7.1, we will obtain a number of exact solutions to equations of the form (2.7.4.1), where two functional coefficients, a(x) and f (u), are set arbitrarily and the others are expressed in terms of them. Just as previously, for brevity, we often omit the arguments of the functions appearing in transformation (2.7.1.2) and equation (2.7.4.1). On making the change of variable (2.7.1.2), we substitute the derivatives (2.7.1.3) into (2.7.4.1) and rearrange the terms to obtain 1−m −ϑt + (aϑm + aϑ1+m (f ζ −m )′u + bgζ = 0. x )x f ζ x

(2.7.4.2)

Equation (2.7.4.2) can be represented in the bilinear form (2.7.1.4) with N = 4: 4 X

Φn Ψn = 0,

(2.7.4.3)

n=1

where 1+m , Φ1 = −ϑt , Φ2 = (aϑm x )x , Φ3 = aϑx

Ψ1 = 1,

Ψ2 = f ζ

1−m

,

Ψ3 =

(f ζ −m )′u ,

Φ4 = b; Ψ4 = gζ.

(2.7.4.4)

Seeking exact solutions starting from equation (2.7.4.2). Solution 1. Equation (2.7.4.3) can be satisfied identically by using the linear relations Φ1 = −AΦ4 ,

Φ2 = BΦ4 ;

Ψ3 = 0;

Ψ4 = AΨ1 − BΨ2 ,

(2.7.4.5)

where A and B are arbitrary constants. By inserting (2.7.4.4) into (2.7.4.5), we arrive at the equations ϑt = Ab,

(aϑm x )x = Bb;

(f ζ −m )′u = 0,

gζ = A − Bf ζ 1−m .

(2.7.4.6)

The solution to the system composed of the first two equations in (2.7.4.6) with m 6= 0 is given by 1/m Z  k k Bx + C1 b(x) = , ϑ(x, t) = kt + dx + C2 , (2.7.4.7) A A a(x) where a(x) is an arbitrary function, while C1 , C2 , and k are arbitrary constants. The solutions to the other two equations of (2.7.4.6) can be written as g = Af −1/m − B,

ζ = f 1/m ,

(2.7.4.8)

where f = f (u) is an arbitrary function. Setting A = k in (2.7.4.7) and (2.7.4.8), we get the equation k ut = [a(x)f (u)um − B, x ]x + f 1/m (u)

244

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

which admits the generalized traveling wave solution in implicit form 1/m Z Z  Bx + C1 f 1/m (u) du = kt + dx + C2 . a(x) Solution 2. Equation (2.7.4.3) also holds if we set Φ1 = −AΦ2 ,

Φ3 = −Φ4 ;

Ψ2 = AΨ1 ,

Ψ3 = Ψ4 ,

(2.7.4.9)

where A is an arbitrary constant. Substituting (2.7.4.4) into (2.7.4.9) yields ϑt = A(aϑm x )x ,

aϑ1+m = −b; x

f ζ 1−m = A,

(f ζ −m )′u = gζ.

(2.7.4.10)

A solution to the system consisting of the first two equations in (2.7.4.10) will be sought in the form ϑ = kt + r(x). This results in   m+1 1 Z  m kx + C1 m kx + C1 , ϑ(x, t) = kt + dx + C2 , b(x) = −a(x) Aa(x) Aa(x) (2.7.4.11) where a(x) is an arbitrary function, while C1 , C2 , and k are arbitrary constants. The solutions to the last two equations of (2.7.4.10) are given by 1 g= 1−m



f A

 1+m

1−m

fu′ ,

ζ=



f A



1 m−1

(m 6= 1).

(2.7.4.12)

By setting C1 = 0, A = k = 1, and m 6= 1 in (2.7.4.11) and (2.7.4.12), we arrive at the equation ut =

[a(x)f (u)um x ]x

1 + m−1



xm+1 a(x)



1 m



1 m

f

1+m 1−m

(u)fu′ (u),

which admits the exact solution in implicit form Z

f

1 m−1

(u) du = t +

Z 

x a(x)

dx + C2 .

Solution 3. Equation (2.7.4.3) can be satisfied by setting Φ3 = −Φ1 ,

Φ2 = −Φ4 ;

Ψ3 = Ψ1 ,

Ψ2 = Ψ4 .

(2.7.4.13)

Substituting (2.7.4.4) into (2.7.4.13), we get aϑ1+m = ϑt , x

(aϑm x )x = −b;

(f ζ −m )′u = 1,

gζ = f ζ 1−m .

(2.7.4.14)

The solution to the system composed of the first two equations in (2.7.4.14) with m 6= −1 is given by λm − m [a(x)] m+1 a′x (x), m+1 Z − 1 ϑ(x, t) = λm+1 t + λ [a(x)] m+1 dx + C1 ,

b(x) = −

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

245

where a(x) is an arbitrary function, while C1 and λ are arbitrary constants. The solutions to the other two equations of (2.7.4.14) are ζ = (f /u)1/m ,

g = u,

where f = f (u) is an arbitrary function. As a result, we obtain the PDE ut = [a(x)f (u)um x ]x − k[a(x)]

m − m+1

a′x (x)u,

which has the exact solution Z 

f (u) u



1 m

du = λm+1 t + λ

Z

[a(x)]

1 − m+1

1

λ = [k(m + 1)] m .

dx + C1 ,

Solution 4. Equation (2.7.4.3) can be satisfied if we use the relations Φ1 + kΦ3 + Φ4 = 0,

Φ2 = 0;

Ψ3 = kΨ1 ,

Ψ4 = Ψ1 .

(2.7.4.15)

By inserting (2.7.4.4) into (2.7.4.15), we arrive at −ϑt + kaϑ1+m + b = 0, x

(aϑm x )x = 0;

(f ζ −m )′u = k,

gζ = 1. (2.7.4.16)

Integrating the first two equations of (2.7.4.16) yields b(x) = C1 −

kC2m+1 [a(x)]−1/m ,

ϑ(x, t) = C1 t + C2

Z

[a(x)]−1/m dx + C3 ,

(2.7.4.17) where C1 , C2 , and C3 are arbitrary constants. The solutions of the last two equations (2.7.4.16) can be written as g(u) = k 1/m



u f (u)

1/m

,

ζ(u) = k −1/m



f (u) u

1/m

.

(2.7.4.18)

Setting k = 1, C1 = β, and C2m+1 = γ in (2.7.4.17) and (2.7.4.18), we arrive at the equation −1/m ut = [a(x)f (u)um x ]x + β − γ[a(x)]

which admits the exact solution Z 

f (u) u



1 m

1

du = βt + γ m+1

Z

 
u 1/m , f (u) 1

[a(x)]− m dx + C3 .

Solution 5. Equation (2.7.4.3) holds identically if Φ1 = −Φ4 ,

Φ2 = −kΦ3 ;

Ψ1 = Ψ4 ,

kΨ2 = Ψ3 .

(2.7.4.19)

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2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

In view of (2.7.4.4), we obtain the equations ϑt = b,

1+m (aϑm ; x )x = −kaϑx

gζ = 1,

(f ζ −m )′u = kf ζ 1−m .

(2.7.4.20)

From the first two equations in (2.7.4.20) we find that   Z m k −1/m b(x) = β = const, ϑ(x, t) = βt + ln [a(x)] dx + C1 + C2 , k m where C1 and C2 are arbitrary constants. The last two equations in (2.7.4.20) give    −1 Z Z k k g(u) = f −1/m , f 1/m du + C3 , ζ(u) = f 1/m f 1/m du + C3 m m where f = f (u). Further, on putting k = m and C3 = γ, we obtain the equation Z  m −1/m 1/m ut = [a(x)f (u)ux ]x + β[f (u)] [f (u)] du + γ , (2.7.4.21) which has the exact solution Z  Z [f (u)]1/m du + γ = A2 eβt [a(x)]−1/m dx + A1 ,

(2.7.4.22)

where A1 and A2 are arbitrary constants. To represent the solution in the form (2.7.4.22), we used the relation   Z Z k m ln f 1/m du + C3 + const. ζ(u) du = k m Note that the diffusion term [a(x)f (u)um x ]x in equation (2.7.4.21) vanishes by virtue of solution (2.7.4.22). Solution 6. Equation (2.7.4.3) can be satisfied if we take all Φi (i = 1, 2, 3) proportional to Φ4 : Φ1 = k1 Φ4 ,

Φ2 = k2 Φ4 ,

Φ3 = k3 Φ4 ;

Ψ4 = −k1 Ψ1 − k2 Ψ2 − k3 Ψ3 . (2.7.4.23)

By inserting (2.7.4.4) into (2.7.4.23), we get ϑt = −k1 b,

(aϑm x )x = k2 b,

aϑ1+m = k3 b; x

(2.7.4.24)

gζ = −k1 − k2 f ζ 1−m − k3 (f ζ −m )′u .

Let us consider two cases. 1◦ . The simplest solution to the first four equations of (2.7.4.24) is a(x) = b(x) = 1,

θ(x, t) = −k1 t + λx + C1 ,

k2 = 0,

k3 = λ1+m .

It leads to a traveling wave solution of the original porous medium type equation (2.7.4.1); this solution will not be discussed here.

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

247

2◦ . The first four equations of (2.7.4.24) also admit another solution a(x) = xm+1 ,

b(x) = 1,

ϑ(x, t) = −k1 t + λ ln x + C1 ,

k2 = λm ,

k3 = λ1+m .

(2.7.4.25)

Setting k = k1 and λ = 1 in (2.7.4.25) and using the last equation of (2.7.4.24), we arrive at the porous medium type equation ut = [xm+1 f (u)um x ]x + g(u),

(2.7.4.26)

where g(u) = −

1  k + f (u)ζ 1−m (u) + [f (u)ζ −m (u)]′u , h(u)

(2.7.4.27)

while f = f (u) and ζ = ζ(u) are arbitrary functions. Equation (2.7.4.26) admits the exact solution Z ζ(u) du = −kt + ln x + C1 . (2.7.4.28) Remark 2.75. The invariant solution (2.7.4.28) to equation (2.7.4.26) can be found by seeking it in the standard way as u = U (z), where z = −kt + ln x, in which case relation (2.7.4.27) is not used. The function U (z) is described by the ODE

[f (U )(Uz′ )m ]′z + f (U )(Uz′ )m + kUz′ + g(U ) = 0.

Seeking exact solutions with the help of equivalent equations. Some other exact solutions to equation (2.7.4.1) can be obtained if, instead of (2.7.4.3) and (2.7.4.4), we use equivalent equations reducible to (2.7.4.3) and (2.7.4.4) on the set of functions satisfying relation (2.7.1.2). Solution 7. Let us revisit the class of porous medium type equations (2.7.4.1). After making the change of variable (2.7.1.2), we will look at, instead of (2.7.4.2), the more complex equation 1−m −eλϑ e−λZ ϑt + (aϑm + aϑ1+m (f ζ −m )′u + bgζ = 0, (2.7.4.29) x )x f ζ x R where Z = ζ du and λ is an arbitrary constant. Equations (2.7.4.2) and (2.7.4.29) are equivalent, since ϑ = Z by virtue of transformation (2.7.1.2). Equation (2.7.4.29) can be represented in the bilinear form (2.7.4.3) where 1+m Φ1 = −eλϑ ϑt , Φ2 = (aϑm , x )x , Φ3 = aϑx

Ψ1 = e

−λZ

,

Ψ2 = f ζ

1−m

,

Ψ3 =

(f ζ −m )′u ,

Φ4 = b; Ψ4 = gζ.

(2.7.4.30)

Just as previously, equation (2.7.4.3) can be satisfied by using relations (2.7.4.5). Inserting (2.7.4.30) into (2.7.4.5) gives −m ′ eλϑ ϑt = Ab, (aϑm )u = 0, gζ = Ae−λZ −Bf ζ 1−m . (2.7.4.31) x )x = Bb; (f ζ

248

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

These equations coincide with (2.7.4.6) at λ = 0. A solution to system (2.7.4.31) is given by C1 1 exp[λr(x)], ϑ(x, t) = ln(C1 λt + C2 ) + r(x), A λ   Z −1/m 1/m g = Af exp −λ f du − B, ζ = f 1/m ,

b(x) =

(2.7.4.32)

where f = f (u) is an arbitrary function and r = r(x) satisfies the ordinary differential equation BC1 λr [a(rx′ )m ]′x = e . (2.7.4.33) A Formulas (2.7.4.32) and equation (2.7.4.33) define the functional coefficients of equation (2.7.4.1) and its solution (2.7.1.2). It is noteworthy that for a(x) = a0 xk , equation (2.7.4.33) admits the exact solution r(x) = σ ln x + µ,

σ=

k−m−1 , λ

µ=

1 Aa0 σ m (k − m) . ln λ BC1

Solution 8. Substituting (2.7.4.30) into (2.7.4.23), we arrive at the equations eλϑ ϑt = −k1 b,

(aϑm x )x = k2 b,

aϑ1+m = k3 b; x

(2.7.4.34)

gζ = −k1 e−λZ − k2 f ζ 1−m − k3 (f ζ −m )′u .

The first three equations of system (2.7.4.34) admit the following solution for exponential functional coefficients: a(x) = b(x) = eλx , k1 = −

1 , λ

k2 = λ,

ϑ(x, t) =

1 ln t + x, λ

(2.7.4.35)

k3 = 1.

Using the last equation in (2.7.4.34), we obtain the nonlinear PDE λx ut = [eλx f (u)um x ]x + e g(u),

(2.7.4.36)

where

  1 1 −λZ 1−m −m ′ g(u) = − − e + k2 f ζ − k3 (f ζ )u , ζ λ

Z=

Z

ζ du, (2.7.4.37)

while f = f (u) and ζ = ζ(u) are arbitrary functions, which admits the exact solution in implicit form Z 1 ζ(u) du = ln t + x. (2.7.4.38) λ

Remark 2.76. The invariant solution (2.7.4.38) to equation (2.7.4.36) can be obtained by seeking it in the standard form u = U (z) with z = λ1 ln t+x, in which case relation (2.7.4.37) is not used. The function U (z) is described by the ODE

1 ′ Uz = [eλz f (U )(Uz′ )m ]′z + eλz g(U ). λ

2.7. General Functional Separation of Variables. Implicit Representation of Solutions

249

Solution 9. The first three equations of system (2.7.4.34) also admit the following solution for power-law coefficients: a(x) = xm+n−1 , λ = n − 2,

b(x) = xn−2 ,

k1 = −

1 , n−2

ϑ(x, t) =

k2 = n − 1,

1 ln t + ln x, n−2

(2.7.4.39)

k3 = 1.

Using the last equation in (2.7.4.34), we arrive at the nonlinear PDE n−2 ut = [xm+n−1 f (u)um g(u), x ]x + x

(2.7.4.40)

where   1 1 −(n−2)Z 1−m −m ′ g(u) = − − e + (n − 1)f ζ + (f ζ )u , ζ n−2

Z=

Z

ζ du,

(2.7.4.41) while f = f (u) and ζ = ζ(u) are arbitrary functions, which admits the exact solution Z 1 ln t + ln x. (2.7.4.42) ζ(u) du = n−2 Remark 2.77. The self-similar solution (2.7.4.42) of equation (2.7.4.40) can be obtained 1

by seeking it in the standard form u = U (z) with z = xt n−2 , in which case relation (2.7.4.41) is not used. The function U (z) is described by the ODE 1 zU ′ = [z m+n−1 f (U )(Uz′ )m ]′z + z n−2 g(U ). n−2 z

Solution 10. Now let us look at the equation 1−m −(Z/ϑ)ϑt + (Z/ϑ)m (aϑm + aϑ1+m (f ζ −m )′u + bgζ = 0, (2.7.4.43) x )x f ζ x R with Z = ζ(u) du, which is equivalent to (2.7.4.2) by virtue of transformation (2.7.1.2). Equation (2.7.4.43) can be rewritten in the bilinear form (2.7.4.3) where 1+m Φ1 = −ϑt /ϑ, Φ2 = ϑ−m (aϑm , x )x , Φ3 = aϑx

Ψ1 = Z,

Ψ2 = f ζ

1−m

m

Z ,

Ψ3 =

(f ζ −m )′u ,

Φ4 = b; Ψ4 = gζ.

(2.7.4.44)

Taking into account that equation (2.7.4.3) can be satisfied with relations (2.7.4.5) and substituting the expressions (2.7.4.44) into (2.7.4.5), we arrive at the system of equations ϑt /ϑ = Ab,

ϑ−m (aϑm x )x = Bb;

(f ζ −m )′u = 0,

gζ = AZ − Bf ζ 1−m Z m .

It admits the solution b(x) = λ, ϑ(x, t) = CeAλt r(x); g = Af −1/m Z − BZ m , ζ = f 1/m , (2.7.4.45)

250

2. M ETHODS OF F UNCTIONAL S EPARATION OF VARIABLES

where f = f (u) is an arbitrary function, C and λ are arbitrary constants, Z = and r = r(x) is a function satisfying the ODE [a(rx′ )m ]′x = Bλrm .

R

ζ du,

(2.7.4.46)

Using (2.7.4.45), we find that the nonlinear equation Z m Z −1/m 1/m 1/m ut = [a(x)f (u)um ] + Aλf (u) f (u) du − Bλ f (u) du x x admits the exact solution in implicit form Z f 1/m (u) du = CeAλt r(x), where a(x) and f (u) are arbitrary functions, while r(x) is a function satisfying ODE (2.7.4.46).

3. Direct Method of Symmetry Reductions. Weak Symmetries 3.1. Direct Method of Symmetry Reductions 3.1.1. Simplified Scheme. Generalized Burgers–Korteweg–de Vries Equation Before we proceed to describe the general case of the direct method of symmetry reductions,∗ which is also known as the Clarkson–Kruskal direct method, we will first consider a simplified scheme. The main idea of the simplified scheme is the following: exact solutions to partial differential equations with two independent variables, x and t, are sought in the form u = f (t)w(z) + g(x, t),

z = ϕ(t)x + ψ(t).

(3.1.1.1)

The functions f (t), g(x, t), ϕ(t), and ψ(t) are determined in the course of the solution; these are chosen so as to ensure that the unknown function w(z) satisfies a single ordinary differential equation [72, 275]. Below we consider a few specific examples of constructing exact solutions of the form (3.1.1.1) to nonlinear equations of mathematical physics. ◮ Example 3.1. Let us look at the generalized nth-order Burgers–Korteweg–de Vries equation [275]: ut = au(n) (3.1.1.2) x + buux .

We seek its exact solutions in the form (3.1.1.1). Substituting (3.1.1.1) into equation (3.1.1.2) gives af ϕn wz(n) + bf 2 ϕwwz′ + f (bgϕ − ϕ′t x − ψt′ )wz′

+ (bf gx − ft′ )w + agx(n) + bggx − gt = 0.

(3.1.1.3)

(n)

Equating the functional coefficients of wz and wwz′ in (3.1.1.3) with each other, we get f = ϕn−1 . (3.1.1.4) Further, equating the coefficient of wz′ with zero, we find that g= ∗ Sometimes,

1 (ϕ′ x + ψt′ ). bϕ t

(3.1.1.5)

we will also call it the direct reduction method for short.

DOI: 10.1201/9781003042297-3

251

252

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

On substituting (3.1.1.4) and (3.1.1.5) into (3.1.1.3), we arrive at the equation ϕ2n−1 (awz(n) +bwwz′ )+(2−n)ϕn−2 ϕ′t w+

 1  (2ϕ2t −ϕϕtt )x+2ϕt ψt −ϕψtt = 0. 2 bϕ

On dividing by ϕ2n−1 and on eliminating x using the formula x = (z − ψ)/ϕ, which follows from the second relation in (3.1.1.1), we obtain awz(n) + bwwz′ + (2 − n)ϕ−n−1 ϕ′t w + +

1 −2n−2 ϕ [2(ϕ′t )2 − ϕϕ′′tt ]z b

1 −2n−2 ′′ ϕ [ϕψϕ′′tt − ϕ2 ψtt + 2ϕϕ′t ψt′ − 2ψ(ϕ′t )2 ] = 0. b

(3.1.1.6)

Now we require that the functional coefficient of w and the last term are constant: ϕ−n−1 ϕ′t = −A,

′′ ϕ−2n−2 [ϕψϕ′′tt − ϕ2 ψtt + 2ϕϕ′t ψt′ − 2ψ(ϕ′t )2 ] = B,

where A and B are arbitrary constants. As a result, we arrive at the following system of ordinary differential equations for ϕ and ψ: ϕ′t = −Aϕn+1 ,

′′ ψtt + 2Aϕn ψt′ + A2 (1 − n)ϕ2n ψ = −Bϕ2n .

(3.1.1.7)

Using (3.1.1.6) and (3.1.1.7), we obtain the ODE for w(z): awz(n) + bwwz′ + A(n − 2)w +

B A2 (1 − n)z + = 0. b b

(3.1.1.8)

For A 6= 0, the general solution to equations (3.1.1.7) is expressed as 1 ϕ(t) = (Ant + C1 )− n ,

ψ(t) = C2 (Ant + C1 )

n−1 n

1 + C3 (Ant + C1 )− n +

B , − 1)

(3.1.1.9)

A2 (n

where C1 , C2 , and C3 are arbitrary constants. Formulas (3.1.1.1), (3.1.1.4), (3.1.1.5), and (3.1.1.9) in conjunction with equation (3.1.1.8) describe an exact solution to the generalized Burgers–Korteweg–de ◭ Vries equation (3.1.1.2). ◮ Example 3.2. Following [72], consider the Boussinesq equation

utt + (uux )x + auxxxx = 0.

(3.1.1.10)

As in Example 3.1, we look for a solution in the form (3.1.1.1); the functions f (t), g(x, t), ϕ(t), and ψ(t) will be determined in the subsequent analysis. On substituting (3.1.1.1) into (3.1.1.10), we get ′′ ′′′′ ′′ + f (zt2 + gϕ2 )wzz af ϕ4 wzzzz + f 2 ϕ2 wwzz

+ f 2 ϕ2 (wz′ )2 + (f ztt + 2f gx ϕ + 2ft zt )wz′ + (f gxx + ftt )w + gtt + ggxx + gx2 + agxxxx = 0.

(3.1.1.11)

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3.1. Direct Method of Symmetry Reductions

′′′′ ′′ Equating the functional coefficients of wzzzz and wwzz with each other, we find

that f = ϕ2 .

(3.1.1.12)

′′ Equating the functional coefficient of wzz with zero and taking into account (3.1.1.12), we obtain 1 g = − 2 (ϕ′t x + ψt′ )2 . (3.1.1.13) ϕ

Substituting (3.1.1.12) and (3.1.1.13) into (3.1.1.11) yields the equation ′′′′ ′′ ′′ ϕ6 [awzzzz + wwzz + (wz′ )2 ] + ϕ2 (xϕ′′tt + ψtt )wz′ + 2ϕϕ′′tt w   −2 ′ − ϕ (ϕt x + ψt′ )2 tt + 6ϕ−4 ϕ2t (ϕ′t x + ψt′ )2 = 0.

Performing the double differentiation of the expression in square brackets in the second line, then dividing all terms by ϕ6 , and eliminating x using the formula x = (z − ψ)/ϕ, we arrive at the equation ′′′′ ′′ ′′ awzzzz + wwzz + (wz′ )2 + ϕ−5 (ϕ′′tt z + ϕψtt − ψϕ′′tt )wz′ + 2ϕ−5 ϕ′′tt w + · · · = 0. (3.1.1.14) Now we require that the functional coefficient of wz′ is a function of the single variable z, so that ′′ ′′ ϕ−5 (ϕ′′tt z + ϕψtt − ψϕ′′tt ) = ϕ−5 ϕ′′tt z + ϕ−5 (ϕψtt − ψϕ′′tt ) ≡ Az + B,

where A and B are arbitrary constants. This results in the following system of ordinary differential equations for ϕ and ψ: ϕ′′tt = Aϕ5 , ′′ ψtt = (Aψ + B)ϕ4 .

(3.1.1.15)

Using (3.1.1.15), we eliminate the second and third derivatives of ϕ and ψ from (3.1.1.14) to arrive at the ordinary differential equation for w = w(z) ′′′′ ′′ awzzzz + wwzz + (wz′ )2 + (Az + B)wz′ + 2Aw − 2(Az + B)2 = 0. (3.1.1.16)

Formulas (3.1.1.1), (3.1.1.12), and (3.1.1.13) together with equations (3.1.1.15) and (3.1.1.16) describe an exact solution to the Boussinesq equation (3.1.1.10). ◭ ◮ Example 3.3. Let us look at the nonlinear third-order equation with a mixed

derivative uxt + uuxx − u2x = νuxxx + q(t)ux + p(t).

(3.1.1.17)

It describes a wide class of exact solutions to the three-dimensional Navier–Stokes equations [15], where ν is the kinematic viscosity of the fluid. The functions p = p(t) and q = q(t), which appear in equation (3.1.1.17), can be chosen arbitrarily. Equation (3.1.1.17) is derived below in Section 4.4.3 (see Example 4.27). We look for exact solutions in the form [15, 16] u = f (t)w(z) + g(t)x + h(t),

z = ϕ(t)x + ψ(t),

(3.1.1.18)

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where the functions f = f (t), g = g(t), h = h(t), ϕ = ϕ(t), and ψ = ψ(t) are to be determined in the subsequent analysis. On substituting (3.1.1.18) into (3.1.1.17), we get ′′ ′′′ [a(t)w + b(t)z + c(t)]wzz − a(t)(wz′ )2 = νwzzz + qe(t)wz′ + pe(t),

(3.1.1.19)

where

f 1 1 , b = 3 (gϕ + ϕ′t ), c = 3 (hϕ2 − gϕψ + ϕψt′ − ψϕ′t ), ϕ ϕ ϕ (3.1.1.20) 1 1 ′ 2 ′ [f qϕ + 2f gϕ − (f ϕ) ], p e = (p + gq + g − g ). qe = t t f ϕ3 f ϕ3

a=

Now setting

a = νC1 ,

b = νC2 ,

c = νC3 ,

qe = νC4 ,

pe = νC5 ,

(3.1.1.21)

where C1 , . . . , C5 are arbitrary constants, we find from (3.1.1.19) the ordinary differential equation for w = w(z): e1 w + C2 z + C3 )w′′ − C1 (w′ )2 = w′′′ + C4 w′ + C5 . (C zz z zzz z

In this case, relations (3.1.1.20) under conditions (3.1.1.21) form a mixed system of algebraic and ordinary differential equations for the functional parameters of solution (3.1.1.18) and functional coefficients p = p(t) and q = q(t) of equation (3.1.1.17). The functions f = f (t) and ψ = ψ(t) appearing in this system can be treated as arbitrary, while the other ones, ϕ = ϕ(t), g = g(t), h = h(t), p = p(t), and q = q(t), ◭ are expressed in terms of f (t) and ψ(t) without quadrature. Remark 3.1. Subsections 3.2.3–3.2.5 give examples of utilizing the simplified scheme of the direct reduction method for finding exact solutions to nonlinear third-order PDEs with three independent variables, which arise in fluid dynamics.

3.1.2. Special Form of Reductions. The Boussinesq Equation The procedure for seeking exact solutions to nonlinear PDEs by the direct method of reductions of special form consists of several consecutive stages [72], which are outlined below. 1◦ . One seeks exact solutions to a partial differential equation with two independent variables, x and t, in the form [72] u(x, t) = f (x, t)w(z) + g(x, t),

z = z(x, t).

(3.1.2.1)

The functions f (x, t), g(x, t), z(x, t) are to be determined during the solution process so as to ensure that the function w(z) satisfies a single ordinary differential equation. Importantly, the functions u and w in formulas (3.1.1.1) and (3.1.2.1) are connected linearly.

3.1. Direct Method of Symmetry Reductions

255

2◦ . On substituting expression (3.1.2.1) into the nonlinear partial differential equation in question, which has a quadratic or power-law nonlinearity, one obtains Φ1 (x, t)Ψ1 [w] + Φ2 (x, t)Ψ2 [w] + · · · + Φm (x, t)Ψm [w] = 0.

(3.1.2.2)

Here, Ψk [w] are differential forms representing the products of nonnegative integer ′′ powers of the unknown w and its derivatives, wz′ , wzz , etc., while Φk (x, t) are differential forms that depend on f (x, t), g(x, t), and z(x, t) and their partial derivatives with respect to x and t. Suppose the differential form Ψ1 [w] contains the highest derivative with respect to z. Then the function Φ1 (x, t) is used as the normalizing factor. This means that the relations Φk (x, t) = Γk (z) Φ1 (x, t),

k = 1, . . . , m,

(3.1.2.3)

must hold; the functions Γk (z) are to be determined, with Γ1 (z) ≡ 1. 3◦ . In practice, to simplify the determination of the functions f , g, z, u, and Γk , one can take advantage of the following properties [72]: (a) if f = f (x, t) has the form f = f0 (x, t)Ω(z), one can set Ω ≡ 1 [this corresponds to renaming w(z) ⇒ w(z)/Ω(z)]; (b) if g = g(x, t) has the form g = g0 (x, t) + f (x, t)Ω(z), one can set Ω ≡ 0 [this corresponds to renaming w(z) ⇒ w(z) − Ω(z)]; (c) if z = z(x, t) is defined by an algebraic equation of the form Ω(z) = h(x, y), where Ω(z) is any invertible function, then one can set Ω(z) = z [this corresponds to renaming z ⇒ Ω−1 (z)]. 4◦ . Once Γk (z) are all determined, on substituting (3.1.2.3) into (3.1.2.2), one arrives at an ordinary differential equation for w = w(z): Ψ1 [w] + Γ2 (z)Ψ2 [w] + · · · + Γm (z)Ψm [w] = 0.

(3.1.2.4)

Let us illustrate the characteristic features of applying the direct reduction method with a specific example. ◮ Example 3.4. We will seek solutions to the Boussinesq equation (3.1.1.10) in the form (3.1.2.1). We have ′′′ ′′ ′′′′ + · · · = 0. + f 2 zx2 wwzz + a(6f zx2 zxx + 4fx zx3 )wzzz af zx4 wzzzz

(3.1.2.5)

The first three terms are only written out here and the arguments of the functions f ′′′′ ′′ and z are omitted. The functional coefficients of wzzzz and wwzz must satisfy the following condition [see (3.1.2.3)]: f 2 zx2 = af zx4 Γ3 (z), where Γ3 (z) is to be determined. Then, taking advantage of property (a) from Item 3◦ , we choose f = zx2 , Γ3 (z) = 1/a. (3.1.2.6)

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′′′′ ′′′ Likewise, the functional coefficients of wzzzz and wzz must satisfy the condition

6f zx2 zxx + 4fx zx3 = f zx4 Γ2 (z),

(3.1.2.7)

where Γ2 (z) is a new function that has to be determined. Then, in view of (3.1.2.6), we get 14 zxx/zx = Γ2 (z)zx . Integrating with respect to x yields ln zx = I(z) + ln ϕ(t), ˜

1 I(z) = 14

Z

Γ2 (z) dz,

where ϕ(t) e is an arbitrary function. Integrating once more gives Z e e−I(z) dz = ϕ(t)x e + ψ(t),

e is an arbitrary function. The left-hand side is a function of z; hence, where ψ(t) taking advantage of property (c) from Item 3◦ , we find that z = xϕ(t) + ψ(t),

(3.1.2.8)

where the functions ϕ(t) and ψ(t) are to be determined. It follows from formulas (3.1.2.6) to (3.1.2.8) that f = ϕ2 (t),

Γ2 (z) = 0.

(3.1.2.9)

On substituting (3.1.2.8) and (3.1.2.9) into (3.1.2.1), we obtain a solution in the form (3.1.1.1), with the function f defined by formula (3.1.1.12). It follows that using the general approach based on representing solutions in the form (3.1.2.1) leads ◭ ultimately to exactly the same result as using the simpler formula (3.1.1.1). Remark 3.2. In a similar fashion, we can show that constructing exact solutions to the nth-order generalized Burgers–Korteweg–de Vries equation (3.1.1.2) on the basis of formulas (3.1.1.1) and (3.1.2.1) leads to identical results.

◮ Example 3.5. Consider the nonlinear wave equation with two space variables anisotropic in one of the directions

utt = auxx + [(bu + c)uy ]y .

(3.1.2.10)

It is noteworthy that the special case of equation (3.1.2.10) with a = 1, b < 0, and c > 0 describes spatial transonic flows of an ideal polytropic gas [241]. We seek exact solutions to equation (3.1.2.10) in the form u = w(z) + f (x, t),

z = y + g(x, t).

(3.1.2.11)

On substituting (3.1.2.11) into equation (3.1.2.10), we get [(bw + agx2 − gt2 + bf + c)wz′ ]′z + (agxx − gtt )wz′ + afxx − ftt = 0.

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3.1. Direct Method of Symmetry Reductions

We assume that the functions f and g satisfy the overdetermined system of equations afxx − ftt = C1 , agxx − gtt = C2 ,

agx2 − gt2 + bf = C3 ,

(3.1.2.12) (3.1.2.13) (3.1.2.14)

where C1 , C2 , and C3 are arbitrary constants. Then the function w = w(z) is described by the autonomous ordinary differential equation [(bw + c + C3 )wz′ ]′z + C2 wz′ + C1 = 0.

(3.1.2.15)

The general solutions to PDEs (3.1.2.12) and (3.1.2.13) are expressed as f = ϕ1 (ξ) + ψ1 (η) −

2 1 2 C1 t , 2 1 2 C2 t ,

g = ϕ2 (ξ) + ψ2 (η) − √ √ ξ = x + t a, η = x − t a.

Let us substitute these expressions into equation (3.1.2.14) and eliminate t using the ξ−η formula t = √ . On rearranging, we obtain the functional differential equation 2 a with two arguments bϕ1 (ξ) + C2 ξϕ′2 (ξ) − kξ 2 − C3 + bψ1 (η) + C2 ηψ2′ (η) − kη 2 + ψ2′ (η)[4aϕ′2 (ξ) − C2 ξ] + η[2kξ − C2 ϕ′2 (ξ)] = 0,

(3.1.2.16)

where

1 (bC1 + 2C22 ). 8a Equation (3.1.2.16) can be solved using the splitting method (see Section 1.5) by setting bϕ1 (ξ) + C2 ξϕ′2 (ξ) − kξ 2 − C3 = A1 , 4aϕ′2 (ξ) − C2 ξ = A2 , (3.1.2.17) 2kξ − C2 ϕ′2 (ξ) = A3 , k=

where A1 , A2 , and A3 are some constants. The compatible solution to the overdetermined system (3.1.2.17) is ϕ1 (ξ) = − ϕ2 (ξ) =

BC2 A1 + C3 C22 2 ξ − ξ+ , 8ab b b

C2 2 ξ + Bξ. 8a

(3.1.2.18)

The corresponding values of the constants are A1 is any, A2 = 4aB, A3 = −BC2 , B is any, C2 C2 C1 = − 2 , C2 and C3 are any, k = 2 . b 8a

(3.1.2.19)

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From relations (3.1.2.16) and (3.1.2.17) we get the equation connecting the functions ψ1 and ψ2 : A1 + bψ1 (η) + C2 ηψ2′ (η) − kη 2 + A2 ψ2′ (η) + A3 η = 0. In view of (3.1.2.19), we obtain 1 1 ψ1 (η) = − (C2 η + 4aB)ψ2′ (η) + b b ψ2 (η) is an arbitrary function.



 C22 2 η + BC2 η − A1 , 8a

Finally, we find the functions that define solution (3.1.2.11): √ C22 C22 2 2 a BC2 C3 1 f (x, t) = − √ xt + t − t+ − (C2 η + 4aB)ψ2′ (η), 2 ab 2b b b b
√ √ C2 2 2 x + 2 a xt − 3at + B(x + a t) + ψ2 (η), g(x, t) = 8a √ ◭ where η = x − t a. Remark 3.3. On setting f (x, t) = 1 and g(x, t) = 0 in (3.1.2.1), we arrive at the representation of solutions in the form (2.6.1.1), where w is renamed to U . Therefore, the functional separable solutions to nonlinear diffusion and wave type PDEs described in Subsections 2.6.2 to 2.6.4, that reduce to a single ODE for U , can be obtained by the direct reduction method.

3.1.3. General Form of Reductions. The Harry Dym Equation The main idea of the general scheme for utilizing the direct reduction method is the following: one looks for exact solutions to partial differential equations with two independent variables, x and t, in the form [72]
u(x, t) = F x, t, w(z) , z = z(x, t). (3.1.3.1) The functions F (x, t, w) and z(x, t) should be chosen so as to ensure that the function w(z) satisfies a single ordinary differential equation. Unlike the representation of solutions in the form (3.1.1.1) or (3.1.2.1), the connection between w and u in (3.1.3.1) can be nonlinear. Below we illustrate the characteristic features of utilizing the direct reduction method for seeking exact solutions in the form (3.1.3.1). ◮ Example 3.6. Consider once again the Boussinesq equation (3.1.1.10). On substituting (3.1.3.1) into (3.1.1.10), we get ′′′ ′′′ ′′′′ +· · · = 0. (3.1.3.2) +4aFww zx4 wz′ wzzz +a(4Fxw zx3 +6Fw zx2 zxx )wzzz aFw zx4 wzzzz

The first three principal terms are only written out here and the arguments of the functions F and z are omitted. For equation (3.1.3.2) to reduce to an ordinary differential ′′′ ′′′ equation for w = w(z), the ratios of the functional coefficients of wz′ wzzz , wzzz , ...

259

3.1. Direct Method of Symmetry Reductions

′′′′ to the functional coefficient of the highest derivative wzzzz must be functions of z and w, so that

4aFww zx4 = Γ2 (z, w), aFw zx4

a(4Fxw zx3 + 6Fw zx2 zxx ) = Γ3 (z, w), aFw zx4

... .

The first relation gives 4Fww /Fw = Γ2 (z, w). Integrating twice with respect to w, we obtain F (x, t, w) = f (x, t) Θ(z, w) + g(x, t),

(3.1.3.3)

where f (x, t) and g(x, t) are arbitrary functions of two arguments and  Z  Z 1 Θ = exp Γ2 dw dw. 4 Assuming that Θ(z, w(z)) = U (z) in (3.1.3.3) and using the representation (3.1.3.1), we arrive at a solution that, up to renaming, coincides with (3.1.2.1). Therefore, seeking exact solutions to the Boussinesq equation (3.1.1.10) with the help of the general representation (3.1.3.1) leads to the simpler special form of ◭ solutions (3.1.2.1). ◮ Example 3.7. Consider the Harry Dym equation

ut + 2(u−1/2 )xxx = 0.

(3.1.3.4)

We seek its exact solutions in the form (3.1.3.1). On substituting this expression into equation (3.1.3.4), we get
′′′ ′′ −F −3/2 Fw zx3 wzzz + −3F −3/2 Fww + 92 F −5/2 Fw2 zx3 wz′ wzz + · · · = 0.

′′ ′′′ The ratio between the functional coefficients of wz′ wzz and wzzz must be a function of z and w, so that 9 Fw Fww − = Γ(z, w). 3 Fw 2 F Integrating twice gives

F −1/2 (x, t, w) = f (x, t)Θ(z, w) + g(x, t),

(3.1.3.5)

where f (x, t) and g(x, t) are arbitrary functions with two arguments and   Z Z 1 Θ = − exp Γ dw dw. 3 It follows from formulas (3.1.3.1) and (3.1.3.5) that exact solutions to the Harry Dym equation (3.1.3.4) can be sought in the form u−1/2 (x, t) = f (x, t)U (z) + g(x, t),

z = z(x, t).

◭

Remark 3.4. The studies [20, 73, 140, 141, 223, 275] present the results of applying the direct reduction method to construct exact solutions for various nonlinear partial differential equations.

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3.2. Direct Method of Weak Symmetry Reductions 3.2.1. General Description of the Method. Steady-State Boundary Layer Equations Preliminary remarks. In using the various modifications of the direct reduction method, based on formulas (3.1.1.1), (3.1.2.1), and (3.1.3.1), the function w = w(z) stands out, since the other functions must be chosen to ensure that w(z) satisfies a single ordinary differential equation. This requirement imposes serious restrictions on the capacity of the method and its effective use to construct many exact solutions that can be obtained by other methods. In particular, it fails to find the overwhelming majority of the generalized and functional separable solutions considered previously in Chapter 2. The effectiveness of the direct reduction method would greatly increase if it was combined with the methods of generalized and functional separation of variables, where all determining functions are treated as equally important and the function w(z) can be described not only by a single ODE but also by an overdetermined system of several ODEs. The direct method of weak symmetry reductions. To construct exact solutions to nonlinear PDEs, we use relation (3.1.2.1). On substituting it into the nonlinear partial differential equation in question, we arrive at relation (3.1.2.2). In developing the algorithm outlined in Subsection 3.2.1, we will also permit the function w(z) to satisfy an overdetermined system of several ODEs, with the functions f = f (x, t) and g = g(x, t) allowed to satisfy overdetermined systems of PDEs. Considering the fact that equation (3.1.2.2) coincides, up to obvious renaming, with (2.7.1.4), we will use the generalized splitting principle (see Subsections 2.7.1 and 2.7.2) to construct exact solutions to the bilinear equation (3.1.2.2). We will call this combination of the direct reduction method and the method of functional separation of variables the direct method of weak symmetry reductions. Importantly, there is a qualitative difference in the representation of the results obtained by the direct method of symmetry reductions and the direct method of weak symmetry reductions. The solutions obtained with the former method are usually expressed in terms of solutions to nonlinear ODEs, while those obtained with the latter one often admit a representation in a closed form (in terms of quadratures). ◮ Example 3.8. Consider the equation of an extended steady-state axisymmetric laminar hydrodynamic boundary layer

uy uxy − ux uyy = a(yuyy )y + F (x),

(3.2.1.1)

where u is the stream function, x is the coordinate along the symmetry axis, y = 14 r2 , r is the radial coordinate, and F (x) is a pressure function. The longitudinal and transverse components of the fluid velocity, v1 and v2 , are expressed in terms of the stream function as v1 = 2r−1 ur and v2 = −2r−1 ux . We seek a solution to equation (3.2.1.1) in the form u(x, y) = af (x)w(z) + ag(x),

z = ϕ(x)y + ψ(x).

(3.2.1.2)

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3.2. Direct Method of Weak Symmetry Reductions

The factor a is taken for convenience. Substituting (3.2.1.2) into the original equation (3.2.1.1), eliminating y with the help of ϕ(x)y = z − ψ(x), and dividing by a2 ϕ2 f , we arrive at the following functional differential equation with two arguments of the form (1.2.2.1)–(1.2.2.2) from Subsection 1.2.2, where k = 6: (f ϕ)′x ′ 2 F (wz ) + 2 2 = 0. (3.2.1.3) ϕ a fϕ

′′ ′ ′′′ ′′ ′′ (zwzz )z − ψwzzz + fx′ wwzz + gx′ wzz −

Following [48], we use the simplified scheme for constructing exact solutions. ′′ ′′ We assume that the functional coefficients of wwzz , wzz , (wz′ )2 , and 1 are lin′′ ′ ear combinations of the coefficients, 1 and ψ, of the highest-order terms (zwzz )z ′′′ and wzzz . Then we have fx′ = A1 + B1 ψ, gx′ = A2 + B2 ψ, ′ −(f ϕ)x /ϕ = A3 + B3 ψ,

(3.2.1.4)

F /(a2 f ϕ2 ) = A4 + B4 ψ,

where A1 , . . . , A4 and B1 , . . . , B4 are arbitrary constants. Substituting (3.2.1.4) into equation (3.2.1.3), collecting the terms proportional to ψ (we assume that ψ 6= const), and equating the functional coefficient of ψ with zero, we obtain the following overdetermined system of two ordinary differential equations for w = w(z): ′′ ′ ′′ ′′ (zwzz )z + A1 wwzz + A2 wzz + A3 (wz′ )2 + A4 = 0, ′′′ −wzzz

+

′′ B1 wwzz

+

′′ B2 wzz

+

B3 (wz′ )2

Consider three cases. Case 1. We put A1 = A3 = A4 = 0,

(3.2.1.5)

+ B4 = 0.

(3.2.1.6)

A2 = −n.

(3.2.1.7)

Then, equation (3.2.1.5) has the solution w(z) =

C1 z n+1 + C2 z + C3 , n(n + 1)

(3.2.1.8)

where C1 , C2 , and C3 are constants of integration. Solution (3.2.1.8) to equation (3.2.1.5) is simultaneously a solution to equation (3.2.1.6) as well, provided that n = −2,

B1 = B3 ,

C1 = −

4 , B1

C22 = −

B4 , B1

C3 = −

B2 . B1

(3.2.1.9)

Substituting the coefficients (3.2.1.7) and (3.2.1.9) into system (3.2.1.4) and integrating, we find that g(x) = 2x − C3 f,

ϕ=

C4 , f2

ψ=−

C1 ′ f , 4 x

F = −(aC2 C4 )2

fx′ , (3.2.1.10) f3

where f = f (x) is an arbitrary function. Formulas (3.2.1.2), (3.2.1.8), and (3.2.1.10) provide an exact solution to the axisymmetric boundary layer equation (3.2.1.1).

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Case 2. If A2 = B1 = B3 = B4 = 0,

B2 = −λ,

A3 = −A1 ,

A4 = λ2/A1 , (3.2.1.11)

the compatible solution to system (3.2.1.5)–(3.2.1.6) is w(z) =

1 (C1 e−λz + λz − 3). A1

(3.2.1.12)

The solution to system (3.2.1.4) with coefficients (3.2.1.11) is expressed as f = A1 x + C2 ,

ϕ = C3 ,

ψ=−

1 ′ g , λ x

F=

(aC3 λ)2 (A1 x + C2 ), (3.2.1.13) A1

where C1 , C2 , and C3 are arbitrary constants and g = g(x) is an arbitrary function. Formulas (3.2.1.2), (3.2.1.12), and (3.2.1.13) define an exact solution to the boundary layer equation (3.2.1.1). Case 3. System (3.2.1.5)–(3.2.1.6) also admits solutions of the form w(z) = C1 z 2 + C2 z + C3 , with the constants C1 , C2 , and C3 connected to A1 , . . . , A4 and B1 , . . . , B4 . The easiest way to obtain the associated solution is to substitute u = ϕ2 (x)y 2 + ϕ1 (x)y + ϕ0 (x) directly into the original equation (3.2.1.1), which is equivalent to using the method of generalized separation of variables. As a result, we arrive at the solution [275] Z 1 2 1 2 u(x, y) = C1 y + ϕ(x)y + ϕ (x) − F (x) dx − x + C3 , 4C1 2C1 where F (x) and ϕ(x) are arbitrary functions, while C1 and C3 are arbitrary con◭ stants.

3.2.2. Burgers–Huxley Equation (Diffusion Type Equation with a Cubic Nonlinearity) Consider the equation with a cubic nonlinearity ut + σuux = auxx + b3 u3 + b2 u2 + b1 u + b0 .

(3.2.2.1)

For σ = 1 and b0 = b1 = b2 = b3 = 0, it is the Burgers equation, which describes the propagation of waves in nonlinear dissipative systems [368]. For σ = b0 = 0, it coincides with Huxley’s equation, which models the propagation of an electric pulse along a nerve fiber [363]. For b0 = 0, (3.2.2.1) is an unnormalized Burgers–Huxley equation, which describes wall motion of the fluid in liquid crystals as well as the dynamics of populations taking into account reproduction, mortality, nutrition, and diffusion motion [180]. We will seek solutions to equation (3.2.2.1) in the form u(x, t) = f (x, t)w(z) + λ,

z = z(x, t),

(3.2.2.2)

263

3.2. Direct Method of Weak Symmetry Reductions

where the functions f = f (x, t), z = z(x, t), and w = w(z) as well as the constant λ are to be determined. On substituting (3.2.2.2) into (3.2.2.1), we get an equation that can conveniently be represented in the bilinear form 7 X

Φn [x, t]Ψn [z] = 0.

(3.2.2.3)

n=1

Here, the forms Φn = Φn [x, t] depend on the coefficients of equation (3.2.2.1) as well as the functions (and their derivatives) appearing in solution (3.2.2.2); these are expressed as Φ1 = b3 λ3 + b2 λ2 + b1 λ + b0 , Φ2 = 3b3 λ2 f + 2b2 λf + b1 f + afxx − σλfx − ft ,

Φ3 = 3b3 λf 2 + b2 f 2 − σf fx ,

Φ4 = b 3 f 3 ,

Φ5 = af zxx + 2afx zx − σλf zx − f zt ,

Φ6 = −σf 2 zx ,

Φ7 = af zx2 . (3.2.2.4)

The functions Ψn = Ψn [z] are given by Ψ1 = 1, Ψ2 = w, Ψ3 = w2 , Ψ4 = w3 , ′′ Ψ5 = wz′ , Ψ6 = wwz′ , Ψ7 = wzz .

(3.2.2.5)

Further, we use the direct method of week symmetry reductions. It is easy to verify that if w(z) = 1/z, (3.2.2.6) the following three linear relations between the functions (3.2.2.5) hold: Ψ7 = 2Ψ4 ,

Ψ6 = −Ψ4 ,

Ψ5 = −Ψ3 .

(3.2.2.7)

On substituting (3.2.2.7) into (3.2.2.3), we get Φ1 Ψ1 + Φ2 Ψ2 + (Φ3 − Φ5 )Ψ3 + (Φ4 − Φ6 + 2Φ7 )Ψ4 = 0.

(3.2.2.8)

Equating the functional coefficient of Ψ4 in (3.2.2.8) with zero and cancelling by f , we arrive at the equation b3 f 2 + σf zx + 2azx2 = 0, whose solution is f = βzx ,

(3.2.2.9)

where β is a root of the quadratic equation b3 β 2 + σβ + 2a = 0.

(3.2.2.10)

Equating the functional coefficients of Ψ1 , Ψ2 , and Ψ3 in (3.2.2.8) with zero, taking into account (3.2.2.4) and (3.2.2.9), performing simple mathematical transformations, and rearranging the terms, we obtain zt − (3a + βσ)zxx + (σλ + b2 β + 3b3 βλ)zx = 0 (coefficient of Ψ3 ),

zxt − azxxx + σλzxx − (b1 + 2λb2 + 3b3 λ2 )zx = 0 (coefficient of Ψ2 ),

b3 λ3 + b2 λ2 + b1 λ + b0 = 0 (coefficient of Ψ1 ). (3.2.2.11)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

The first two linear partial differential equations make up an overdetermined system for z = z(x, t). The third (cubic) equation serves to determine the constant λ. Using (3.2.2.2), (3.2.2.6), and (3.2.2.9), we write a solution to equation (3.2.2.1) as β u(x, t) = zx + λ. (3.2.2.12) z Let β be a root of the quadratic equation (3.2.2.10) and let λ be a root of the last (cubic) equation in (3.2.2.11). Depending on the value of b3 , we will consider two cases. 1◦ . Case b3 6= 0. From the first two equations in (3.2.2.11) we get zt + p1 zxx + p2 zx = 0, zxxx + q1 zxx + q2 zx = 0,

(3.2.2.13)

where p1 = −βσ − 3a, q1 = −

p2 = λσ + βb2 + 3βλb3 ,

βb2 + 3βλb3 , βσ + 2a

q2 = −

3b3 λ2 + 2b2 λ + b1 . βσ + 2a

(3.2.2.14)

Below are solutions to the overdetermined system of linear equations (3.2.2.13) with (3.2.2.14), which, in conjunction with formula (3.2.2.12) and quadratic equation (3.2.2.10), determine exact solutions of the original PDE (3.2.2.1). Five main situations are possible, which are considered below in order: 1.1. For q2 6= 0 and q12 > 4q2 , we have z(x, t) = C1 exp(k1 x + s1 t) + C2 exp(k2 x + s2 t) + C3 , p kn = − 12 q1 ± 21 q12 − 4q2 , sn = −kn2 p1 − kn p2 , n = 1, 2,

where C1 , C2 , and C3 are arbitrary constants. 1.2. For q2 6= 0 and q12 < 4q2 , we find that

z(x, t) = [C1 sin(k1 x + s1 t) + C2 cos(k1 x + s1 t)] exp(k2 x + s2 t) + C3 , p p k1 = 21 4q2 − q12 , s1 = 12 (p1 q1 − p2 ) 4q2 − q12 ,
k2 = − 12 q1 , s2 = p1 q2 − 12 q12 + 12 p2 q1 ,

where C1 , C2 , and C3 are arbitrary constants. 1.3. For q2 6= 0 and q12 = 4q2 , we get

z(x, t) = C1 exp(kx + s1 t) + C2 (kx + s2 t) exp(kx + s1 t) + C3 , k = − 12 q1 , s1 = − 14 p1 q12 + 12 p2 q1 , s2 = − 12 p1 q12 + 12 p2 q1 . 1.4. For q2 = 0 and q1 6= 0, z(x, t) = C1 (x − p2 t) + C2 exp[−q1 x + q1 (p2 − p1 q1 )t] + C3 .

3.2. Direct Method of Weak Symmetry Reductions

265

1.5. For q2 = q1 = 0, z(x, t) = C1 (x − p2 t)2 + C2 (x − p2 t) − 2C1 p1 t + C3 . 2◦ . Case b3 = 0, b2 6= 0. The solutions are defined by formulas (3.2.2.12) where    b1 σ 2ab2 z(x, t) = C1 + C2 exp Ax + A t , + 2b2 σ σ(b1 + 2b2 λ) 2a β=− , A= , σ 2ab2 and λ = λ1,2 are roots of the quadratic equation b2 λ2 + b1 λ + b0 = 0. Remark 3.5. The above solutions to equation (3.2.2.1) with b0 = 0 were obtained in [94, 95, 176] using the Weiss–Tabor–Carnevalle method [369], which is based on truncated Painlev´e expansions (see also [65, 180]).

3.2.3. Unsteady Plane and Axisymmetric Boundary Layer Equations Plane boundary layer equations. The hydrodynamic boundary layer equations describe flows of viscous fluids in a thin layer near solid, fluid, or gas surfaces of various shape at large Reynolds numbers [191, 192, 259, 329]. The system of equations of an unsteady plane laminar boundary layer for an incompressible Newtonian fluid (classical fluid model) is written as [192, 329]: Ut + U Ux + V Uy = νUyy + F (t, x), Ux + Vy = 0,

(3.2.3.1) (3.2.3.2)

where t is time, x and y are the longitudinal and transverse coordinates (y = 0 correspond to the surface of the body), U and V are the longitudinal and transverse fluid velocity components, F (t, x) = −px /ρ is a given function (proportional to the pressure gradient), p is pressure, ρ is mass density, and ν is the kinematic viscosity. With the stream function W defined by the formulas U = Wy ,

V = −Wx ,

(3.2.3.3)

system (3.2.3.1)–(3.2.3.2) reduces to a single third-order PDE [192, 329]: Wty + Wy Wxy − Wx Wyy = νWyyy + F (t, x).

(3.2.3.4)

Exact solutions and transformations for the system of PDEs (3.2.3.1)–(3.2.3.2) and equation (3.2.3.4) as well as various boundary layer problems have been considered in numerous studies (e.g., see [6, 47–50, 153, 191, 192, 196, 197, 231, 235, 243, 244, 259, 271, 275, 323, 329, 355, 372, 373]). For invariant and noninvariant exact solutions to these equations in the steady-state case (with Wt = 0), see [6, 47, 48, 50, 153, 191, 192, 235, 243, 259, 275, 323, 329]. For some exact solutions and transformations of the unsteady plane boundary layer equations (3.2.3.4), see [49, 196, 197, 231, 244, 271, 275, 280, 281, 355, 373].

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Remark 3.6. The studies [48, 275, 323] present some exact solutions to the axisymmetric boundary layer equations on the surface of an extended body of revolution; these dealt with the equation that can formally be obtained from (3.2.3.4) by replacing Wyyy with (yWyy )y (see also Subsection 3.2.4).

Axisymmetric boundary layer equations. The system of unsteady axisymmetric laminar boundary layer equations is written as [191, 329]: Ut + U Ux + V Uy = νUyy + F (t, x), (rU )x + (rV )y = 0,

r = r(x),

(3.2.3.5) (3.2.3.6)

where x and y are the longitudinal and transverse coordinates (the corresponding unit vectors, ex and ey , are tangent and normal to the surface of the body of revolution), U and V are the longitudinal and transverse fluid velocity components, F (t, x) = −px /ρ is a given function, and r = r(x) is a dimensionless cross-sectional radius perpendicular to the axis of rotation, which defines the shape of the body. The other notations are the same as in equation (3.2.3.4). The choice of the unit length for r = r(x) does not affect the form of the equations. Introducing the stream function W by the formulas U = Wy ,

V = −Wx −

rx′ W, r

r = r(x),

(3.2.3.7)

one converts system (3.2.3.5)–(3.2.3.6) to a single third-order equation [191, 329]: Wty + Wy Wxy − Wx Wyy −

rx′ W Wyy = νWyyy + F (t, x). r

(3.2.3.8)

For r(x) = const, equation (3.2.3.8) coincides with (3.2.3.4). The case r(x) 6= const is more complicated. The study [7] described a number of specific functions r(x) that allow the unsteady boundary layer equation with three independent variables (3.2.3.8) to be reduced to a single ODE or a single PDE with two independent variables; the analysis relied on a modification of the direct method of symmetry reductions [72]. In what follows, we will consider the general case of equation (3.2.3.8) with arbitrary r = r(x). The admissible forms of the pressure function F (t, x) will, as usual, be determined in the subsequent investigation. Reduction to a plane boundary layer equation with a variable viscosity. By introducing the new variables z = r(x)y,

w = r(x)W,

(3.2.3.9)

we convert equation (3.2.3.8) to a more convenient form for the further analysis [293]: wtz + wz wxz − wx wzz = νr2 (x)wzzz + F (t, x).

(3.2.3.10)

This equation can be treated as an unsteady plane boundary layer equation with a variable viscosity, νe = νr2 (x),

267

3.2. Direct Method of Weak Symmetry Reductions

dependent on the longitudinal coordinate x. The pressure function F (t, x) remains unchanged. In the special case r(x) ≡ 1, equation (3.2.3.10) coincides with the plane boundary layer equation (3.2.3.4). At the end of Subsection 3.2.3, we give a physical interpretation of a boundary layer equation with a variable viscosity, νe = νe (x), unrelated to system (3.2.3.5)– (3.2.3.6). General form of solutions. Determining equations. Reduction to an ODE. We seek exact solutions to equation (3.2.3.10) in the form [293] w = f u(ξ) + gz + h,

ξ = ϕz + ψ,

(3.2.3.11)

where the six functions f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), ψ = ψ(t, x), and u = u(ξ) are to be determined in the analysis. On substituting (3.2.3.11) into (3.2.3.10), we obtain an equation that can conveniently be represented in the bilinear form 7 X

Φn [x, t]Ψn [ξ] = 0.

(3.2.3.12)

n=1

The functions Φn = Φn [x, t] depend on the functional coefficients (and their derivatives) appearing in equation (3.2.3.10) and solution (3.2.3.11); these are expressed as Φ1 = gt + ggx − F,

Φ2 = (f ϕ)t + (f gϕ)x ,

Φ3 = f ϕ(f ϕ)x ,

Φ4 = f (ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 ),

Φ5 = f (ϕt + gϕx − gx ϕ),

2

(3.2.3.13)

2

Φ6 = −f fx ϕ ,

3

Φ7 = −νr f ϕ .

The functions Ψn = Ψn [ξ] are given by Ψ1 = 1,

Ψ2 = u′ξ ,

Ψ5 = ξu′′ξξ ,

Ψ3 = (u′ξ )2 ,

Ψ6 = uu′′ξξ ,

Ψ4 = u′′ξξ ,

(3.2.3.14)

Ψ7 = u′′′ ξξξ .

Equation (3.2.3.12)–(3.2.3.14) reduces to a single ODE for u = u(ξ) if all Φn (n = 1, . . . , 6) are set proportional to Φ7 , so that Φn = −an Φ7

(n = 1, . . . , 6),

(3.2.3.15)

where a1 , . . . , a6 are free parameters. On substituting (3.2.3.13) into (3.2.3.15), we arrive at a linear system of PDEs for the functions f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), and ψ = ψ(t, x): gt + ggx − F = a1 νr2 f ϕ3 , 2

3

(f ϕ)t + (f gϕ)x = a2 νr f ϕ ,

(3.2.3.17)

2

2

(3.2.3.18)

2

3

(3.2.3.19)

2

3

(3.2.3.20)

(f ϕ)x = a3 νr ϕ , 2

(3.2.3.16)

ϕψt + gϕψx + gx ϕψ − ϕt ψ − gϕx ψ − hx ϕ = a4 νr ϕ , ϕt + gϕx − gx ϕ = a5 νr ϕ , 2

fx = −a6 νr ϕ.

(3.2.3.21)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

As a result, equation (3.2.3.12) reduces to the following ODE for u = u(ξ): a1 + a2 u′ξ + a3 (u′ξ )2 + a4 u′′ξξ + a5 ξu′′ξξ + a6 uu′′ξξ = u′′′ ξξξ .

(3.2.3.22)

Any compatible solution to the determining system of first-order PDEs (3.2.3.16) to (3.2.3.21) in conjunction with the associated solution to the third-order ODE (3.2.3.22) generates an exact solution (3.2.3.11) to equation (3.2.3.10). Analysis and solutions to the determining system (3.2.3.16)–(3.2.3.21) for the axisymmetric boundary layer with an arbitrary r(x). The system of equations (3.2.3.16)–(3.2.3.21) can be split up into a few simpler subsystems, which can be treated independently. The subsystem consisting of equations (3.2.3.18) and (3.2.3.21) allows us to express f and ϕ in terms of r = r(x). For given f and ϕ, the subsystem consisting of equations (3.2.3.17) and (3.2.3.20) serves to determine g and r, whence it follows that the function r = r(x) cannot generally be treated as arbitrary. Once g and r are determined, the pressure function is evaluated as F = gt + ggx − a1 νr2 f ϕ3 ,

(3.2.3.23)

which follows from equation (3.2.3.16). The function ψ in equation (3.2.3.19) can be treated as defined arbitrarily. Integrating with respect to x gives Z
1 ϕψt + gϕψx + gx ϕψ − ϕt ψ − gϕx ψ − a4 νr2 ϕ3 dx + p(t), (3.2.3.24) h= ϕ2 where p(t) is an arbitrary function. In what follows, we consider r = r(x) to be arbitrary. We start the analysis with the subsystem consisting of equations (3.2.3.17) and (3.2.3.20). On isolating the derivative gx , we rewrite these equations in the form (f ϕ)x (f ϕ)t g = a2 νr2 ϕ2 − , fϕ fϕ ϕt ϕx g = −a5 νr2 ϕ2 + . gx − ϕ ϕ

gx +

(3.2.3.25) (3.2.3.26)

Let us find the conditions under which equations (3.2.3.25) and (3.2.3.26) coincide. In this case, r = r(x) can be treated as arbitrary. Then, g is determined from (3.2.3.26) and, hence, is expressed in terms of ϕ and r. Equating the functional coefficients of g as well as the right-hand sides of equations (3.2.3.25) and (3.2.3.26) with each other, we find that ϕx (f ϕ)x + = 0, ϕ fϕ

ϕt (f ϕ)t + = (a2 + a5 )νr2 ϕ2 . ϕ fϕ

(3.2.3.27)

Integrating the first equation of (3.2.3.27) yields f = λ(t)ϕ−2 ,

(3.2.3.28)

3.2. Direct Method of Weak Symmetry Reductions

269

where λ(t) is an arbitrary function. Substituting (3.2.3.28) into the second equation of (3.2.3.27) gives [ln λ(t)]′t = (a2 + a5 )νr2 (x)ϕ2 . (3.2.3.29) Equation (3.2.3.29) holds in the following two cases: (a) a5 = −a2 , (b) ϕ =

σ(t) , r(x)

λ(t) = λ0 ,

ϕ = ϕ(t, x) is any function;   Z λ(t) = λ0 exp (a2 + a5 ) ν σ 2 (t) dt ,

(3.2.3.30) (3.2.3.31)

where λ0 is an arbitrary constant and σ(t) is an arbitrary function. Case (a). Without loss of generality, we can set λ0 = 1. In view of (3.2.3.28), we get f = ϕ−2 . Equations (3.2.3.18) and (3.2.3.21) become ϕ′x = −a3 νr2 ϕ4 ,

ϕ′x =

2 4 1 2 a6 νr ϕ .

(3.2.3.32)

These equations are compatible if a6 = −2a3 . Integrating the first equation of (3.2.3.32) and then (3.2.3.26), we obtain an exact solution to equation (3.2.3.10) of the form (3.2.3.11), in which one should set  2/3 Z 2 f = 3a3 ν r (x) dx + b(t) ,  −1/3 Z 2 ϕ = 3a3 ν r (x) dx + b(t) ,  Z  ϕt g = c(t)ϕ + ϕ + a2 νr2 ϕ dx, ϕ2

(3.2.3.33)

where b = b(t), c = c(t), and r = r(x) are arbitrary functions, h is defined by (3.2.3.24), ψ = ψ(t, x) is an arbitrary function, and u = u(ξ) is a function satisfying the single ODE (3.2.3.22) with a5 = −a2 and a6 = −2a3 [293]. The pressure function is given by (3.2.3.23). It is noteworthy that for a3 6= 0, the function g can be represented as Z 1 a2 −1 g = c(t)ϕ − b′t (t)ϕ ϕ2 dx + ϕ . 3 2a3

Case (b). It follows from formulas (3.2.3.28) and (3.2.3.31) as well as equations (3.2.3.18) and (3.2.3.21) that the conditions a6 = −2a3 and r(x) = αx + β must hold, which means that r(x) is not arbitrary. We do not consider this case here. Solutions to the determining system (3.2.3.16)–(3.2.3.21) for the plane boundary layer with r(x) ≡ 1. Following [293], we will now consider the case of the plane boundary layer equation (3.2.3.10) with r(x) = 1 and describe some of its exact solutions of the form (3.2.3.11). In all the cases considered below, we will only give the expressions of ϕ = ϕ(t, x), f = f (t, x), and g = g(t, x). The function ψ = ψ(t, x) is arbitrary, while the functions h = h(t, x) and F = F (t, x) are defined by formulas (3.2.3.24) and (3.2.3.23) with r(x) = 1. The function u = u(ξ) satisfies ODE (3.2.3.22).

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

1◦ . Case of a5 = −a2 , a6 = −2a3 , and a3 6= 0. We obtain an exact solution of the form (3.2.3.11) by setting r(x) = 1 in (3.2.3.33), (3.2.3.23), and (3.2.3.24). As a results, we get  −1/3 ϕ = 3a3 νx + b(t) ,  2/3 f = 3a3 νx + b(t) , 1 ′ a2 −1 g = c(t)ϕ − b (t) + ϕ , 3a3 ν t 2a3

(3.2.3.34)

where b = b(t) and c = c(t) are arbitrary functions. 2◦ . Case of a5 = −a2 and a3 = a6 = 0. Formulas (3.2.3.33) with a3 = 0 results in the exact solution (3.2.3.11) in which one should put ϕ = ϕ(t) is an arbitrary function, f = ϕ−2 ,   ϕt g= + a2 νϕ2 x + c(t)ϕ, ϕ

(3.2.3.35)

where c = c(t) is an arbitrary function, while the other functions are described above. 3◦ . Case of a3 = a6 = 0. The exact solution of the form (3.2.3.11) is given by ϕ = ϕ(t) is an arbitrary function,   Z k 2 f = 2 exp (a2 + a5 ) ν ϕ dt , ϕ   ϕt 2 g= − a5 νϕ x + c(t)ϕ, ϕ

(3.2.3.36)

where c(t) is an arbitrary function and k is an arbitrary constant. The three functions (3.2.3.36) satisfy four equations (3.2.3.17), (3.2.3.18), (3.2.3.20), and (3.2.3.21) with r = 1 and a3 = a6 = 0; for a5 = −a2 and k = 1, they become (3.2.3.35). For order reduction and exact solutions of the nonlinear ODE (3.2.3.22), see [293]. Utilizing the method of weak symmetry reductions to construct exact solutions. The boundary layer equation (3.2.3.10) with arbitrary r(x) has a lot more exact solutions of the form (3.2.3.11) than those obtained above by direct reduction to a single ODE, (3.2.3.22), with a5 = −a2 and a6 = −2a3 . Following [293], we outline a procedure for constructing other exact solutions of the form (3.2.3.11). The determining system (3.2.3.16)–(3.2.3.21) was obtained under the assumption that the first six functions Ψi in (3.2.3.14) are linearly independent. If some of the Ψi ’s are linearly dependent, one should choose a linearly independent subsystem, {Ψj }, and express the other functions via the elements of this subsystem. In this case, one should rewrite relation (3.2.3.12) as a linear combination of the functions Ψj and then equate the functional coefficients of Ψj with zero. The

3.2. Direct Method of Weak Symmetry Reductions

271

resulting modified determining system will not only differ from system (3.2.3.16)– (3.2.3.21) but will also consist of fewer equations. The fewer linearly independent functions enter (3.2.3.14), the fewer equations will enter the determining system and the greater number of arbitrary functions can appear in its solution. A number of other solutions of the form (3.2.3.11) than those described by equations (3.2.3.16)–(3.2.3.22) arise when, among the function Ψi , there are two or more linearly independent subsystems one of which involves Ψ7 . As functions u = u(ξ) that determine Ψi in (3.2.3.14), one should use particular solutions of ODE (3.2.3.22) with suitable values of the parameters ai . Table 3.1 presents, in accordance with [256, 293], twelve functions u = u(ξ) that lead to two or three linear relations between the functions (3.2.3.14). The function in row 1 gives Ψ7 = 0, which corresponds to a degenerate solution. The other rows contain functions that lead to nondegenerate solutions of equation (3.2.3.10). Table 3.1. Generating functions u and associated linear relations for Ψn . Notation: Ψ1 = 1, Ψ2 = u′ξ , Ψ3 = (u′ξ )2 , Ψ4 = u′′ξξ , Ψ5 = ξu′′ξξ , Ψ6 = uu′′ξξ , Ψ7 = u′′′ ξξξ . No. Generating functions u

Linear relations for Ψn

1

u=ξ

2

Ψ4 = 2Ψ1 , Ψ5 = 32 Ψ2 , Ψ6 = 12 Ψ3

2

u = ξ3

Ψ5 = 2Ψ2 , Ψ6 = 32 Ψ3 , Ψ7 = 6Ψ1

3

u=ξ

−1

4

u = exp ξ

5

u = cosh ξ

6

u = sinh ξ

7

u = cos ξ

8

u = sin ξ

9

u = tanh ξ

10

u = coth ξ

11

u = tan ξ

12

u = cot ξ

Ψ5 = −2Ψ2 , Ψ6 = 2Ψ3 , Ψ7 = −6Ψ3 Ψ2 = Ψ4 = Ψ7 , Ψ6 = Ψ3 Ψ6 = Ψ1 +Ψ3 , Ψ7 = Ψ2 Ψ6 = Ψ3 −Ψ1 , Ψ7 = Ψ2

Ψ6 = Ψ3 −Ψ1 , Ψ7 = −Ψ2 Ψ6 = Ψ3 −Ψ1 , Ψ7 = −Ψ2

Ψ6 = −2Ψ2 +2Ψ3 , Ψ7 = −2Ψ2 −3Ψ6 Ψ6 = −2Ψ2 +2Ψ3 , Ψ7 = −2Ψ2 −3Ψ6 Ψ6 = −2Ψ2 +2Ψ3 , Ψ7 = 2Ψ2 +3Ψ6 Ψ6 = 2Ψ2 +2Ψ3 , Ψ7 = 2Ψ2 −3Ψ6

Examples of constructing exact solutions. Below we give a few examples illustrating the construction of exact solutions to the third-order nonlinear PDE (3.2.3.10) by using some generating functions specified in Table 3.1. ◮ Example 3.9. Let us use the function u = ξ 3 from row 2 of Table 3.1 and

construct an exact solution. In view of formulas (3.2.3.14) and the linear relations for Ψi (see Table 3.1), we rewrite equation (3.2.3.12)–(3.2.3.14) in the form   gt + ggx − F − 6νr2 f ϕ3 Ψ1   + (f ϕ)t + (f gϕ)x + 2f (ϕt + gϕx − gx ϕ) Ψ2   + f ϕ(f ϕ)x − 23 f fx ϕ2 Ψ3 + f (ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 )Ψ4 = 0.

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Equating the functional coefficients of Ψ1 , . . . , Ψ4 with zero, we arrive at the system of PDEs gt + ggx − F − 6νr2 f ϕ3 = 0, (f ϕ)t + (f gϕ)x + 2f (ϕt + gϕx − gx ϕ) = 0, 1 3 fx ϕ − hx ϕ2

f ϕx +

ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ

(3.2.3.37) (3.2.3.38)

= 0,

(3.2.3.39)

= 0.

(3.2.3.40)

Equation (3.2.3.39) has been rewritten in a more convenient form and the last two equations have been divided by the nonzero quantities f ϕ and f . System (3.2.3.37)–(3.2.3.40) has two equations fewer than system (3.2.3.16)– (3.2.3.21). Moreover, the resulting equations do not require a compatibility analysis. Equations (3.2.3.37)–(3.2.3.39) allow one to find f , g, and F for arbitrary r(x). The functions ϕ = ϕ(t, x) and ψ = ψ(t, x) remain arbitrary, while h, as follows from (3.2.3.40), is defined by formula (3.2.3.24) with a4 = 0. The general solution to equation (3.2.3.39) is f = a(t)ϕ−3 ,

(3.2.3.41)

where a(t) is an arbitrary function. On substituting this into (3.2.3.38) and on integrating, we obtain  1  ′ g= a (t)x + b(t) , (3.2.3.42) a(t) t where b(t) is an arbitrary function. On substituting (3.2.3.41) and (3.2.3.42) into (3.2.3.37), we find the pressure function: 1 [a′′ (t)x + b′t (t)]. F (t, x) = −6νa(t)r2 (x) + a(t) tt For ψ = h = 0, substituting (3.2.3.41) and (3.2.3.42) into (3.2.3.11) gives the solution 1 [a′ (t)x + b(t)]z, u = a(t)z 3 + a(t) t where a(t) and b(t) are arbitrary functions. Interestingly, the arbitrary function ϕ, which appears in (3.2.3.41), does not enter the final result, because it is cancelled out ◭ after the substitution into (3.2.3.11). ◮ Example 3.10. Now let us take the function u = exp ξ from row 4 of Table 3.1. Using formulas (3.2.3.14) and the linear relations for Ψi from the table, we rewrite equation (3.2.3.12)–(3.2.3.14) as   (gt + ggx − F )Ψ1 + (f ϕ)t + (f gϕ)x − f ϕ2 hx − νr2 f ϕ3 Ψ2   + f ϕ (f ϕ)x − fx ϕ Ψ3 + f (ϕt + gϕx − gx ϕ)Ψ5 = 0,

where we set ψ = 0 for simplicity. On equating the functional coefficients of Ψ1 , Ψ2 , Ψ3 , and Ψ5 with zero, we get gt + ggx − F = 0,

(f ϕ)t + (f gϕ)x − f ϕ2 hx − νr2 f ϕ3 = 0, ϕx = 0, ϕt + gϕx − gx ϕ = 0.

(3.2.3.43)

3.2. Direct Method of Weak Symmetry Reductions

273

It follows from the first, third, and fourth equations that ϕ = ϕ(t),

g=

 1 ′ ϕ x + b(t) , ϕ t

F =

 1  ′′ ϕ x + b′t (t) , ϕ tt

(3.2.3.44)

where ϕ = ϕ(t) and b = b(t) are arbitrary functions. The second equation in (3.2.3.43) gives  Z  2ϕ′t 1 fx ft 2 2 h= 2 x+ (3.2.3.45) +g − νr ϕ dx + c(t), ϕ ϕ f f where c(t) is an arbitrary function. Formulas (3.2.3.11), (3.2.3.44), and (3.2.3.45) for arbitrary f = f (t, x) and ψ = 0 define an exact solution to equation (3.2.3.10) for arbitrary r(x), that is, for any shape ◭ of the surface of the body. Remark 3.7. Importantly, the direct method of weak symmetry reductions outlined in Subsection 3.2.1 allows one to find a lot more exact solutions to equation (3.2.3.10) than the direct method of symmetry reductions described in Section 3.1.

A physical interpretation of the plane boundary layer equation with a variable viscosity. Consider the plane boundary layer equation with a variable viscosity, ν = ν(x), Wty + Wy Wxy − Wx Wyy = ν(x)Wyyy + F (t, x),

(3.2.3.46)

where x and y are the longitudinal and transverse coordinates, and W is the stream function, in terms of which the velocity components U and V are expressed by formulas (3.2.3.3). Such equations arise in studying connected thermohydrodynamic problems when the following conditions hold: (a) kinematic viscosity depends significantly on temperature in the temperature range of interest: ν = ν˜(T ); (b) surface that is in contact with the flow is heated unevenly: T |y=0 = T0 (x); (c) Prandtl number Pr is small; this is true for liquid metals and melts, where 5 × 10−3 ≤ Pr ≤ 5 × 10−2 [212, 259, 296]; (d) in the temperature range of interest, the relative change of the fluid density is negligible compared to that of viscosity; this is the case for liquid metals and melts [212, 296] (in particular, the kinematic viscosity of sodium changes by 52% across the temperature range from 100 to 200 ◦ C, while its density changes by only 2.8%). Condition (c) implies that the hydrodynamic boundary layer is much thinner than the thermal boundary layer. This means that the temperature in the hydrodynamic boundary layer is approximately equal to the leading term of the asymptotic expansion of the temperature near the surface in accordance with condition (b): T ≈ T0 (x). It follows from the above and condition (a) that viscosity only depends on the longitudinal coordinate: ν = ν˜(T0 (x)). Up to notation, equation (3.2.3.10) coincides with (3.2.3.46). A Blasius type problem for an unevenly heated flat plate. Consider a uniform steady-state flow of an incompressible viscous fluid past an unevenly heated flat

274

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

plate. We assume that conditions (a)–(d) hold. The corresponding hydrodynamic problem is described by the boundary layer equation (3.2.3.46) with W = W (x, y), Wt = 0, F (t, x) = 0, and ν(x) = ν˜(T0 (x)) and the boundary conditions Wx = Wy = 0 at y = 0,

Wy → U∞

as y → ∞,

(3.2.3.47)

where U∞ is the unperturbed fluid velocity far away from the plate, while y = 0 corresponds to the plate surface. In the special case ν = const, problem (3.2.3.46)– (3.2.3.47) becomes the Blasius problem [192, 329]. Problem (3.2.3.46)–(3.2.3.47) has the following solution for arbitrary ν = ν(x): r Z x p U∞ W = U∞ X u(ξ), ξ = y , X= ν(s) ds. (3.2.3.48) X 0

The function u = u(ξ) satisfies the equation and boundary conditions u′′′ ξξξ +

1 2

u = u′ξ = 0 at ξ = 0,

uu′′ξξ = 0; u′ξ → 1

as ξ → ∞.

(3.2.3.49)

For numerical and approximate analytical solutions to problem (3.2.3.49), see [191, 192, 329]. In view of the above results, the local shear stress (friction drag) at the surface of the plate can be evaluated as Z x 3/2 ′′ τ (x) = (µUy )y=0 = ρU∞ u (0)ν(x)X −1/2 , X = ν(s) ds, (3.2.3.50) 0

where u′′ (0) ≈ 0.332. The corresponding friction drag coefficient equals Z x 0.664 U∞ L(x) 1 τ (x) √ = , Rex = , L(x) = ν(s) ds, cf = 1 2 ν(x) ν(x) 0 Rex 2 ρU∞ (3.2.3.51) where Rex is the local Reynolds number and L(x) is the effective distance from the front edge of the plate. In the special case of constant viscosity (ν = const), formulas (3.2.3.50) and (3.2.3.51) coincide with those presented in [191, 192, 329]. ◮ Example 3.11. Measurements of temperature dependences of the kinematic viscosity, ν = ν˜(T ) show that many liquid metals and melts are characterized by a viscosity hysteresis in wide temperature ranges [130]; this means that the ν˜(T ) curves are different in heating and cooling. The heating curve usually has a sophisticated shape, while an Arrhenius type equation, ν = αeβ/T , well describes the cooling curve, where α and β are some constants characterizing the physical properties of the fluid and T is thermodynamic temperature. At temperatures above a certain threshold, which is fluid specific, the heating and cooling curves practically coincide, to within the measurement errors [130]. Rather than the Arrhenius equation, we will use the simple approximation ν = ν0 eγ(T0 −T ) , which works well in a wide range of temperatures.

275

3.2. Direct Method of Weak Symmetry Reductions

We will compare two scenarios to estimate the effect of the temperature dependence of viscosity on the local shear stress and friction drag coefficient: (i) the plate is maintained at a constant temperature, T0 = 200 ◦ C, and (ii) the plate temperature decreases linearly from T0 = 200 ◦ C at x = 0 to Tl = 100 ◦ C at a certain length x = l. Neglecting the changes in the fluid density, we will estimate the ratio cf2 (l)/cf1 (l) ≈ τ2 (l)/τ1 (l), where the subscripts 1 and 2 refer to cases (i) and (ii), respectively. It follows from formulas (3.2.3.50) and (3.2.3.51) that s Z l cf2 (l) τ2 (l) ν(l) ν0 l = = ; ν(x) = ν0 eγ(T0 −T (x)) , X(l) = ν(x) dx. cf1 (l) τ1 (l) ν0 X(l) 0 Since temperature is linearly dependent on the coordinate, we have T = T0 −

T0 − Tl x l

=⇒

dx = −

l dT. T0 − Tl

(3.2.3.52)

Hence, passing from x to the variable of integration T , we get Z

Tl


l ν0 l dT = eγ(T0 −Tl ) − 1 Tl − T0 γ(T0 − Tl ) T0 ν  ν0 l l = −1 , νl = ν(l). ln(νl /ν0 ) ν0

X(l) =

ν0 eγ(T0 −T )

As a result, we obtain [293] νl cf2 (l) = cf1 (l) ν0

s

νl ln ν0



 νl −1 . ν0

(3.2.3.53)

Table 3.2 lists the ratios of the friction drag coefficients (penultimate column) calculated by formula (3.2.3.53) for three liquid metals and one melt under the above ◭ assumptions. The results obtained are discussed at the end of Example 3.12. Table 3.2. Increment in the friction drag coefficient due to temperature dependence of viscosity. The superscript ‘e’ refers to the exponential approximation ν = ν0 eγ(T0 −T ) , while the superscript ‘p’ refers to the power-law approximation ν = ν0 (T0 /T )k . Liquid metal/melt ν0 , 108 m2/s

k

cef2 (l)/cef1 (l) cpf2 (l)/cpf1 (l)

Hg

7.9

0.260

1.13

1.15

Na

50.5

0.615

1.37

1.39

K

41.2

0.527

1.22

1.32

25% Na + 75% K

43.7

0.540

1.25

1.33

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

◮ Example 3.12. Now we assume that viscosity is a power-law function of temperature, ν = ν0 (T0 /T )k ; a detailed justification of this formulas is given in [296]. Consider the two scenarios described in Example 3.11. Neglecting the changes in the fluid density, we find from formulas (3.2.3.50) and (3.2.3.51) that s  k Z l cf2 (l) τ2 (l) νl ν0 l T0 ≈ = ; ν = ν0 , X(l) = ν(T (x)) dx. cf1 (l) τ1 (l) ν0 X(l) T 0

Since, by assumption, temperature is linearly dependent on the coordinate, we have (3.2.3.52), and hence Z Tl −l ν0 l T0 − T0k Tl1−k ν0 (T0 /T )k X(l) = dT = , 0 < k < 1. T0 − Tl 1−k T0 − Tl T0 As a result, we get [296]: s r  T k (T0 /Tl ) − 1 1−k cf2 (l) 0 k = (1 − k) =2 . k cf1 (l) Tl (T0 /Tl ) − (T0 /Tl ) 2 − 2k

(3.2.3.54)

Table 3.2 lists the ratios of the friction drag coefficients (last column) evaluated by formula (3.2.3.54) for three liquid metals and one melt under the adopted assumptions. It is apparent that due to the temperature dependence of viscosity, the drag coefficient increases substantially. The increment is 15, 39, 32, and 33% for Hg, Na, K, and 25% Na + 75% K, respectively. These figures are slightly greater than the respective figures (fourth column) obtained with the exponential approximation of viscosity’s temperature dependence. It is noteworthy that formulas (3.2.3.53) and (3.2.3.54) hold true for the laminar flow mode, in which Rel = U∞ X(l)/νl2 . 105 [192, 329]. ◭

3.2.4. Axisymmetric Boundary Layer Equations for an Extended Body of Revolution Reduction of the system of boundary layer equations to a single PDE. The system of boundary layer equations that describes an unsteady axisymmetric laminar flow of a Newtonian fluid past an extended body of revolution is expressed as [77, 329]:
Ut + U Ux + V Ur = ν Urr + r−1 Ur + F (t, x), (3.2.4.1) Ux + Vr + r−1 V = 0,

(3.2.4.2)

where t is time, U and V are the axial and radial fluid velocity components, respectively, x and r are the axial and radial coordinates, F (t, x) = −px /ρ is a given function (proportional to the axial pressure gradient), p is pressure, ρ is mass density, and ν is kinematic viscosity. For F ≡ 0, system (3.2.4.1)–(3.2.4.2) describes an axisymmetric jet flow at large Reynolds numbers.

3.2. Direct Method of Weak Symmetry Reductions

277

A self-similar solution to a jet flow problem described by the steady-state system of equations (3.2.4.1)–(3.2.4.2) with F (t, x) ≡ 0 was obtained in [329]. Introducing a modified stream function, w, by the formulas U = 2r−1 wr ,

V = −2r−1 (wx − ν),

z=

1 2 4r ,

(3.2.4.3)

we reduce system (3.2.4.1)–(3.2.4.2) to the single nonlinear third-order equation wtz + wz wxz − wx wzz = νzwzzz + F (t, x).

(3.2.4.4)

Remark 3.8. Formulas (3.2.4.3) differ slightly from those used in [48, 275, 323] and result in a simpler equation, (3.2.4.4). The studies [48, 275, 323] present a number of transformations as well as some exact solutions to the equation obtained from (3.2.4.4) by the change of variable w = w ¯ + νx.

Note that equations (3.2.4.4) and (3.2.3.10) differ from each other in the functional coefficient of the highest derivative. Following [295], we will further construct a few exact solutions to the unsteady nonlinear third-order PDE (3.2.4.4), in which the admissible pressure gradient functions F (t, x) will, as usual, be determined in the subsequent analysis. General form of solutions. The determining equations. Reduction to an ODE. Just as in Subsection 3.2.3, we will seek exact solutions to equation (3.2.4.4) in the form w = f u(ξ) + gz + h, ξ = ϕz + ψ, (3.2.4.5) where the functions f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), ψ = ψ(t, x), and u = u(ξ) are to be determined in the subsequent analysis. On substituting (3.2.4.5) into (3.2.4.4), we obtain an equation that can conveniently be represented in the bilinear form 8 X

Φn [x, t]Ψn [ξ] = 0.

(3.2.4.6)

n=1

The functions Φn = Φn [x, t] depend on the functional coefficients (and their derivatives) appearing in equation (3.2.4.4) and solution (3.2.4.5); these are expressed as Φ1 = gt + ggx − F,

Φ2 = (f ϕ)t + (f gϕ)x ,

Φ3 = f ϕ(f ϕ)x ,

Φ4 = f (ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 ),

Φ5 = f (ϕt + gϕx − gx ϕ),

Φ7 = νf ϕ2 ψ,

Φ6 = −f fxϕ2 ,

(3.2.4.7)

Φ8 = −νf ϕ2 .

The functions Ψn = Ψn [ξ] are Ψ1 = 1,

Ψ2 = u′ξ ,

Ψ5 = ξu′′ξξ ,

Ψ3 = (u′ξ )2 ,

Ψ6 = uu′′ξξ ,

Ψ4 = u′′ξξ ,

Ψ7 = u′′′ ξξξ ,

Ψ8 = ξu′′′ ξξξ .

(3.2.4.8)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Equation (3.2.4.6)–(3.2.4.8) reduces to a single ODE for u = u(ξ) if all Φn (n = 1, . . . , 7) are set proportional to Φ8 , or Φn = −an Φ8

(n = 1, . . . , 7),

(3.2.4.9)

where a1 , . . . , a7 are free parameters. On substituting the functions (3.2.4.7) into (3.2.4.9), we arrive at a nonlinear system of PDEs for f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), and ψ = ψ(t, x): gt + ggx − F = a1 νf ϕ2 , 2

(f ϕ)t + (f gϕ)x = a2 νf ϕ , (f ϕ)x = a3 νϕ, ϕψt + gϕψx + gx ϕψ − ϕt ψ − gϕx ψ − hx ϕ2 = a4 νϕ2 , 2

ϕt + gϕx − gx ϕ = a5 νϕ , fx = −a6 ν, ψ = a7 .

(3.2.4.10) (3.2.4.11) (3.2.4.12) (3.2.4.13) (3.2.4.14) (3.2.4.15) (3.2.4.16)

As a result, equation (3.2.4.6) reduces to the following ODE for u = u(ξ): ′′′ a1 + a2 u′ξ + a3 (u′ξ )2 + a4 u′′ξξ + a5 ξu′′ξξ + a6 uu′′ξξ + a7 u′′′ ξξξ = ξuξξξ . (3.2.4.17)

Any compatible solution to the determining system of first-order PDEs (3.2.4.10) to (3.2.4.15) as well as relation (3.2.4.16) and the corresponding solution to the thirdorder ODE (3.2.4.17) generate an exact solution to equation (3.2.4.4) of the form (3.2.4.5). Remark 3.9. The article [48] considered exact solutions of the form (3.2.4.5) with g ≡ 0 for steady-state extended axisymmetric boundary layer equations (see also Example 3.8).

The analysis and solutions of the determining system (3.2.4.10)–(3.2.4.16). The system (3.2.4.10)–(3.2.4.16) consists of one simple independent equation and a few subsystems, which can be analyzed sequentially. From equation (3.2.4.16) we have ψ = a7 = const. The subsystem composed of equations (3.2.4.12) and (3.2.4.15) allows us to determine f and ϕ. The overdetermined subsystem consisting of PDEs (3.2.4.11) and (3.2.4.14) requires a compatibility analysis and serves to determine g. Then the pressure function can be evaluated by the formula F = gt + ggx − a1 νf ϕ2 , (3.2.4.18) which is a corollary of equation (3.2.4.10). Integrating (3.2.4.13) gives Z
1 h = −a4 νx + a7 gx ϕ − ϕt − gϕx dx + h0 (t), 2 ϕ where h0 (t) is an arbitrary function.

(3.2.4.19)

3.2. Direct Method of Weak Symmetry Reductions

279

Let us analyze the overdetermined system of PDEs (3.2.4.11) and (3.2.4.14). Isolating the derivative gx , we rewrite the equations as (f ϕ)t (f ϕ)x g = a2 νϕ − , fϕ fϕ ϕx ϕt gx − g = −a5 νϕ + . ϕ ϕ

gx +

(3.2.4.20) (3.2.4.21)

Let us find the conditions under which PDEs (3.2.4.20) and (3.2.4.21) coincide with each other. Then the function g can be determined from (3.2.4.21). On equating the functional coefficients of g as well as the right-hand sides of equations (3.2.4.20) and (3.2.4.21) with each other, we obtain (f ϕ)x ϕx + = 0, ϕ fϕ

ϕt (f ϕ)t + = (a2 + a5 )νϕ. ϕ fϕ

(3.2.4.22)

Integrating the first equation gives f = λ(t)ϕ−2 ,

(3.2.4.23)

where λ(t) is an arbitrary function. Substituting this expression into the second equation of (3.2.4.22) yields [ln λ(t)]′t = (a2 + a5 )νϕ.

(3.2.4.24)

Equation (3.2.4.24) holds in the following two cases: (a) a5 = −a2 ,

λ(t) = λ0 , ϕ = ϕ(t, x) is arbitrary,   Z (b) λ(t) = λ0 exp (a2 + a5 ) ν ϕ(t) dt , ϕ = ϕ(t) is arbitrary,

(3.2.4.25) (3.2.4.26)

where λ0 is an arbitrary constant. Case (a). In the first case, we can put λ0 = 1 without loss of generality. In view of (3.2.4.23), we have f = ϕ−2 . Then, from (3.2.4.12) and (3.2.4.15) it follows that ϕ′x = −a3 νϕ3 ,

ϕ′x =

3 1 2 a6 νϕ .

(3.2.4.27)

These two equations are compatible if a6 = −2a3 . Integrating the first equation as well as equation (3.2.4.21), we obtain an exact solution to equation (3.2.4.4) of the form (3.2.4.5) in which f = 2a3 νx + b(t),  −1/2 ϕ = 2a3 νx + b(t) ,   1 ′ g = a2 νx + c(t) ϕ − b (t), 2a3 ν t

(3.2.4.28)

where b = b(t) and c = c(t) are arbitrary functions. The function h is given by (3.2.4.19), ψ = a7 , and u = u(ξ) satisfies ODE (3.2.4.17) with a5 = −a2 and a6 = −2a3 . The pressure function is expressed by formula (3.2.4.18).

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Case (b). In the second case, it follows from formulas (3.2.4.23) and (3.2.4.26) as well as equations (3.2.4.12) and (3.2.4.15) that the conditions a3 = a6 = 0 must hold. Moreover, equation (3.2.4.17) becomes linear. Integrating (3.2.4.21) results in an exact solution to equation (3.2.4.4) of the form (3.2.4.5) in which −2

f = λ(t)ϕ

,

  Z λ(t) = λ0 exp (a2 + a5 ) ν ϕ(t) dt ,

ϕ = ϕ(t) is an arbitrary function,  ′  ϕt g= − a5 νϕ x + c(t), ϕ

(3.2.4.29)

where c = c(t) is an arbitrary function. The function h is found from (3.2.4.19), ψ = a7 , and u = u(ξ) satisfies ODE (3.2.4.17) with a3 = a6 = 0. The pressure function is defined by (3.2.4.18). For exact solutions to the nonlinear ODE (3.2.4.17), see [297]. Utilizing the direct method of weak symmetry reductions to construct exact solutions. The boundary layer equation (3.2.4.4) admits a lot more exact solutions of the form (3.2.4.5) than those obtained previously by reduction to a single ODE. Following [295], we will outline a procedure for constructing some other exact solutions of the form (3.2.4.5). The determining system (3.2.4.10)–(3.2.4.16) was obtained under the assumption that the first seven functions Ψi from (3.2.4.8) are all linearly independent. If some of the Ψi ’s are linearly dependent, one should choose a linearly independent subsystem, {Ψj }, and express the other functions in terms of its elements. Then, equation (3.2.4.6) should be rewritten as a linear combination of the functions Ψj and their functional coefficients must be equated with zero. As a result, one arrives at a modified determining system that involves fewer equations than system (3.2.4.10)– (3.2.4.16). As the functions u = u(ξ) determining Ψi in (3.2.4.8), one should take particular solutions to ODE (3.2.4.17) for some specific values of ai . Solutions of the form (3.2.4.5) other than those described by equations (3.2.4.10)–(3.2.4.17) arise when there are at least two linearly dependent subsystems amongst the functions of (3.2.4.8). For Ψ8 = 0, we have a degenerate case. Table 3.3 lists some functions u = u(ξ) that generate three or four linear relations between the functions (3.2.4.8). Each row contains a linearly dependent subsystem involving Ψ8 . The second and third rows highlight the special power-law functions that generate an extra linearly dependent subsystem as compared to the generic case u = ξ k with k 6= −1, 0, 3. Examples of constructing exact solutions. Below we construct several exact solutions to the nonlinear third-order PDE (3.2.4.4) by using the generating functions specified in Table 3.3. Solution 1. Let us take u = ξ k from the first row of Table 3.3 and construct a related exact solution. Taking into account (3.2.4.8) and the linear combinations of the functions Ψi from the first row of the table, we rewrite equation (3.2.4.6)–

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3.2. Direct Method of Weak Symmetry Reductions

Table 3.3. Generating functions u that produce linear relations between the functions Ψn . Notation: Ψ1 = 1, Ψ2 = u′ξ , Ψ3 = (u′ξ )2 , Ψ4 = u′′ξξ , Ψ5 = ξu′′ξξ , Ψ6 = uu′′ξξ , Ψ7 = u′′′ ξξξ , Ψ8 = ξu′′′ ξξξ . No. Generating function u 1

u = ξ k , k 6= 0 u=ξ

2 3

u=ξ

3

Linear relations for Ψn Ψ5 = (k − 1)Ψ2, Ψ6 = Ψ5 = 2Ψ2 , Ψ6 =

−1

k−1 Ψ3 , k

2 Ψ , 3 3

Ψ8 = (k − 2)Ψ4

Ψ7 = 6Ψ1, Ψ8 = Ψ4

Ψ5 = −2Ψ2, Ψ6 = 2Ψ3 , Ψ7 = −6Ψ3 , Ψ8 = −3Ψ4

4

u = exp ξ

Ψ2 = Ψ4 = Ψ7 , Ψ6 = Ψ3 , Ψ8 = Ψ5

5

u = ln ξ

Ψ3 = −Ψ4 = 12 Ψ8, Ψ5 = −Ψ2

(3.2.4.8) in the form (gt + ggx − F )Ψ1

+ [(f ϕ)t + (f gϕ)x + (k − 1)f (ϕt + gϕx − gx ϕ)]Ψ2

+ f ϕ[(f ϕ)x −

k−1 k

fx ϕ]Ψ3

+ f [ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 − ν(k − 2)ϕ2 ]Ψ4 + νf ϕ2 ψΨ7 = 0.

(3.2.4.30)

Equating the functional coefficients of Ψi with zero, we obtain the determining system gt + ggx − F = 0,

(3.2.4.31)

(f ϕ)t + (f gϕ)x + (k − 1)f (ϕt + gϕx − gx ϕ) = 0, kf ϕx + fx ϕ = 0,

(3.2.4.32) (3.2.4.33)

ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 − ν(k − 2)ϕ2 = 0,

(3.2.4.34)

ψ = 0.

(3.2.4.35)

The last three equations have been divided by nonzero factors. In equation (3.2.4.33), the terms have been rearranged. System (3.2.4.31)–(3.2.4.35) has two equations fewer than system (3.2.4.10)– (3.2.4.16) and does not require a compatibility analysis. The general solution to system (3.2.4.31)–(3.2.4.35) can be written as 1 a′t (t) x + b(t), h = ν(2 − k)x + c(t), k − 2 a(t) (3.2.4.36) ϕ = ϕ(t, x) is an arbitrary function, ψ = 0, F = gt + ggx ,

f = a(t)ϕ−k ,

g=

where a(t), b(t), and c(t) are arbitrary functions. On substituting (3.2.4.36) into (3.2.4.5), we arrive at the exact solution   1 a′t (t) w = a(t)z k + x + b(t) z + ν(2 − k)x + c(t). (3.2.4.37) k − 2 a(t)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Interestingly, the arbitrary function ϕ, which appears in the first formula of solution (3.2.4.36), is cancelled out and does not enter the final result. The pressure function corresponding to solution (3.2.4.37) is expressed as   ′ 2  3−k ba′t 1 a′′t at F = + x + b′t + , a = a(t), b = b(t). 2 k−2 a (k − 2) a (k − 2)a (3.2.4.38) Solution 2. We take u = ξ 3 from row 2 of Table 3.3. Now we set k = 3 in (3.2.4.30) and take into account the additional constraint Ψ7 = 6Ψ1 . Reasoning further as in Solution 1, we obtain the determining system of PDEs gt + ggx + 6νf ϕ2 ψ − F = 0,

(3.2.4.39)

(f ϕ)t + (f gϕ)x + 2f (ϕt + gϕx − gx ϕ) = 0, 3f ϕx + fx ϕ = 0,

(3.2.4.40) (3.2.4.41)

ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 − νϕ2 = 0.

(3.2.4.42)

Without loss of generality, we can set ϕ = 1, which is equivalent to a renormalization of ψ. On solving system (3.2.4.39)–(3.2.4.42), we ultimately arrive at the following exact solution to equation (3.2.4.4): Z w = a(t)(z + ψ)3 + gz − νx + (ψt + gψx + gx ψ) dx + c(t), (3.2.4.43) a′ (t) g = t x + b(t), a(t) where ψ = ψ(t, x), a = a(t), b = b(t), and c = c(t) are arbitrary functions. The corresponding pressure function is expressed as F =

a′′tt a′ b x + b′t + t + 6νaψ. a a

(3.2.4.44)

It is apparent that F = F (t, x) can be chosen arbitrarily and ψ can be expressed via F . Solution 3. If we take u = ξ −1 (row 3 of Table 3.3), we should set k = −1 in (3.2.4.30). Considering the additional constraint Ψ7 = −6Ψ3 , we arrive at a determining system of PDEs (not written out here), whose solution can represented as f = f (t, x) is an arbitrary function, Z 1 g = a(t)f −1/3 − f −1/3 f −2/3 ft dx, 3 1 (3.2.4.45) h= (f + gfx ) + 3νx + b(t), 6ν 1 ϕ = 1, ψ = fx , 6ν F = gt + ggx , where a(t) and b(t) are arbitrary functions. In this case, we can treat f as an arbitrary function and set ϕ = 1 (which is equivalent to a renormalization of f ).

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3.2. Direct Method of Weak Symmetry Reductions

Formulas (3.2.4.5) and (3.2.4.45) define an exact solution to equation (3.2.4.4). Solution 4. Let us take u = exp ξ and consider the linear relations for Ψi from row 4 of Table 3.3. As a result, we obtain a determining system of PDEs, whose solution is expressed as f = f (t, x) is an arbitrary function,  ′  ϕt g= − νϕ x + a(t), ϕ Z 1 (f ϕ)t + ϕ(f g)x h= 2 dx + b(t), ϕ f ϕ = ϕ(t) is an arbitrary function, ψ = 0,

(3.2.4.46)

F = gt + ggx , where a(t) and b(t) are arbitrary functions. Here, we treat f as an arbitrary function and have set ψ = 0 (which is equivalent to a renormalization of f ). Solution 5. Let us take u = ln ξ (row 5 of Table 3.3). Omitting the intermediate calculations, we will write out the resulting exact solution to equation (3.2.4.4):   f ′ (t) w = f (t) ln z + a(t) − t x z + 2νx + b(t), (3.2.4.47) 2f (t) where f = f (t), a = a(t), and b = b(t) are arbitrary functions. The corresponding pressure function is expressed as   ′ 2  1 f′ 3 ft 1 ftt′′ F = a′t − a t + − x. 2 f 4 f 2 f Importantly, the second, third, and fourth solutions involve an arbitrary function with two arguments. Furthermore, all five of the above solutions are represented in closed form. Remark 3.10. The direct method of weak symmetry reductions outlined in Subsection 3.2.1 allows one to obtain a lot more exact solutions to the nonlinear equation (3.2.4.4) than the direct method of symmetry reductions described in Section 3.1.

One class of generalized separable solutions. 1◦ . The determining system of equations. Suppose the pressure function F is linear in x: F (t, x) = f1 (t)x + f0 (t), (3.2.4.48) where f1 (t) and f0 (t) are arbitrary functions. Then, equation (3.2.4.4) admits generalized separable solutions of the form w = Φ(t, z)x + Ψ (t, z),

(3.2.4.49)

with the functions Φ = Φ(t, z) and Ψ = Ψ (t, z) satisfying the system of PDEs Φtz + Φ2z − ΦΦzz = νzΦzzz + f1 (t), Ψtz + Φz Ψz − ΦΨzz = νzΨzzz + f0 (t).

(3.2.4.50) (3.2.4.51)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Equation (3.2.4.50) involves one unknown function, Φ, and does not depend on equation (3.2.4.51). With the change of variable ψ = Ψz , the third-order equation (3.2.4.51) reduces to a parabolic second-order equation linear in ψ. 2◦ . A formula for constructing exact solutions to equation (3.2.4.51). The following statement holds true. Let Φ = Φ(t, z) be a solution to equation (3.2.4.50). Then, equation (3.2.4.51) also admits exact solutions of the form Ψ = A(t)Φ − A′t (t)z + B(t),

(3.2.4.52)

where B(t) is an arbitrary function and A = A(t) is a solution to the linear secondorder ODE A′′tt − f1 (t)A + f0 (t) = 0. (3.2.4.53) Formula (3.2.4.52) allows one to construct exact solutions to equation (3.2.4.51) once a solution to equation (3.2.4.50) is known. 3◦ . Reduction to a linear second-order PDE. Equation (3.2.4.50) admits degenerate solutions of the form Φ = a(t)z + b(t), (3.2.4.54) where b(t) is an arbitrary function and a = a(t) is a function satisfying the firstorder ODE a′t + a2 = f1 (t), which is a Riccati equation [276]. In view of (3.2.4.54), equation (3.2.4.51) becomes ψt + a(t)ψ − [a(t)z + b(t)]ψz = νzψzz + f0 (t),

ψ = Ψz .

If b(t) = 0, the transformation  Z   Z ψ = exp − a(t) dt Ω(τ, ζ) + f0 (t)σ(t) dt , Z  Z τ = ν σ(t) dt, ζ = zσ(t), σ(t) = exp a(t) dt

(3.2.4.55)

(3.2.4.56)

converts (3.2.4.55) to a simpler equation Ωτ = ζΩζζ .

(3.2.4.57)

The handbook [273] (see page 95) describes a lot of exact solutions to equation (3.2.4.57). In particular, equation (3.2.4.57) has the solutions Ω = 2C1 τ ζ + C1 ζ 2 + C2 ,   ζ Ω = C1 exp − + C3 , τ + C2 where C1 , C2 , and C3 are arbitrary constants.

3.2. Direct Method of Weak Symmetry Reductions

285

4◦ . Solutions involving an exponential function. Equation (3.2.4.50) admits nondegenerate solutions involving an exponential function: Φ = a(t)eλ(t)z + b(t)z + c(t),

(3.2.4.58)

where the functional coefficients a = a(t), b = b(t), c = c(t), and λ = λ(t) satisfy three ODEs λ′t − bλ − νλ2 = 0, a′t + 3ab + νaλ − acλ = 0,

b′t

(3.2.4.59)

2

+ b = f1 (t).

The second equation has been transformed using the first equation. The functions a and λ in (3.2.4.59) can be treated as arbitrary. Then, we find b and c from the first two equations without integrating. The last equation determines f1 . Suppose the function Φ is determined by equations (3.2.4.58) and (3.2.4.59), and hence, it solves equation (3.2.4.50). Then, equation (3.2.4.51) has the exact solution Ψ = α(t)eλ(t)z + β(t)z + γ(t),

(3.2.4.60)

where γ(t) is an arbitrary function, while α = α(t) and β = β(t) satisfy the system of ODEs α′t + 2bα + aβ + ναλ − cαλ = 0, (3.2.4.61) βt′ + bβ = f0 (t). Considering the function α to be arbitrary, one can easily find β from the first equation without integrating. The second equation determines f0 . The functions (3.2.4.58) and (3.2.4.60) the coefficients of which satisfy equations (3.2.4.59) and (3.2.4.61) provide an exact solution to system (3.2.4.50)–(3.2.4.51). These formulas and equations define exact solutions of the form (3.2.4.49) to the unsteady axisymmetric boundary layer equation (3.2.4.4).

3.2.5. Plane and Axisymmetric Boundary Layer Equations for Non-Newtonian Fluids Plane boundary layer equations for non-Newtonian fluids. Apart from the classical Newtonian fluid model (see Subsections 3.2.3 and 3.2.4 for boundary layer equations for this model), more complicated rheological models (non-Newtonian fluids) quite frequently arise in applications (e.g., see [4, 38, 133, 259, 331, 332]). The system of equations of an unsteady plane laminar boundary layer for a power-law non-Newtonian fluid [4, 259, 331] is expressed as Ut + U Ux + V Uy = κUyn−1 Uyy + F (t, x), Ux + Vy = 0,

(3.2.5.1) (3.2.5.2)

where t is time, x and y are the longitudinal and transverse coordinates (y = 0 corresponds to the surface of the body), U and V are the longitudinal and transverse

286

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

fluid velocity components, F (t, x) = −px /ρ is a given function (proportional to the longitudinal pressure gradient), p is pressure, ρ is mass density, κ is an effective viscosity, and n > 0 is a rheological parameter characterizing the fluid (n = 1 correspond to the Newtonian fluid). We assume the fluid to be incompressible and Uy ≥ 0. The introduction of a stream function, W , by formulas (3.2.3.3) converts the system of PDEs (3.2.5.1)–(3.2.5.2) to a single nonlinear third-order PDE: n−1 Wty + Wy Wxy − Wx Wyy = κWyy Wyyy + F (t, x).

(3.2.5.3)

The studies [234, 243, 259, 271, 323, 331, 375, 386] present exact solutions and transformations for the steady-state and unsteady equations (3.2.5.1)–(3.2.5.2) and (3.2.5.3) for power-law fluids (n 6= 1). Axisymmetric boundary layer equations for non-Newtonian fluids. The system of equations for the axisymmetric unsteady laminar boundary layer for a powerlaw non-Newtonian fluid is expressed as Ut + U Ux + V Uy = κUyn−1 Uyy + F (t, x), (rU )x + (rV )y = 0,

r = r(x),

(3.2.5.4) (3.2.5.5)

where x and y are the longitudinal and transverse coordinates (the corresponding unit vectors, ex and ey , are tangent and normal to the surface of the body of revolution), U and V are the longitudinal and transverse fluid velocity components, F (t, x) = −px /ρ is a given function, and r = r(x) is the dimensionless cross-sectional radius perpendicular to the axis of rotation and defining the shape of the body. The other notations are the same as in equation (3.2.5.3). The introduction of a stream function, W , by formulas (3.2.3.7) converts system (3.2.5.4)–(3.2.5.5) to a single nonlinear third-order PDE: Wty + Wy Wxy − Wx Wyy −

rx′ n−1 W Wyy = κWyy Wyyy + F (t, x). r

(3.2.5.6)

Equation (3.2.5.6) coincides with (3.2.5.3) for r(x) = 1. The presence of r(x) 6= const on the left-hand side of equation (3.2.5.6) complicates its analysis substantially (especially if r(x) is arbitrary). In what follows, we will deal with the general case of equation (3.2.5.6) where r(x) is an arbitrary function. The admissible pressure functions F (t, x) and admissible values of the rheological parameter n are to be determined in the subsequent analysis. Reduction to a plane boundary layer equation with a variable viscosity. Introducing the new variables z = r(x)y,

w = r(x)W,

(3.2.5.7)

we rewrite equation (3.2.5.6) in the form [297] n−1 wtz + wz wxz − wx wzz = κrn+1 (x)wzz wzzz + F (t, x).

(3.2.5.8)

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3.2. Direct Method of Weak Symmetry Reductions

This equation can be treated as an unsteady plane boundary layer equation with a variable viscosity κe = κrn+1 (x), which depends on the longitudinal coordinate x. The pressure function F (t, x) remains unchanged. In the special case r(x) ≡ 1, equation (3.2.5.8) coincides with the plane boundary layer equation (3.2.5.3). Remark 3.11. In [248], the third-order equation (3.2.5.8) was first reduced to a secondorder equation with the help of a Crocco type transformation in which µ = wz and u = wzz were used as the new variables instead of z and w, and then a few exact solutions were obtained for the resulting equation.

Formulas that allow a generalization of exact solutions. The unsteady axisymmetric boundary layer equations for power-law fluids possess remarkable properties, which are stated below as two propositions [248, 297]. Proposition 1. Let w(t, x, z) be a solution to equation (3.2.5.8). Then the function Z ∂ w1 = w(t, x, ζ) + φ(t, x) dx + p(t), ζ = z + φ(t, x), (3.2.5.9) ∂t where φ(t, x) and p(t) are arbitrary functions, is also a solution to this equation. Proposition 2. Let W (t, x, y) be a solution to the unsteady axisymmetric boundary layer equation (3.2.5.6). Then the function   Z ∂ 1 r(x)φ(t, x) dx + p(t) , η = y + φ(t, x), W1 = W (t, x, η) + r(x) ∂t (3.2.5.10) where φ(t, x) and p(t) are arbitrary functions, is also a solution to this equation. Propositions 1 and 2 can be proved by direct verification. Both are true for any functions r(x) and F (t, x) in equations (3.2.5.6) and (3.2.5.8). The propositions allow one to generalize exact solutions by including additional arbitrary functions. In [248], these propositions were used to construct exact solutions to the nonlinear PDEs under consideration. General form of solutions. The determining equation. Reduction to an ODE. Just as in Subsections 3.2.3 and 3.2.4, we seek exact solutions to equation (3.2.5.8) in the form w = f u(ξ) + gz + h,

ξ = ϕz + ψ,

(3.2.5.11)

where the functions f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), ψ = ψ(t, x), and u = u(ξ) are to be determined in the subsequent analysis. To be specific, we will assume in what follows that f > 0 and g > 0.

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

On substituting (3.2.5.11) into (3.2.5.8), we obtain an equation that can conveniently be represented in the bilinear form 7 X

Φn [x, t]Ψn [ξ] = 0.

(3.2.5.12)

n=1

The functions Φn = Φn [x, t] depend on the functional coefficients (and their derivatives) of equation (3.2.5.8) and solution (3.2.5.11); these are given by Φ1 = gt + ggx − F,

Φ2 = (f ϕ)t + (f gϕ)x ,

Φ3 = f ϕ(f ϕ)x ,

Φ4 = f (ϕψt + gϕψx + gx ϕψ − ψϕt − gϕx ψ − hx ϕ2 ), Φ5 = f (ϕt + gϕx − gx ϕ),

Φ6 = −f fx ϕ2 ,

Φ7 = −κrn+1 f n ϕ2n+1 . (3.2.5.13)

The functions Ψn = Ψn [ξ] are expressed as Ψ1 = 1,

Ψ2 = u′ξ ,

Ψ5 = ξu′′ξξ ,

Ψ3 = (u′ξ )2 ,

Ψ6 = uu′′ξξ ,

Ψ4 = u′′ξξ ,

(3.2.5.14)

Ψ7 = (u′′ξξ )n−1 u′′′ ξξξ .

Equation (3.2.5.12)–(3.2.5.14) reduces to a single ODE for u = u(ξ) if Φ1 , . . . , Φ6 are all set proportional to Φ7 , so that Φn = −an Φ7

(n = 1, . . . , 6),

(3.2.5.15)

where a1 , . . . , a6 are free parameters. On substituting (3.2.5.13) into (3.2.5.15), we arrive at the following nonlinear system of PDEs for f = f (t, x), g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), and ψ = ψ(t, x): gt + ggx − F = a1 κrn+1 f n ϕ2n+1 ,

(f ϕ)t + (f gϕ)x = a2 κr

n+1 n 2n+1

(f ϕ)x = a3 κr

n+1 n−1 2n

2

ϕψt − ϕt ψ + gϕψx + gx ϕψ − gϕx ψ − hx ϕ = a4 κr ϕt + gϕx − gx ϕ = a5 κr

f ϕ

f

(3.2.5.16)

,

(3.2.5.17)

ϕ ,

(3.2.5.18)

n+1 n−1 2n+1

f

ϕ

, (3.2.5.19)

n+1 n−1 2n+1

fx = −a6 κr

f

ϕ

, (3.2.5.20)

n+1 n−1 2n−1

f

ϕ

. (3.2.5.21)

As a result, equation (3.2.5.12) reduces to a nonlinear ODE for u = u(ξ): a1 + a2 u′ξ + a3 (u′ξ )2 + a4 u′′ξξ + a5 ξu′′ξξ + a6 uu′′ξξ = (u′′ξξ )n−1 u′′′ ξξξ .

(3.2.5.22)

Any compatible solution to the determining system of first-order PDEs (3.2.5.16) to (3.2.5.21) in conjunction with the associated solution to the third-order ODE (3.2.5.22) generate an exact solution to equation (3.2.5.8) of the form (3.2.5.11). For order reduction and particular solutions to the nonlinear ODE (3.2.5.22), see the article [297].

3.2. Direct Method of Weak Symmetry Reductions

289

Analysis and solutions of the determining system (3.2.5.16)–(3.2.5.21). System (3.2.5.16)–(3.2.5.21) breaks up into a few simpler subsystems that can be analyzed step by step. The subsystem composed of equations (3.2.5.18) and (3.2.5.21) allows one to express f and ϕ via r = r(x). Once f and ϕ are known, the subsystem of equations (3.2.5.17) and (3.2.5.20), serves to determine g and r; whence it follows that r = r(x) cannot generally be an arbitrary function. Given g and r, the pressure function is found as F = gt + ggx − a1 κrn+1 f n ϕ2n+1 , (3.2.5.23) which follows from (3.2.5.16). The function ψ in equation (3.2.5.19) can be treated as arbitrary. Integrating with respect to x, we obtain the following expression of h: Z
1 ϕψt − ϕt ψ + gϕψx + gx ϕψ − gϕx ψ − a4 κrn+1 f n−1 ϕ2n+1 dx + p(t), h= 2 ϕ (3.2.5.24) where p(t) is an arbitrary function. In what follows, we assume that r = r(x) is an arbitrary function. We start with the analysis of the subsystem composed of equations (3.2.5.17) and (3.2.5.20). On isolating the derivative gx , we rewrite the equations as (f ϕ)t (f ϕ)x g = a2 κrn+1 f n−1 ϕ2n − , fϕ fϕ ϕx ϕt gx − g = −a5 κrn+1 f n−1 ϕ2n + . ϕ ϕ gx +

(3.2.5.25) (3.2.5.26)

Let us find the conditions under which equations (3.2.5.25) and (3.2.5.26) coincide. In this case, r = r(x) can be considered to be arbitrary and g is determined from equation (3.2.5.26) and, hence, is expressed in terms of ϕ and r. Equating the functional coefficients of g with each other as well as the right-hand sides of equations (3.2.5.25) and (3.2.5.26), we obtain (f ϕ)x ϕx + = 0, ϕ fϕ

ϕt (f ϕ)t + = (a2 + a5 )κrn+1 f n−1 ϕ2n . ϕ fϕ

(3.2.5.27)

Integrating the first equation of (3.2.5.27) gives f = λ(t)ϕ−2 ,

(3.2.5.28)

where λ(t) is an arbitrary function. Substituting this expression into the second equation of (3.2.5.27) yields [ln λ(t)]′t = (a2 + a5 )κrn+1 (x)λn−1 ϕ2 .

(3.2.5.29)

Equation (3.2.5.29) holds in the following two cases: (a) a5 = −a2 , (b) ϕ =

λ(t) = λ0 ,

σ(t) r(n+1)/2 (x)

,

ϕ = ϕ(t, x) is arbitrary; (3.2.5.30)  1  Z 1−n 2 λ(t) = (a2 + a5 )(1 − n) κ σ (t) dt + C0 ,

(3.2.5.31)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

where C0 is an arbitrary constant and σ(t) is an arbitrary function. Case (a). Without loss of generality, we can restrict ourselves to the case λ0 = 1. In view of (3.2.5.28), we get f = ϕ−2 . Then, equations (3.2.5.18) and (3.2.5.21) become (3.2.5.32) ϕ′x = −a3 κrn+1 ϕ4 , ϕ′x = 21 a6 κrn+1 ϕ4 . These two equations are consistent if a6 = −2a3 . Integrating the first equation of (3.2.5.32) and then (3.2.5.26), we obtain an exact solution to equation (3.2.5.8) of the form (3.2.5.11) in which one should set 

f = 3a3 κ 

Z

r

n+1

2/3 (x) dx + b(t) ,

−1/3 ϕ = 3a3 κ r (x) dx + b(t) ,  Z  ϕt n+1 g = c(t)ϕ + ϕ + a2 κr (x)ϕ dx, ϕ2 Z

n+1

(3.2.5.33)

where b = b(t) and c = c(t) are arbitrary functions, h is defined by (3.2.5.24), ψ = ψ(t, x) is an arbitrary function, and u = u(ξ) is a function satisfying ODE (3.2.5.22) with a5 = −a2 and a6 = −2a3 . The pressure function F is found by formula (3.2.5.23). It is noteworthy that for a3 6= 0, the function g can be represented as Z a2 −1 1 ′ ϕ . g = c(t)ϕ − bt (t)ϕ ϕ2 dx + 3 2a3

Case (b). It follows from formulas (3.2.5.28) and (3.2.5.31) as well as equations (3.2.5.18) and (3.2.5.21) that the conditions a6 = −2a3 and r(x) = (αx + β)2/(n+1) must hold, where α and β are some constants. This implies that r(x) is not an arbitrary function. We skip this special case. Solutions of the determining system (3.2.5.16)–(3.2.5.21) for a plane boundary layer with r(x) ≡ 1. Now we will consider the special case of the plane boundary layer equations (3.2.5.8) with r(x) = 1 and describe some of its exact solutions of the form (3.2.5.11). In the two cases below, we will only present the expressions of the function ϕ = ϕ(t, x), f = f (t, x), and g = g(t, x). The function ψ = ψ(t, x) is arbitrary, while h = h(t, x) and F = F (t, x) are defined by formulas (3.2.5.24) and (3.2.5.23) with r(x) = 1. The function u = u(ξ) satisfies ODE (3.2.5.22). 1◦ . Case of a5 = −a2 and a6 = −2a3 6= 0. Exact solutions of the form (3.2.5.11) follow from (3.2.5.33), (3.2.5.23), and (3.2.5.24) if we put r(x) = 1 therein. As a result, we get  −1/3 ϕ = 3a3 κx + b(t) ,  2/3 f = 3a3 κx + b(t) , 1 ′ a2 −1 g = c(t)ϕ − ϕ , b (t) + 3a3 κ t 2a3

(3.2.5.34)

291

3.2. Direct Method of Weak Symmetry Reductions

where b = b(t) and c = c(t) are arbitrary functions. 2◦ . Case a3 = a6 = 0. Exact solutions of the form (3.2.5.11) are given by ϕ = ϕ(t) is an arbitrary function,   1 Z 1−n −2 2 f =ϕ (a2 + a5 )(1 − n) κ ϕ (t) dt + C0 ,   ϕt − a5 κf n−1 ϕ2n x + c(t), g= ϕ

(3.2.5.35)

where c(t) is an arbitrary function, C0 is an arbitrary constant, and n 6= 1. The three functions (3.2.5.35) satisfy four equations (3.2.5.17), (3.2.5.18), (3.2.5.20), and (3.2.5.21) with r = 1 and a3 = a6 = 0. Utilizing the direct method of weak symmetry reductions to construct exact solutions. The boundary layer equation (3.2.5.8) for arbitrary r(x) has a lot more exact solutions of the form (3.2.5.11) than those obtained previously by reduction to the single ODE (3.2.5.22) with a5 = −a2 and a6 = −2a3 . Following [297], we will outline a procedure for constructing some other exact solutions of the form (3.2.5.11). The determining system (3.2.5.16)–(3.2.5.21) was obtained under the assumption that the first six functions Ψi from (3.2.5.14) are all linearly independent. If some of the functions Ψi are linearly dependent, one should select a linearly independent subsystem, {Ψj }, and express the other functions in terms of the elements of this subsystem. Then, one should rewrite relation (3.2.5.12) as a linear combination of the functions Ψj and equate the functional coefficients of Ψj with zero. The resulting modified determining system of PDEs will not only differ from system (3.2.5.16)–(3.2.5.21), but will also contain fewer equations. The more linear relations there are between the functions Ψi from (3.2.5.14), the fewer equations will appear in the determining system and the more arbitrary functions will enter its solution. As the functions u = u(ξ) determining Ψi in (3.2.5.14), one should take particular solutions to ODE (3.2.5.22) at some values of the parameters ai . Solutions of the form (3.2.5.11) other than those described by equations (3.2.5.16)–(3.2.5.22) arise when there are at least two linearly independent subsystems of functions from (3.2.5.14). Table 3.1 presents some functions u = u(ξ) that generate two or three linear relations between the functions of (3.2.5.14) for a Newtonian fluid (with n = 1). Construction of exact solutions for u = εξ m with n 6= 1. For power-law non-Newtonian fluids, the substitution of the power-law function u = εξ m , where ξ > 0 and ε = sign[m(m − 1)], into formulas (3.2.5.14) gives Ψ1 = 1, Ψ2 = εmξ m−1 , Ψ3 = m2 ξ 2m−2 , Ψ4 = εm(m − 1)ξ m−2 , Ψ5 = εm(m − 1)ξ m−1 , Ψ6 = m(m − 1)ξ

2m−2

(3.2.5.36) n mn−2n−1

, Ψ7 = (m − 2)[εm(m − 1)] ξ

.

292

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Remark 3.12. The coefficient ε = ±1 is introduced to make sure that the expression εm(m − 1) = |m(m − 1)| is nonnegative. This allows us to avoid a negative radicand in Ψ7 , which can arise at fractional values of n; for example, if 0 < n < 12 , it follows from rows 3 and 4 of Table 3.4 that m(m − 1) < 0.

From (3.2.5.36) it follows that for any n, there are two linear relations: Ψ5 = (m − 1)Ψ2 ,

mΨ6 = (m − 1)Ψ3 .

(3.2.5.37)

In general, there are degenerate solutions that correspond to m = 1 and m = 2 and lead to Ψ7 = 0; we will skip these solutions. The nondegenerate solutions are characterized by the linear constraints Ψ7 = ki Ψi , where i = 1, 2, 3, 4. Table 3.4 lists the power-law functions u = εξ m that generate three linear constraints between the functions (3.2.5.14) for non-Newtonian fluids. Table 3.4. Power-law generating functions u = εξ m , where ε = sign[m(m − 1)], and the associated linear constraints of the form Ψ7 = ki Ψi for non-Newtonian fluids (n 6= 1); two more constraints are specified in (3.2.5.37). No.

Power m

Linear constraints Ψ7 = (m−2)|m(m−1)|n Ψ1

m=

2n+1 n

2

m=

2n n−1

Ψ7 = ε

3

m=

2n−1 n−2

Ψ7 =

4

m=

2n−1 n−1

1

Restrictions

m−2 m

m−2 m2

n

n 6= 0

|m(m−1)| Ψ2

n 6= 0, 1

|m(m−1)|n Ψ3

n 6= 12 , 2

Ψ7 = (m−2)|m(m−1)|n−1 Ψ4

n 6= 12 , 1

Solution 1 for u = εξ (2n+1)/n . Let us construct a family of exact solutions 2n+1

corresponding to u = εξ n (see row 1 from Table 3.4). In view of (3.2.5.14) and the linear relations for Ψi (see formulas (3.2.5.36)–(3.2.5.37) and Table 3.4 with m = 2n+1 n ), equation (3.2.5.12)–(3.2.5.14) can be rewritten as   gt +ggx −F −κ|m(m−1)|n(m−2)rn+1f nϕ2n+1 Ψ1     + (f ϕ)t +(f gϕ)x +(m−1)f (ϕt +gϕx −gxϕ) Ψ2 +f ϕ (f ϕ)x − m−1 m fx ϕ Ψ3 +f (ϕψt −ψϕt +gϕψx +gxϕψ−gϕxψ−hxϕ2)Ψ4 = 0,

m = 2n+1 n .

Equating the functional coefficients of all Ψi with zero, we obtain gt + ggx − F − κ|m(m − 1)|n (m − 2)rn+1 f n ϕ2n+1 = 0,

(3.2.5.38)

(f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ) = 0, mf ϕx + fx ϕ = 0,

(3.2.5.39) (3.2.5.40)

ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 = 0.

(3.2.5.41)

The last two equations have been divided by a nonzero factor and equation (3.2.5.40) has been simplified.

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3.2. Direct Method of Weak Symmetry Reductions

The system of PDEs (3.2.5.38)–(3.2.5.41) has two fewer equations than system (3.2.5.16)–(3.2.5.21). Furthermore, there is no need to carry out a compatibility analysis. Equations (3.2.5.38)–(3.2.5.40) allow us to find the functions f , g, and F for arbitrary r(x). The functions ϕ = ϕ(t, x) and ψ = ψ(t, x) remain arbitrary, while h is evaluated by formula (3.2.5.24) with a4 = 0. The general solution to equation (3.2.5.40) is f = a(t)ϕ−m = a(t)ϕ−

2n+1 n

,

(3.2.5.42)

where a(t) is an arbitrary function. On substituting this into (3.2.5.39) and on solving the resulting equation, we find that g=

a′ (t) a′t (t) x + b(t) = n t x + b(t), (m − 2)a(t) a(t)

(3.2.5.43)

where b(t) is an arbitrary function. On substituting (3.2.5.42) and (3.2.5.43) into equation (3.2.5.38), we obtain the pressure function F = gt + ggx − κ|m(m − 1)|n (m − 2)rn+1 f n ϕ2n+1 ,

m=

2n + 1 . (3.2.5.44) n 2n

Solution 2 for u = εξ 2n/(n−1) . Now let us take the function u = εξ n−1 from row 2 of Table 3.4. In view of formulas (3.2.5.14) and using the linear constraints 2n for Ψi (see relations (3.2.5.36)–(3.2.5.37) and Table 3.4 with m = n−1 ), we rewrite equation (3.2.5.12)–(3.2.5.14) in the form (gt + ggx − F )Ψ1  + (f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ)  m−2 − κε |m(m − 1)|n rn+1 f n ϕ2n+1 Ψ2 m   + f ϕ (f ϕ)x − m−1 m fx ϕ Ψ3

+ f (ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 )Ψ4 = 0,

m=

2n . n−1

Equating the functional coefficients of Ψi with zero, we arrive at the determining system gt + ggx − F = 0, (f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ) m−2 − κε |m(m − 1)|n rn+1 f n ϕ2n+1 = 0, m mf ϕx + fx ϕ = 0,

(3.2.5.45)

ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 = 0. The last equation in system (3.2.5.45) coincides with (3.2.5.41) and the third equation in (3.2.5.45) differs from (3.2.5.40) in only the value of m. The functions

294

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

ϕ = ϕ(t, x) and ψ = ψ(t, x) remain arbitrary, and h is expressed by (3.2.5.24) with a4 = 0. The functions f , g, and F are given by 2n − n−1

f = a(t)ϕ−m = a(t)ϕ

,

a′t (t)

κε x− |m(m − 1)|n an−1 (t) g= (m − 2)a(t) m F = gt + ggx ,

Z

rn+1 (x) dx + b(t), (3.2.5.46)

2n where a(t) and b(t) are arbitrary functions and m = n−1 . Formulas (3.2.5.11), (3.2.5.24) with a4 = 0, and (3.2.5.46) with arbitrary ϕ = ϕ(t, x) and ψ = ψ(t, x) describe an exact solution to equation (3.2.5.8) for arbitrary body shape function r(x). 2n−1

Solution 3 for u = εξ (2n−1)/(n−2) . Let us take u = εξ n−2 from the third row of Table 3.4. Considering formulas (3.2.5.14), (3.2.5.36), and (3.2.5.37) with m = 2n−1 n−2 and reasoning as in constructing Solution 1, we obtain the following determining system from (3.2.5.12)–(3.2.5.14): gt + ggx − F = 0,

(f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ) = 0, m−2 mf ϕx + fx ϕ − κ |m(m − 1)|n rn+1 f n−1 ϕ2n = 0, m ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 = 0.

(3.2.5.47)

The functions ϕ = ϕ(t, x) and ψ = ψ(t, x) in system (3.2.5.47) can be treated as arbitrary. The third equation in (3.2.5.47) is easy to integrate, since it is a Bernoulli ODE for f , in which x is treated as the independent variable and t as a parameter. The second equation in (3.2.5.47) is a first-order linear ODE for g with independent variable x and parameter t. We omit the solutions of these two ODEs. The functions F and h are easy to find from the first and last equations of (3.2.5.47). Solution 4 for u = εξ (2n−1)/(n−1) . We will construct a solution for u = 2n−1

εξ n−1 from row 4 of Table 3.4. Considering formulas (3.2.5.14), (3.2.5.36), and (3.2.5.37) with m = 2n−1 n−1 and reasoning as above, we derive the following determining system from equation (3.2.5.12)–(3.2.5.14): gt + ggx − F = 0, (f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ) = 0,

mf ϕx + fx ϕ = 0,

ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2

− κ(m − 2)|m(m − 1)|n−1 rn+1 f n−1 ϕ2n+1 = 0.

295

3.2. Direct Method of Weak Symmetry Reductions

Its general solution is written as 2n−1 − n−1

f = a(t)ϕ−m = a(t)ϕ g=

a′t (t)

,

x + b(t) = (n − 1)

a′t (t) x + b(t), a(t)

(m − 2)a(t) Z 1  h= ϕψt − ϕt ψ + gϕψx + gx ϕψ − gϕx ψ ϕ2 − κ(m − 2)|m(m − 1)|n−1 rn+1 f n−1 ϕ2n+1 dx + c(t), F = gt + ggx , where ϕ = ϕ(t, x), ψ = ψ(t, x), a(t), b(t), and c(t) are arbitrary functions and m = 2n−1 n−1 . Solution 5 for u = − ln |ξ|. On substituting the logarithmic function u = − ln |ξ| into (3.2.5.14), we get Ψ1 = 1, Ψ5 = ξ −1 ,

Ψ2 = −ξ −1 ,

Ψ3 = ξ −2 ,

Ψ6 = −ξ −2 ln |ξ|,

Ψ4 = ξ −2 ,

Ψ7 = −2ξ −2n−1 .

(3.2.5.48)

It is apparent that there are two linear relations: Ψ5 = −Ψ2 ,

Ψ4 = Ψ3 .

(3.2.5.49)

Nontrivial solution arise when Ψ7 = ki Ψi , where i = 1, 2, 3. It follows from (3.2.5.48) that there are three admissible values of the rheological parameter: n = − 12 , 0, 12 . The first two do not have a physical meaning and, therefore, we omit them. For n = 12 , we have Ψ7 = −2Ψ3 . Using formulas (3.2.5.14), (3.2.5.48), and (3.2.5.49) and equation (3.2.5.12)–(3.2.5.14), we reason as above to obtain the determining system of PDEs: gt + ggx − F = 0, (f ϕ)t + (f gϕ)x − f (ϕt + gϕx − gx ϕ) = 0,

ϕ(f ϕ)x + ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 + 2κr3/2 f −1/2 ϕ2 = 0, fx = 0.

Its general solution can be written as f ′ (t) f = f (t), g = − t x + b(t), 2f (t) Z  1  f ϕϕx +ϕψt −ψϕt +gϕψx +gxϕψ−gϕxψ+2κr3/2f −1/2ϕ2 dx+c(t), h= 2 ϕ F = gt + ggx , where f (t), b(t), c(t), ϕ = ϕ(t, x), and ψ = ψ(t, x) are arbitrary functions.

296

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Solution 6 for u = exp ξ with n = 2. On substituting the exponential function u = eξ into (3.2.5.14), we get Ψ1 = 1, Ψ2 = eξ , Ψ3 = e2ξ , Ψ4 = eξ , Ψ5 = ξeξ , Ψ6 = e2ξ , Ψ7 = e2ξ . (3.2.5.50) It is apparent that the following three linear relations hold: Ψ2 = Ψ4 ,

Ψ3 = Ψ6 = Ψ7 .

(3.2.5.51)

Using formulas (3.2.5.14), (3.2.5.50), and (3.2.5.51) and equation (3.2.5.12), we proceed in the same way as in constructing Solution 1 to arrive at the determining system gt + ggx − F = 0, ϕt + gϕx − gx ϕ = 0,

(3.2.5.52) (3.2.5.53)

ϕx = κr3 ϕ4 ,

(3.2.5.54) 2

(f ϕ)t + (f gϕ)x + f (ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ ) = 0. (3.2.5.55) The functions f = f (t, x) and ψ = ψ(t, x) in system (3.2.5.52)–(3.2.5.55) can be treated as arbitrary. Then, it follows from equations (3.2.5.53) and (3.2.5.54) that  Z −1/3 ϕ = − 3κ r3 dx + a(t) ,

Z ϕt g = b(t)ϕ + ϕ dx, ϕ2

(3.2.5.56)

where a(t) and b(t) are arbitrary functions. The functions F and h are determined from equations (3.2.5.52) and (3.2.5.55); specifically, we get F = gt + ggx , Z  1  h= (f ϕ)t + (f gϕ)x + f (ϕψt − ψϕt + gϕψx + gxϕψ − gϕxψ) dx + c(t), 2 fϕ

where c(t) is an arbitrary function. The plane boundary layer equation for the general model of non-Newtonian fluids. For the general model of a non-Newtonian fluid, the unsteady plane boundary layer equation for the stream function W is written as [243, 271, 275]: Wty + Wy Wxy − Wx Wyy = [G(Wyy )]y + F (t, x).

(3.2.5.57)

The function G(Uy ) = 1ρ µ(Uy )Uy is determined by the rheological fluid model, where µ(Uy ) > 0 is the non-Newtonian viscosity; G is the shear stress per unit density of the fluid. The other notations are the same as in equation (3.2.5.3). For power-law fluids, we have G(Uy ) = (κ/n)|Uy |n−1 Uy , in which case, equation (3.2.5.57) becomes (3.2.5.3) provided that Uy > 0. A few other models of a nonNewtonian fluid will be outlined below (see also [38, 133, 259, 331]).

297

3.2. Direct Method of Weak Symmetry Reductions

For some exact solutions and transformations of equation (3.2.5.57), see [243, 271, 275]. We look for exact solutions to equation (3.2.5.57) in the form W = ϕ−2 u(ξ) + gy + h,

ξ = ϕy + ψ,

(3.2.5.58)

where the functions g = g(t, x), h = h(t, x), ϕ = ϕ(t, x), ψ = ψ(t, x), and u = u(ξ) are to be determined in the subsequent analysis. Substituting expression (3.2.5.58) into equation (3.2.5.57) and rearranging, we obtain gt + ggx − F − ϕ−2 (ϕt + gϕx − gx ϕ)u′ξ − ϕ−3 ϕx (u′ξ )2

+ ϕ−2 (ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2 )u′′ξξ

+ ϕ−2 (ϕt + gϕx − gx ϕ)ξu′′ξξ + 2ϕ−3 ϕx uu′′ξξ = ϕ[G(u′′ξξ )]′ξ .

(3.2.5.59)

For equation (3.2.5.59) to reduce to an ODE for u = u(ξ), one has to set gt + ggx − F = a1 ϕ,

(3.2.5.60)

3

ϕt + gϕx − gx ϕ = a2 ϕ ,

(3.2.5.61) 4

ϕx = −a3 ϕ , 2

3

ϕψt − ϕt ψ + gϕψx + gx ϕψ − gϕx ψ − hx ϕ = a4 ϕ ,

(3.2.5.62) (3.2.5.63)

where a1 , . . . , a4 are arbitrary constants. As a result, we arrive at the nonlinear ODE a1 − a2 u′ξ + a3 (u′ξ )2 + a4 u′′ξξ + a2 ξu′′ξξ − 2a3 uu′′ξξ = [G(u′′ξξ )]′ξ .

(3.2.5.64)

Integrating (3.2.5.61) and (3.2.5.62) gives  −1/3 ϕ = 3a3 x + b(t) , a2 −1 1 ′ bt (t) − ϕ , g = c(t)ϕ − 3a3 2a3 where b = b(t) and c = c(t) are arbitrary functions. The functions F and h are easy to find from (3.2.5.60) and (3.2.5.63); their expressions are not written out here. The function h = h(t, x) remains arbitrary. If a3 6= 0, the introduction of a new variable, θ = θ(ξ), by the formula θ = a4 + a2 ξ − 2a3 u converts ODE (3.2.5.64) to an autonomous equation, which is reduced, with the substitution ω(θ) = θξ′ , to a second-order ODE. If a3 = 0, the change of variable q(ξ) = u′ξ reduces equation (3.2.5.64) to a second-order ODE. The boundary layer equation for a three-parameter fluid model. The boundary layer equation for a three-parameter polynomial rheological model is written as 2 Wty + Wy Wxy − Wx Wyy = (κ1 + κ2 Wyy + κ3 Wyy )Wyyy + F (t, x), (3.2.5.65)

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3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

where W is the stream function, which is introduced by formulas (3.2.3.3). Equation (3.2.5.65) is a special case of equation (3.2.5.57) with G(Uy ) = κ1 Uy +

1 2

κ2 Uy2 +

3 1 3 κ3 U y .

(3.2.5.66)

If κ3 = 0, we get a special case of the Sisko model [259, 332]. Remark 3.13. Formally, expression (3.2.5.66) can be obtained by expanding the nonNewtonian viscosity µ in a series in powers of Uy and retaining the first three terms. The presence of three parameters, κi , allows one to use the representation (3.2.5.66) in a wide range fluid velocities.

Consider the axisymmetric boundary layer equation for the three-parameter polynomial rheological model: rx′ W Wyy r 2 = (κ1 + κ2 Wyy + κ3 Wyy )Wyyy + F (t, x), (3.2.5.67)

Wty + Wy Wxy − Wx Wyy −

where the body shape function r = r(x) treated as arbitrary. For r(x) = 1, equation (3.2.5.67) coincides with (3.2.5.65). In equation (3.2.5.67), we change to the variables (3.2.5.7) to obtain 2 wtz + wz wxz − wx wzz = (κ1 r2 + κ2 r3 wzz + κ3 r4 wzz )wzzz + F (t, x). (3.2.5.68)

This equation admits a generalized separable solution as a polynomial of degree 3 in z: w = Az 3 + Bz 2 + Cz + D, (3.2.5.69) where A = A(t) is an arbitrary function; the functions B = B(t, x), C = C(t, x), and D = D(t, x) are to be determined in the subsequent analysis. On substituting (3.2.5.69) into (3.2.5.68) and on rearranging, we obtain an equation of the form F z 2 + Gz + H = 0. Equating the functional coefficients F , G, and H with zero, we arrive at the system of PDEs 3A′t − 3ACx + 2BBx = 216κ3r4 A3 ,

Bt + CBx − 3ADx = 18κ2 r3 A2 + 72κ3 r4 A2 B, 2

3

(3.2.5.70) 4

2

Ct + CCx − 2BDx = 6κ1 r A + 12κ2 r AB + 24κ3 r AB + F.

Considering B = B(t, x) to be arbitrary, we find from the first two equations of (3.2.5.70) that Z A′t 1 2 2 B + x − 72κ3 A r4 dx + p(t), C= 3A A Z
1 D= Bt + CBx − 18κ2 r3 A2 − 72κ3 r4 A2 B dx + q(t), 3A

where A = A(t), p = p(t), and q = q(t) are arbitrary functions. The pressure function F is determined from the last equation of (3.2.5.70) without integrating.

3.2. Direct Method of Weak Symmetry Reductions

299

The boundary layer equation for the generalized Sisko model of a nonNewtonian fluid. Consider the following axisymmetric boundary layer equation for the generalized Sisko model of a non-Newtonian fluid: rx′ W Wyy r n1 −1 n2 −1 = (κ1 Wyy + κ2 Wyy )Wyyy + F (t, x), (3.2.5.71)

Wty + Wy Wxy − Wx Wyy −

where r = r(x) is an arbitrary function. It generalizes equation (3.2.5.6) and involves four parameters: n1 , n2 , κ1 , and κ2 . The special case n1 = 1 (or n2 = 1) represents the three-parameter Sisko model [259, 332]. We rewrite equation (3.2.5.71) in terms of the variables (3.2.5.7) to obtain wtz + wz wxz − wx wzz


n1 −1 n2 −1 = κ1 rn1 +1 wzz + κ2 rn2 +1 wzz wzzz + F (t, x). (3.2.5.72)

The analysis shows that equation (3.2.5.72) admits functional separable solutions of the form (3.2.5.11) for six power-law functions u(ξ) = εξ m , which are displayed in Table 3.5 [297]. The first six functions Ψi (i = 1, . . . , 6) are the same as in (3.2.5.14) and (3.2.5.36), while the other two are expressed as n1 mn1 −2n1 −1 Ψ7 = (u′′ξξ )n1 −1 u′′′ , ξξξ = (m − 2)[εm(m − 1)] ξ

n2 mn2 −2n2 −1 Ψ8 = (u′′ξξ )n2 −1 u′′′ , ξξξ = (m − 2)[εm(m − 1)] ξ

where ε = sign[m(m − 1)]. Apart from the relations specified in the last column in Table 3.5, the linear relations (3.2.5.37) also hold. Table 3.5. The exponents n1 , n2 , and m as well as the respective linear relations Ψ7 = αi Ψi and Ψ8 = βi Ψi for the generalized Sisko model of a non-Newtonian fluid with generating power-law functions u = εξ m . No. Relation between n1 and n2

m

n1

n2

1

n2 = 2n1 + 1

2n1 +1 n1

1 m−2

m m−2

2

n2 = 3n1 + 2

2n1 +1 n1

1 m−2

2m−1 m−2

3

n2 = n1 + 1

2n1 +1 n1

1 m−2

m−1 m−2

2n1 n1 −1

m m−2

2m−1 m−2

1 2

4

n2 =

(3n1 + 1)

5

n2 =

1 2

(n1 + 1)

2n1 n1 −1

m m−2

m−1 m−2

6

n2 =

1 3

(n1 + 1)

2n1 −1 n1 −2

2m−1 m−2

m−1 m−2

Relations with Ψ7 and Ψ8 Ψ7 =(m−2)|m(m−1)|n1Ψ1, Ψ8 =ε m−2 |m(m−1)|n2Ψ2 m Ψ7 =(m−2)|m(m−1)|n1Ψ1, Ψ8 = m−2 |m(m−1)|n2Ψ3 m2 Ψ7 =(m−2)|m(m−1)|n1Ψ1, Ψ8 =(m−2)|m(m−1)|n2−1Ψ4 Ψ7 =ε m−2 |m(m−1)|n1Ψ2, m m−2 Ψ8 = m2 |m(m−1)|n2Ψ3 Ψ7 =ε m−2 |m(m−1)|n1Ψ2, m Ψ8 =(m−2)|m(m−1)|n2−1Ψ4 Ψ7 = m−2 |m(m−1)|n1Ψ3, m2 Ψ8 =(m−2)|m(m−1)|n2−1Ψ4

300

3. D IRECT M ETHOD OF S YMMETRY R EDUCTIONS . W EAK S YMMETRIES

Omitting the analysis of all the cases listed in Table 3.5, we restrict ourselves to one specific example. ◮ Example 3.13. Consider the solution corresponding to row 3 in Table 3.5. In view of relations (3.2.5.37), we get

2n + 1 , n m−1 Ψ5 = (m − 1)Ψ2 , Ψ6 = Ψ3 , m n Ψ7 = (m − 2)|m(m − 1)| Ψ1 , Ψ8 = (m − 2)|m(m − 1)|n Ψ4 .

n1 = n,

n2 = n + 1,

m=

(3.2.5.73)

Substituting (3.2.5.11) with u(ξ) = εξ m into equation (3.2.5.72), taking into account relations (3.2.5.73), and reasoning as above in Solution 1, we arrive at the determining system gt + ggx − F − κ1 (m − 2)|m(m − 1)|n rn+1 f n ϕ2n+1 = 0, (f ϕ)t + (f gϕ)x + (m − 1)f (ϕt + gϕx − gx ϕ) = 0, mf ϕx + fx ϕ = 0, ϕψt − ψϕt + gϕψx + gx ϕψ − gϕx ψ − hx ϕ2

− κ2 (m − 2)|m(m − 1)|n rn+2 f n ϕ2n+3 = 0.

(3.2.5.74) (3.2.5.75) (3.2.5.76) (3.2.5.77)

The solutions to equations (3.2.5.75) and (3.2.5.76) are defined by formulas (3.2.5.42) and (3.2.5.43). The pressure function F is evaluated by formula (3.2.5.44) with κ = κ1 . The function h is found as Z 1  h= ϕψt − ϕt ψ + gϕψx + gx ϕψ − gϕx ψ ϕ2 − κ2 (m − 2)|m(m − 1)|n rn+2 f n ϕ2n+3 dx + c(t). All in all, this solution involves five arbitrary functions: ϕ = ϕ(t, x), ψ = ψ(t, x), ◭ a(t), b(t), and c(t).

◮ Example 3.14. (Special two-parameter Sisko model.) We will now consider the axisymmetric boundary layer for the two-parameter rheological model of a nonNewtonian fluid described by equation (3.2.5.67) with κ3 = 0. This is a special twoparameter Sisko model. In terms of the variables (3.2.5.7), this equation has the form (3.2.5.68) with κ3 = 0. We look for a functional separable solution in the form

w = f eϕz + gz + h,

(3.2.5.78)

where the functions f = f (t, x), g = g(t, x), h = h(t, x), and ϕ = ϕ(t, x) are to be determined in the subsequent analysis. On substituting (3.2.5.78) into equation (3.2.5.68) with κ3 = 0 and on rearranging, we arrive at an equation of the form

3.2. Direct Method of Weak Symmetry Reductions

301

Ae2ϕz + Bzeϕz + Ceϕz + D = 0 (the coefficient of ze2ϕz is identically zero). Equating the functional coefficients A, B, C, and D with zero, we obtain the determining system ϕx = κ2 r3 ϕ4 , ϕt + gϕx − ϕgx = 0, (3.2.5.79) (f ϕ)t + (f gϕ)x − f ϕ2 hx = κ1 r2 f ϕ3 , gt + ggx = F. Considering f = f (t, x) to be arbitrary, we find the general solutions of the first three equations (3.2.5.79): 

−1/3 ϕ = − 3κ2 r dx + 3b(t) , Z g = c(t)ϕ + b′t (t)ϕ ϕ2 dx, Z  1  (f ϕ)t + (f gϕ)x − κ1 r2 f ϕ3 dx + d(t), h= f ϕ2 Z

3

where b(t), c(t), and d(t) are arbitrary functions. The pressure function F is obtained ◭ from the last equation of (3.2.5.79) without integrating.

4. Method of Differential Constraints 4.1. Method of Differential Constraints for Ordinary Differential Equations 4.1.1. Description of the Method. First-Order Differential Constraints Description of the method. Prior to describing the method of differential constraints for partial differential equations (PDEs), let us first consider it as applied to simpler ordinary differential equations (ODEs). The main idea of the method is that exact solutions to a complicated (nonintegrable) differential equation are sought by jointly analyzing this equation and an auxiliary simpler (usually integrable) differential equation, called a differential constraint. The order of a differential constraint is the order of the highest derivative involved. Usually, the order of the differential constraint is less than that of the equation; first-order differential constraints are the simplest and most common. The equation and differential constraint must involve a set of free parameters (or even arbitrary functions) whose values are chosen by ensuring that the equation and the constraint are consistent. After the consistency analysis (also called the compatibility analysis), all solutions obtained by integrating the differential constraint will be simultaneously solutions to the original equation. The method makes it possible to find particular solutions to the original equation for some values of the determining parameters. For simplicity, we first consider autonomous ordinary differential equations of the form∗ F (y, yx′ , . . . , yx(n) ; a) = 0, (4.1.1.1) which do not involve the independent variable x explicitly and depend on a vector of free parameters a = {a1 , . . . , ak }. For equations (4.1.1.1), one should take firstorder differential constraints in the autonomous form G(y, yx′ ; b) = 0,

(4.1.1.2)

dependent on a vector of free parameters b = {b1 , . . . , bs }. ∗ Similar equations often arise in mathematical physics when exact solutions to nonlinear partial differential equations are sought in the form of a traveling wave (see Subsection 2.1.1, Item 3◦ ).

DOI: 10.1201/9781003042297-4

303

304

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

By differentiating relation (4.1.1.2) successively sufficiently many times, one (k) can express higher-order derivatives in terms of y and yx′ : yx = ϕk (y, yx′ ; b). Substituting these expressions into the original equation (4.1.1.1), one arrives at a first-order equation H(y, yx′ ; a, b) = 0. (4.1.1.3) By eliminating the derivative yx′ from (4.1.1.2) and (4.1.1.3), one obtains an algebraic (or transcendental) equation P (y; a, b) = 0. (4.1.1.4) Further, one looks for the values of a and b at which equation (4.1.1.4) is satisfied identically for any y (this may result in some restrictions on the components of the vector a). After this, one expresses the vector b in terms of a, so that b = b(a), and substitutes it back into the differential constraint (4.1.1.2) to obtain a first-order ordinary differential equation g(y, yx′ ; a) = 0

(g = G|b=b(a) ).

(4.1.1.5)

The original equation (4.1.1.1) is consistent with equation (4.1.1.5) and inherits all of its solutions. Finally, by solving for the derivative, equation (4.1.1.5) is reduced to a separable equation, which is integrated to obtain a general solution. The general solution of equation (4.1.1.5) is also an exact solution of the original equation (4.1.1.1). Remark 4.1. If the first-order differential constraint is given in the explicit form yx′ =

h(y; b), then its successive differentiation ′′ yxx = (yx′ )′y yx′ = hh′y ,

′′′ ′′ ′ ′ yxxx = (yxx )y yx = h(hh′y )′y ,

...

(k) yx

allows one to express the highest derivatives in terms of y , so that = ϕk (y; b). Using these expressions and the differential constraint to eliminate the derivatives from (4.1.1.1), one immediately arrives at an algebraic/transcendental equation of the form (4.1.1.4). Remark 4.2. Instead of yx′ , one can eliminate the dependent variable y from (4.1.1.2) and

(4.1.1.3) to obtain an algebraic/transcendental equation for the derivative: Q(yx′ ; a, b) = 0. Then, one looks for the values of the parameters a and b that make sure the equation holds identically for any yx′ .

Examples of nonlinear ODEs and their differential constraints. The structural form of the differential constraint (4.1.1.2) can often be taken to be similar to that of the original equation (4.1.1.1), though with other determining parameters. Let us illustrate this with specific examples of second-, third-, fourth-, and higher-order ODEs. ◮ Example 4.1. Consider the second-order ordinary differential equation with a power-law nonlinearity ′′ yxx − cyx′ = ay + by n , (4.1.1.6)

which arises in the theory of chemical reactors, combustion theory, and mathematical biology.∗ ∗ Equations (4.1.1.6) and (4.1.1.12) describe traveling-wave solutions of the Kolmogorov–Petrovskii– Piskunov equation, ut = uzz − f (u), for some forms of the kinetic function f (u). In this case, we have u = y(x) with x = z + ct.

4.1. Method of Differential Constraints for Ordinary Differential Equations

305

Let us supplement equation (4.1.1.6) with the first-order differential constraint yx′ = αy + βy m ,

(4.1.1.7)

which is a separable equation and is easy to integrate. The form of the right-hand side of (4.1.1.7) has been chosen to be similar to that of the original equation (4.1.1.6). The equation and differential constraint involve seven parameters: a, b, c, n, m, α, and β. The further analysis aims at determining the parameters α, β, and m of the differential constraint to express them in terms of a, b, c, and n. Simultaneously, restrictions on the equation parameters will be found and a solution to the equation will be obtained. Differentiating (4.1.1.7) and replacing the first derivative with the right-hand side of (4.1.1.7), we get ′′ yxx = (α + mβy m−1 )yx′ = (α + mβy m−1 )(αy + βy m )

= α2 y + αβ(m + 1)y m + mβ 2 y 2m−1 .

(4.1.1.8)

Eliminating the first and second derivatives from equation (4.1.1.6) using (4.1.1.7) and (4.1.1.8) and rearranging, we obtain (α2 − αc − a)y + β[α(m + 1) − c]y m + mβ 2 y 2m−1 − by n = 0.

(4.1.1.9)

For this equation to hold identically for all y, one must set α2 − αc − a = 0, α(m + 1) − c = 0, 2m − 1 = n,

(4.1.1.10)

mβ 2 − b = 0.

If conditions (4.1.1.10) hold, then solutions to equation (4.1.1.7) are also solutions to the more complex equation (4.1.1.6). The determining system of four equations (4.1.1.10) contains seven parameters a, b, c, n, m, α, and β. The three parameters b, c, and n of the original equation can be regarded as arbitrary, and the other parameters are expressed as follows: r n+1 2c 2c2 (n + 1) 2b , m= , α= , β=± , (4.1.1.11) a=− (n + 3)2 2 n+3 n+1 where n 6= −1, n 6= −3, and b(n + 1) > 0. It is apparent that for equations (4.1.1.6) and (4.1.1.7) to be consistent, the original equation parameter a must be connected with two other parameters, c and n. In this case, two families of parameters (4.1.1.11) of equation (4.1.1.7) can be identified that determine two different one-parameter solutions to equations (4.1.1.6) and (4.1.1.7). Integrating the differential constraint (4.1.1.7), which represents first-order separable ODE, and taking into account formulas (4.1.1.11), we finally obtain two exact 2 (n+1) solutions of equation (4.1.1.6) with a = − 2k(n+3) 2 :   1 y = Ceα(1−m)x − (β/α) 1−m ,

where C is an arbitrary constant and the constants m, α, and β are expressed in terms ◭ of the parameters of the original equation by formulas (4.1.1.11).

306

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

◮ Example 4.2. To construct exact solutions of the second-order ODE with an exponential nonlinearity ′′ yxx − cyx′ = a + beλy , (4.1.1.12)

we use the first-order differential constraint yx′ = α + βeµy ,

(4.1.1.13)

whose right-hand side is chosen to be similar to that of the original equation. The analysis shows that three parameters, b, c, and λ, of equation (4.1.1.12) can be treated as arbitrary, while the other parameters are expressed as r 2c λ 2c2 2b , α= , β=± , µ= . (4.1.1.14) a=− λ λ λ 2 It is apparent that for equations (4.1.1.12) and (4.1.1.13) to be consistent, the parameter a must be related in a certain way to two other parameters of the equation, c and λ. In this case, two families of parameters (4.1.1.14) of the differential constraint (4.1.1.13) can be identified which result in two different one-parameter solutions to equations (4.1.1.12) and (4.1.1.13). Integrating the differential constraint (4.1.1.13), which is a first-order separable ODE, we finally obtain two exact solutions of equation (4.1.1.12) with a = −2k 2/λ: r   bλ 2 , y = − ln C exp(−cx) ∓ λ 2c2 ◭

where C is an arbitrary constant. ◮ Example 4.3. Consider the nonlinear third-order ODE ′′′ yxxx = ay 4 + by 2 + c

(4.1.1.15)

in conjunction with the first-order differential constraint yx′ = αy 2 + β.

(4.1.1.16)

Using (4.1.1.16), we find the derivatives ′′ yxx = 2αyyx′ = 2αy(αy 2 + β) = 2α2 y 3 + 2αβy, ′′′ yxxx = (6α2 y 2 + 2αβ)yx′ = (6α2 y 2 + 2αβ)(αy 2 + β) = 6α3 y 4 + 8α2 βy 2 + 2αβ 2 . (4.1.1.17) For the third derivative in (4.1.1.17) to coincide with the right-hand side of (4.1.1.15), the following relations must hold:

a = 6α3 ,

b = 8α2 β,

c = 2αβ 2 .

(4.1.1.18)

On solving the first two equations for α and β and on inserting the resulting expressions into the last equation, we obtain  a 1/3  a −2/3 b 3b2 α= , β= , c= . (4.1.1.19) 6 6 8 16a

307

4.1. Method of Differential Constraints for Ordinary Differential Equations

It follows that the third-order equation (4.1.1.15) with c = 3b2/(16a) has a particular solution that results from integrating the first-order separable ODE (4.1.1.16), whose parameters are connected with those of the original equation by the first two ◭ relations in (4.1.1.19). ◮ Example 4.4. Consider the nonlinear fourth-order ODE ′′′′ yxxxx = ay n + by 2n+3 .

(4.1.1.20)

in conjunction with the first-order differential constraint (yx′ )2 = αy m + β.

(4.1.1.21)

Differentiating (4.1.1.21) sequentially gives ′′ yxx = ′′′ yxxx = ′′′′ yxxxx =

=

m−1 1 (after cancelling by yx′ ), 2 αmy m−2 ′ 1 yx , 2 αm(m − 1)y m−2 ′′ 1 yxx + 12 αm(m − 1)(m − 2)y m−3 (yx′ )2 2 αm(m − 1)y m−3 1 + 14 α2 m(m − 1)(3m − 4)y 2m−3 . 2 αβm(m − 1)(m − 2)y

(4.1.1.22) Comparing the right-hand side of (4.1.1.20) and that of the last equation in (4.1.1.22) allows us to draw the following conclusions about the consistency of (4.1.1.20) and (4.1.1.21). 1◦ . For n 6= −1, −2, −3, − 35 and b 6= 0, the values of the parameters of the differential constraint (4.1.1.21) can be expressed as s b 2a m = n+3, α = ±2 , β= . (n + 2)(n + 3)(3n + 5) α(n + 1)(n + 2)(n + 3) (4.1.1.23) 2◦ . For b = 0 and n = − 53 , we have m=

4 , 3

β=−

27a , 4α

α 6= 0 is an arbitrary constant.

(4.1.1.24)

In this case, the solution to equation (4.1.1.21) will depend on two arbitrary constants ◭ (α plays the role of an additional constant of integration). Remark 4.3. For b = 0 and n = − 53 , one can find the general solution of equation

(4.1.1.20) (see page 659 in [273]).

◮ Example 4.5. For the fourth-order ODE with an exponential nonlinearity ′′′′ ′′ yxxxx − cyxx = aeλy + be2λy

(4.1.1.25)

we can use the differential constraint (yx′ )2 = α + βeλy .

(4.1.1.26)

308

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

The analysis shows that for any values of the equation parameters satisfying the condition bλ > 0, there are two families of parameters of the differential constraint (either the upper or the lower signs must be taken) 1/2  1/2  c b 2 a 3λ + 2, β=± . α=± 2 λ b λ λ 3λ These ensure that equations (4.1.1.25) and (4.1.1.26) are compatible, which means that the solutions to the second equation are also solutions to the first equation. ◭ ◮ Example 4.6. Let us show that the nonlinear 2nth-order ODE of the form ′′ yx(2n) + a[yyxx − (yx′ )2 ] = b

(4.1.1.27)

admits the differential constraint (yx′ )2 = αy 2 + βy + γ.

(4.1.1.28)

Indeed, differentiating relation (4.1.1.28) sequentially, we get ′′ yxx = αy +

1 2 β,

...,

yx(2n) = αn y +

1 n−1 β. 2α

Substituting these expressions into (4.1.1.27) and taking into account (4.1.1.28), we find the coefficients of the differential constraint: α is any,

β = 2αn/a,

γ = (α2n−1 − ab)/a2 .

◭

◮ Example 4.7. It is not difficult to verify that exact solutions to the nonlinear ordinary differential equation
′′ yx(2n) = yf yyxx − (yx′ )2 ,

where f (z) is an arbitrary function, can be obtained using the differential constraint ◭ (4.1.1.28) with β = 0. ◮ Example 4.8. Let us show that the nth-order nonlinear equations

yx(n) = aeλy yx(m) , (0)

0 ≤ m < n,

(4.1.1.29)

where yx = y, admit the first-order differential constraint yx′ = beµy .

(4.1.1.30)

Indeed, the successive differentiation of (4.1.1.30) yields yx(m) = bm µm−1 (m − 1)! emµy

for m = 1, 2, . . . .

Substituting the relations obtained in (4.1.1.29) and taking into account (4.1.1.30), we finally find the coefficients of the differential constraint λ µ= , n µ=

λ , n−m

1 n ann−1 b= n−1 λ (n − 1)!   1 a(m − 1)! n−m λ b= n − m (n − 1)! 

for m = 0; for 1 ≤ m < n.

◭

309

4.1. Method of Differential Constraints for Ordinary Differential Equations

′′ Remark 4.4. It follows from (4.1.1.30) that yxx /(yx′ )2 = const. Hence, the equations of

the form


′′ yx(n) = eλy f yxx /(yx′ )2 yx(m) , which are more general than (4.1.1.29), also admit the differential constraint (4.1.1.30). The function f is arbitrary.

Table 4.1 lists the nonlinear ODEs considered in Examples 4.1–4.8 as well as some other second- or higher-order equations whose exact solutions can be found (according to [263]) using first-order differential constraints.

4.1.2. Differential Constraints of Arbitrary Order. General Compatibility Method for two Equations In general, a differential constraint is an ordinary differential equation of arbitrary order. Therefore, it is necessary to be able to analyze overdetermined systems of two ordinary differential equations for consistency. Outlined below is a general algorithm for the analysis of such systems. 1◦ . The case of ODEs of the same order. First, let us consider two ordinary differential equations of the same order F1 (x, y, yx′ , . . . , yx(n) ) = 0,

(4.1.2.1)

F2 (x, y, yx′ , . . . , yx(n) ) = 0.

(4.1.2.2)

Here and henceforth, it is assumed that the equations depend on free parameters, which are omitted for brevity. We eliminate the highest derivative (by solving one (n) of the equations for yx and substituting the resulting expression into the other equation) to obtain an (n − 1)st-order equation G1 (x, y, yx′ , . . . , yx(n−1) ) = 0.

(4.1.2.3) (n)

Differentiating (4.1.2.3) with respect to x and eliminating the derivative yx from the resulting equation using either of the equations (4.1.2.1) or (4.1.2.2), one arrives at another (n − 1)st-order equation G2 (x, y, yx′ , . . . , yx(n−1) ) = 0.

(4.1.2.4)

Thus, the analysis of two nth-order ODEs (4.1.2.1) and (4.1.2.2) is reduced to the analysis of two (n − 1)st-order equations (4.1.2.3) and (4.1.2.4). By reducing the order of equations in a similar manner further, one ultimately arrives at a single algebraic/transcendental equation (since two first-order differential equations are reducible to a single algebraic equation). The analysis of the resulting algebraic equation presents no fundamental difficulties and is performed in the same way as previously in Subsection 4.1.1 for the case of a first-order differential constraint. ◮ Example 4.9. Consider the overdetermined system obtained in Example 1.34 and consisting of two ODEs (see the first two equations in (1.5.3.4)):

(θθx′ )′x = A1 θ + A2 , ′′ θxx = A3 θ + A4 .

(4.1.2.5)

310

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Table 4.1. Some nonlinear ordinary differential equations with parameters and corresponding first-order differential constraints that allow one to find their exact solutions. No.

Ordinary differential equations

Differential constraints

1

′′ yxx = ay n +by 2n+1

yx′ = α+βy n+1

2

′′ yxx = ay n +by m

3

′′ yxx = aeλy +be2λy

(yx′ )2= αy n+1+βy m+1+γ; n, m 6= −1

4

′′ yxx = aeλy +beµy +c

(yx′ )2 = αeλy +βeµy +cy+γ

5

′′ yxx = a cos(ky)+b sin(ky)

(yx′ )2 = α cos(ky)+β sin(ky)+γ

6

′′ yxx −kyx′ = ay+by n

yx′ = αy+βy m for m = 12 (n+1)

′′ yxx −kyx′ = ay n−1 +by n +cy 2n−1

yx′ = α+ky+βy n

′′ yxx −kyx′ = a+beλy +ce2λy

yx′ = α+βeλy

′′ yxx −keλy yx′ = aeλy +be2λy

yx′ = α+βeλy

′′ yxx −k(yx′ )2 = aeλy +beµy +c

(yx′ )2 = αeλy +βeµy +γ

yx′ = α+βeλy

′′ yxx −kyx′ = ay+by n +cy 2n−1

yx′ = αy+βy n

′′ yxx −kyx′ = a+beλy

yx′ = α+βeµy for µ = 21 λ

′′ yxx −ky n−1yx′ = ay+by n +cy 2n−1

yx′ = αy+βy n

′′ yyxx −k(yx′ )2 = ay n +by m +c

(yx′ )2 = αy n +βy m +γ for n, m 6= −1

′′ yxx −k(yx′ )2= a cos(ky)+b sin(ky)+c

(yx′ )2 = α cos(ky)+β sin(ky)+γ

16

′′′ yxxx = a+by 2 +cy 4

yx′ = α+βy 2

17

′′′ yxxx = ay n +by 2n+2 +cy 3n+4

yx′ = α+βy n+2

18

′′′ yxxx = ae3λy +be2λy +ceλy

yx′ = αeλy +β

19

′′′ yxxx = (ay n +b)yx′

(yx′ )2 = αy n+2 +βy 2 +γy+δ

20

′′′ yxxx = (aeλy +beµy +c)yx′

(yx′ )2 = αeλy +βeµy +γy 2 +δy+σ

21

′′′ yxxx = [a cos(λy+µ)+b]yx′

(yx′ )2 = α cos(λy+µ)+βy 2 +γy+δ

22

′′′′ yxxxx = ay n +by 2n+3

(yx′ )2 = αy n+3 +β

23

′′′′ yxxxx = aeλy +be2λy

(yx′ )2 = α+βeλy

24

′′′′ yxxxx = a(yx′ )4 +b(yx′ )2 +c

(yx′ )2 = α+βeµy

25

′′′′ ′′ yxxxx −cyxx = aeλy +be2λy

(yx′ )2 = α+βeλy

7 8 9 10 11 12 13 14 15

′′′′ ′′ yxxxx = a[yyxx −(yx′ )2]+by+c

yx′ = α+βy

27

′′′′ ′′ yxxxx = ayyxx +b(yx′ )2 +cy 2 +dy+p

(yx′ )2 = αy 2 +βy+γ

28

′′′′ ′′ 2 yxxxx = a(yxx ) +by 2 +c

(yx′ )2 = αy 2 +βy+γ

29

(n) yx = aeλy

yx′ = beµy for µ = λ/n

30

(n) yx = aeλyyx′

yx′ = beµy for µ = λ/(n−1)

31

′′ −(yx′ )2]+by+c yx = a[yyxx

26

32

(n) (n)

(m)

yx = aeλy[yx ]k

yx′ = α+βy yx′ = beµy for µ = λ/(n−km)

4.1. Method of Differential Constraints for Ordinary Differential Equations

311

We will show how a compatible solution can be found by studying the system for consistency without solving the second equation in (4.1.2.5) for different values of the determining parameters An . Let us expand the brackets in the first equation of (4.1.2.5) and eliminate the second derivative using the second equation to obtain the first-order ODE (θx′ )2 + A3 θ2 = (A1 − A4 )θ + A2 .

Differentiating this equation and cancelling by θx′ , we arrive at the second-order ODE ′′ 2θxx + 2A3 θ = A1 − A4 .

Eliminating the second derivative using the second equation (4.1.2.5), we obtain a linear algebraic equation for θ: 4A3 θ + 3A4 − A1 = 0.

For this equation to hold identically, we must set A3 = 0 and A4 = 13 A1 . With these values of the parameters, the overdetermined system of ODEs (4.1.2.5) has a compatible solution in the form of a quadratic polynomial: 1 3 θ = A1 x2 + Cx + (A2 − C 2 ), 6 A1 ◭ where C is an arbitrary constant. 2◦ . The case of ODEs of different order. Suppose there are two ordinary differential equations having different orders: F1 (x, y, yx′ , . . . , yx(n) ) = 0,

(4.1.2.6)

F2 (x, y, yx′ , . . . , yx(m) ) = 0

(4.1.2.7)

with m < n. Then, by differentiating (4.1.2.7) n − m times, one reduces system (4.1.2.6)–(4.1.2.7) to a system of the form (4.1.2.1)–(4.1.2.2), in which both equations have the same order n. ◮ Example 4.10. Consider the fourth-order ODEs with a quadratic nonlinearity ′′′′ ′′ 2 yxxxx = a(yxx ) − by 2 + c

(4.1.2.8)

in conjunction with the second-order differential constraint ′′ yxx = αy + β. ′′′′ yxxxx

(4.1.2.9)

2

Differentiating (4.1.2.9) twice gives = α y + αβ. Using this expression and the differential constraint (4.1.2.9) to eliminate the derivatives from (4.1.2.8), one arrives at a quadratic equation for y, which is satisfied identically if the conditions aα2 − b = 0,

α − 2aβ = 0,

c = αβ − aβ 2

hold. Two parameters of the original equation, a and b, can be regarded as arbitrary and the others are expressed in terms of them as follows: r r 1 b b b , α=± , β=± . c= ◭ 4a2 a 2a a

312

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

◮ Example 4.11. Consider another fourth-order ODE with a quadratic nonlin-

earity ′′′′ ′′ 2 yxxxx = a(yxx ) + b(yx′ )2 + c.

(4.1.2.10)

We supplement (4.1.2.10) with the nonlinear second-order differential constraint ′′ yxx = α(yx′ )2 + β,

(4.1.2.11)

which is an autonomous ODE integrable by quadrature. Successively differentiating (4.1.2.11), we find the derivatives ′′′ ′′ yxxx = 2αyx′ yxx = 2αyx′ [α(yx′ )2 + β] = 2α2 (yx′ )3 + 2αβyx′ , ′′′′ ′′ yxxxx = [6α2 (yx′ )2 + 2αβ]yxx = [6α2 (yx′ )2 + 2αβ][α(yx′ )2 + β] 3

= 6α

(yx′ )4

2

+ 8α

β(yx′ )2

(4.1.2.12)

2

+ 2αβ .

Substituting (4.1.2.11) and (4.1.2.12) into (4.1.2.10), we obtain the biquadratic equation with respect to the derivative (6α3 − aα2 )(yx′ )4 + (8α2 β − 2aαβ − b)(yx′ )2 + 2αβ 2 − c = 0, which will hold identically if we put 6α3 − aα2 = 0,

8α2 β − 2aαβ − b = 0,

2αβ 2 − c = 0.

Two parameters of the original equation, a and b, can be considered arbitrary, while the remaining constants are expressed in terms of them as follows: c = 27 a−3 b2 ,

α=

1 6 a,

β = −9 a−2 b.

◭

◮ Example 4.12. The general autonomous second-order differential constraint ′′ yxx = f (y)

is equivalent to the autonomous first-order differential constraint (yx′ )2 = F (y), R where F (y) = 2 f (y) dy + C and C is an arbitrary constant. This fact is proved by differentiating the latter relation and comparing with the original differential ◭ constraint. Remark 4.5. In principle, any differential constraint of arbitrary order (4.1.2.7) can be replaced by a suitable first-order differential constraint. Indeed, the above algorithm for successive order reduction of system (4.1.2.6)–(4.1.2.7) leads, in the nondegenerate case, to a system of first-order equations, one of which can be treated as a first-order differential constraint.

313

4.1. Method of Differential Constraints for Ordinary Differential Equations

4.1.3. Using Point Transformations in Combination with Differential Constraints In some cases, it is first useful to reduce the ODE of interest, with a point transformation, to another equation (simpler or more convenient for investigation), which can then be analyzed using a suitable differential constraint. With this approach, solutions to the autonomous equation (4.1.1.1) are sought in the form y = G(u; b),

(4.1.3.1)

where G is a given function and u = u(x) is a function satisfying the first-order differential equation (the differential constraint) H(u, u′x ; c) = 0.

(4.1.3.2)

The functions G and H in (4.1.3.1) and (4.1.3.2) depend on the vectors of free parameters b and c. The introduction of the new variable u, defined by relation (4.1.3.1), reduces equation (4.1.1.1) to a new ODE with one differential constraint (4.1.3.2); this creates the standard situation discussed in Subsection 4.1.1. ◮ Example 4.13. Following [263], we consider the six-parameter ODE with power-law nonlinearities ′′ yxx + (a1 + a2 y n−1 )yx′ = b1 y + b2 y n + b3 y 2n−1 , p

n 6= 1.

(4.1.3.3)

First, we make the change of variable y = u , where the exponent p is to be determined. Then, on multiplying the resulting equation by u2−p , we get puu′′xx +p(p−1)(u′x)2 +p(a1u+a2uk)u′x = b1u2 +b2uk+1 +b3u2k, k = p(n−1)+1. (4.1.3.4) Case 1. In order to obtain an equation with a quadratic nonlinearity, one must 1

1 set k = 0, whence it follows that p = 1−n . Thus, the change of variable y = u 1−n reduces equation (4.1.3.3) to the form n . uu′′xx +s(u′x)2 +a1uu′x +a2u′x +b1(n−1)u2 +b2(n−1)u+b3(n−1) = 0, s = 1−n (4.1.3.5) Various differential constraints can be used to seek exact solutions to equation (4.1.3.5). These are discussed below. 1.1. We are looking for particular solutions to equation (4.1.3.5) using the linear differential constraint u′x = αu + β. (4.1.3.6)

On eliminating the derivatives in (4.1.3.5) with the help of (4.1.3.6), we obtain an equation of the form Au2 + Bu + C = 0. Equating its coefficients A, B, and C with zero, we arrive at a system of three algebraic equations: (s + 1)α2 + a1 α + b1 (n − 1) = 0, (2s + 1)αβ + a2 α + a1 β + b2 (n − 1) = 0, 2

sβ + a2 β + b3 (n − 1) = 0.

(4.1.3.7)

314

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

The first quadratic equation in system (4.1.3.7) serves to determine α (in a wide range of the parameters a1 , b1 , and n, it has two distinct roots). Similarly, the last quadratic equation in (4.1.3.7) serves to determine β (in a wide range of the parameters a2 , b3 , and n, it has two distinct roots). Therefore, in the general case, the second equation in (4.1.3.7) determines four admissible values of the coefficient b2 , for which there exist solutions of the form u = Ceαx − (β/α) satisfying the differential constraint (4.1.3.6). 1.2. Other particular solutions of equation (4.1.3.5) can be obtained with the help of the differential constraint u′x = αu + βu1/2 + γ.

(4.1.3.8)

We use this relation to eliminate the derivatives from (4.1.3.5) to obtain an algebraic equation of the fourth degree for ξ = u1/2 . Equating its coefficients with zero results in the system consisting of five algebraic equations (s + 1)α2 + a1 α + b1 (n − 1) = 0, β[(2s +

(s +

2 1 2 )(β

3 2 )α

+ a1 ] = 0,

+ 2αγ) + a1 γ + a2 α + b2 (n − 1) = 0, β[(2s +

1 2 )γ

(4.1.3.9)

+ a2 ] = 0,

sγ 2 + a2 γ + b3 (n − 1) = 0. For β = 0, the differential constraint (4.1.3.8) coincides, up to notation, with (4.1.3.6). Therefore, we further assume that β 6= 0. From the second, third, and fourth equations (4.1.3.9), we find the coefficients of the differential constraint (4.1.3.8):   b2(1 − n) − a1γ − a2α − 2αγ(s + 21 ) 1/2 a2 a1 , γ =− , β=± . α=− 2s + 32 2s + 12 s + 21 The first and last equations (4.1.3.9) impose the following two restrictions on the coefficients of equation (4.1.3.5): b1 =

a21(s + 12 ) a22(s + 12 ) 2a21(n + 1) 2a22(n + 1) , b = . = − = − 3 (n + 3)2 (3n + 1)2 (n − 1)(2s + 32 )2 (n − 1)(2s + 21 )2

Here, the relation s = n/(1 − n) has been taken into account.

1.3. For a1 = a2 = 0, we can use the nonlinear differential constraint (u′x )2 = αu2 + βu + γ.

(4.1.3.10)

to equation (4.1.3.5). A simple analysis shows that the coefficients of this differential constraint can be expressed in terms of the coefficients of equation (4.1.3.5) as follows: α = b1 (1 − n)2 ,

β=

2b2 (1 − n)2 , 1+n

γ=

b3 (1 − n)2 . n

315

4.1. Method of Differential Constraints for Ordinary Differential Equations

Remark 4.6. A differential constraint of the form (4.1.3.10) with α = 0 determines the quadratic solution 1 C2 − γ u = βx2 + Cx + , 4 β where C is an arbitrary constant.

Case 2. We now set k = 2 in equation (4.1.3.4), implying that p = corresponds to the substitution y = u fourth-order nonlinearity

1 n−1

1 n−1 ,

which

. As a result, we arrive at the ODE with a

uu′′xx + c(u′x )2 + a1 uu′x + a2 u2 u′x = b1 (n − 1)u2 + b2 (n − 1)u3 + b3 (n − 1)u4 ,

c=

2−n . n−1

(4.1.3.11)

Exact solutions of this equation can be found using the quadratic differential constraint u′x = αu2 + βu + γ.

(4.1.3.12)

We use this relation to eliminate the derivatives from (4.1.3.11) and obtain an algebraic equation of the fourth degree for u. Equating its coefficients with zero results in the system consisting of five algebraic equations (c + 2)α2 + a2 α = b3 (n − 1), (2c + 3)αβ + a1 α + a2 β = b2 (n − 1),

(c + 1)(β 2 + 2αγ) + a1 β + a2 γ = b1 (n − 1), γ[(2c + 1)β + a1 ] = 0,

(4.1.3.13)

cγ 2 = 0. Two cases, c = 0 and γ = 0, must be considered; these follow from the last equation. Let us consider only the first case, c = 0, which corresponds to n = 2. To simplify the calculations, we also set a2 = 0. The coefficients of the differential constraint (4.1.3.12) are determined from the first, third, and fourth equations of (4.1.3.13): α=±

p b3 /2,

β = −a1 ,

b1 . γ =±√ 2b3

(4.1.3.14)

√The second equation of (4.1.3.13) defines the relation between the coefficients: a1 2b3 ± b2 = 0 (either the upper or lower signs must be taken in all formulas). The desired solution is determined by integrating the first-order separable ODE (4.1.3.12) with coefficients (4.1.3.14). Remark 4.7. Setting k = 3 in (4.1.3.5), which implies that p =

can also seek solutions to this equation in the more complex form u = α0 + α1 v + α2 v 2 ,

2 n−1

vx′ = β0 + β1 v + β2 v 2 .

2

and y = u n−1 , we

◭

316

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

◮ Example 4.14. Now we consider the six-parameter ODE with exponential

nonlinearities ′′ yxx + (a1 + a2 eλy )yx′ = b1 + b2 eλy + b3 e2λy .

(4.1.3.15)

Here, we make the change of variable y = (µ/λ) ln u, with the parameter µ to be determined. Then, on multiplying the result by λu2 , we get µuu′′xx − µ(u′x )2 + µ(a1 u + a2uµ+1 )u′x = b1 λu2 + b2 λuµ+2 + b3 u2µ+2 . (4.1.3.16) Denoting µ = k −1 in (4.1.3.16), we obtain an equation of the form (4.1.3.4) with the same nonlinearities but different coefficients. Therefore, in order to construct exact solutions of equation (4.1.3.16), one can consider the same differential constraints as in Example 4.13. In particular, setting µ = −1 in (4.1.3.16), we get the equation with a quadratic nonlinearity uu′′xx − (u′x )2 + (a1 u + a2 )u′x + b1 λu2 + b2 λu + b3 = 0,

(4.1.3.17)

which only differs from equation (4.1.3.5) in coefficients. The differential constraints (4.1.3.6), (4.1.3.8), and (4.1.3.10) allow one to find particular solutions to equation ◭ (4.1.3.17) (the details are omitted).

4.1.4. Using Several Differential Constraints In some cases, the equation in question can be treated using several differential constraints involving an additional unknown function. To be specific, we will revisit the nth-order autonomous ODE (4.1.1.1). We supplement it with two first-order differential constraints y = G(u, u′x ; b), H(u, u′x ; c) = 0,

(4.1.4.1) (4.1.4.2)

where b and c are vectors of free parameters. Substituting (4.1.4.1) into (4.1.1.1) gives an (n + 1)st-order ODE for u = u(x): F1 (u, u′x , . . . , u(n+1) ; a, b) = 0. x

(4.1.4.3)

This equation is analyzed in conjunction with the differential constraint (4.1.4.2) by the method described in Subsection 4.1.1. There is a small but inessential difference here; specifically, the order of equation (4.1.4.3) is higher than that of the original equation (4.1.1.1). ◮ Example 4.15. The studies [177, 178, 181] (see also [180]) chose the differential constraint (4.1.4.2) in one of the following three forms:

u′x + u2 − c1 u − c2 = 0,

(u′x )2 − 4u3 (u′x )2 − u4 − c1 u3

(4.1.4.4)

2

(4.1.4.5)

2

(4.1.4.6)

− c1 u − c2 u − c3 = 0,

− c2 u − c3 u − c4 = 0,

4.2. Description of the Method of Differential Constraints for Partial Differential Equations

317

while the differential constraint (4.1.4.1) was chosen from the class of functions  ′ m K L M X X X ux k ′ l y= c1k u + ux c2l u + c3m . (4.1.4.7) u m=1 k=0

l=0

In (4.1.4.7), for the differential constraint (4.1.4.4), it was assumed that K = M and c2l = 0 (l = 1, . . . , L). As a result, a number of new exact solutions of nonlinear ◭ differential equations of the second, third, and fourth orders were obtained. Remark 4.8. All equations (4.1.4.4)–(4.1.4.6) are reduced to separable ODEs (their solutions are expressed in terms of elementary functions or by quadrature). The solution to equation (4.1.4.5) can be expressed in terms of the Weierstrass function ℘ = ℘(z, g2 , g3 ), and the solution of (4.1.4.6), in terms of the Jacobi elliptic function.

The differential constraints (4.1.4.1) and (4.1.4.2) can involve higher derivatives of u with respect to x. ◮ Example 4.16. With the G′/G-expansion method [21, 26, 361, 378], one looks

for particular solutions to autonomous ODEs using two first- and second-order differential constraints of the special form∗  ′ k n X ux y= bk , (4.1.4.8) u k=0

u′′xx − c1 u′x − c0 u = 0.

(4.1.4.9)

Differential constraints (4.1.4.8)–(4.1.4.9) can be simplified using the substitution ξ = u′x /u. As a result, they are reduced to a polynomial point transformation in combination with a Riccati-type first-order differential constraint: n X y= bk ξk , k=0

ξx′ + ξ 2 − c1 ξ − c0 = 0.

It was shown in [179] that seeking particular solutions to ODEs by the method of G′/G expansions based on the differential constraints (4.1.4.8)–(4.1.4.9) leads to ◭ the same results as the tanh-function method [99, 201, 232].

4.2. Description of the Method of Differential Constraints for Partial Differential Equations∗∗ 4.2.1. Preliminary Remarks. A Simple Example The main idea of the method is to try to find exact solutions of a complicated partial differential equation by jointly analyzing this equation and a simpler auxiliary equation called a differential constraint. ∗ In

the original paper [361] and subsequent publications, the notation u = G was used. proceeding with this section, it is recommended that Section 4.1 be acquainted with first.

∗∗ Before

318

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Subsection 1.1.1 gave a few examples of constructing additive separable solutions to nonlinear PDEs in the form u(x, t) = ϕ(x) + ψ(t).

(4.2.1.1)

At the initial stage, the functions ϕ(x) and ψ(t) are assumed arbitrary and they need to be determined in the subsequent analysis. Differentiating expression (4.2.1.1) with respect to t, we obtain ut = f (t)

(f = ψt′ ).

(4.2.1.2)

Conversely, relation (4.2.1.2) implies a representation of the solution in the form (4.2.1.1). Further, differentiating (4.2.1.2) with respect to x gives uxt = 0.

(4.2.1.3)

Conversely, from (4.2.1.3) we obtain a representation of the solution in the form (4.2.1.1). Thus, the problem of seeking exact solutions of the form (4.2.1.1) for a specific partial differential equation can be replaced by an equivalent problem of seeking exact solutions to the equation in question supplemented with condition (4.2.1.2) or (4.2.1.3). Such supplementary conditions in the form of one or several differential equations are usually referred to as differential constraints. Prior to giving a general description of the differential constraints method, we will demonstrate its features with a simple methodical example. ◮ Example 4.17. Consider the third-order nonlinear PDE

uy uxy − ux uyy = auyyy ,

(4.2.1.4)

which occurs in the theory of the hydrodynamic boundary layer. Let us seek a solution of equation (4.2.1.4) satisfying the linear first-order differential constraint ux = ϕ(y).

(4.2.1.5)

Here, the function ϕ(y) cannot be arbitrary, in general, but must satisfy the condition of compatibility of equations (4.2.1.4) and (4.2.1.5). The compatibility condition is a differential equation for ϕ(y) and is a consequence of equations (4.2.1.4) and (4.2.1.5) as well as equations obtained by their differentiation. Successively differentiating (4.2.1.5) with respect to different variables, we find the derivatives uxx = 0,

uxy = ϕ′y ,

uxxy = 0,

uxyy = ϕ′′yy ,

uxyyy = ϕ′′′ yyy .

(4.2.1.6)

Differentiating (4.2.1.4) with respect to x yields u2xy + uy uxxy − uxx uyy − ux uxyy = auxyyy .

(4.2.1.7)

4.2. Description of the Method of Differential Constraints for Partial Differential Equations

319

On inserting the derivatives (4.2.1.5) and (4.2.1.6) into (4.2.1.7), we arrive at a thirdorder ODE for ϕ: (ϕ′y )2 − ϕϕ′′yy = aϕ′′′ (4.2.1.8) yyy .

It represents a compatibility condition for equations (4.2.1.4) and (4.2.1.5). In order to construct an exact solution, we integrate equation (4.2.1.5) to obtain u = ϕ(y)x + ψ(y).

(4.2.1.9)

The function ψ(y) is found by substituting (4.2.1.9) into (4.2.1.4) and taking into account the condition (4.2.1.8). As a result, we arrive at the ordinary differential equation for ψ(y): ′′ ′′′ ϕ′y ψy′ − ϕψyy = aψyyy . (4.2.1.10) Finally, we obtain an exact solution of the form (4.2.1.9), with the functions ϕ and ψ described by equations (4.2.1.8) and (4.2.1.10).

Remark 4.9. It is easier to obtain the above solution by directly substituting expression (4.2.1.9) into the original equation (4.2.1.4). Remark 4.10. The study [379] used the differential constraint (4.2.1.5) to seek exact solutions to equations describing a steady-state plane flow of a non-Newtonian fluid with viscoelastic properties. ◭

4.2.2. General Description of the Method of Differential Constraints The idea of using differential constraints and compatibility theory to construct exact solutions for nonlinear PDEs was first put forward by N. N. Yanenko [374]. For a description of the method of differential constraints and its relationship with other methods as well as for numerous specific examples of its application, see [12, 81, 107, 163–165, 175, 204, 206, 224, 275, 278, 333, 364]. In general, the procedure for constructing exact solutions to nonlinear partial equations of mathematical physics by the method of differential constraints is significantly more complex than that for ordinary differential equations. It consists of several steps outlined below. 1◦ . Seeking exact solutions to the equation F (x, t, u, ux , ut , uxx , uxt , utt , . . . ) = 0

(4.2.2.1)

is performed by supplementing the equation with a differential constraint (auxiliary differential equation) G(x, t, u, ux , ut , uxx , uxt , utt , . . . ) = 0.

(4.2.2.2)

The form of the differential constraint (4.2.2.2) can be chosen: • from a priori considerations (for example, one can require that the constraint represent a solvable equation) or • on the basis of certain properties of the equation in question (e.g., its symmetries or conservation laws).

320

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

2◦ . In general, the thus obtained overdetermined system (4.2.2.1)–(4.2.2.2) for u requires a compatibility analysis. If the differential constraint (4.2.2.2) is specified on the basis of a priori considerations, it should allow for sufficient functional arbitrariness (i.e., involve arbitrary determining functions). The compatibility analysis of system (4.2.2.1)–(4.2.2.2) should provide conditions that specify the structure of the determining functions. These conditions (compatibility conditions) are written as a system of ordinary differential equations (or a system of partial differential equations). Usually, the compatibility analysis is performed by differentiating (possibly several times) equations (4.2.2.1) and (4.2.2.2) with respect to x and t and eliminating the highest-order derivatives from the resulting differential relations and equations (4.2.2.1) and (4.2.2.2). As a result, one arrives at an equation involving powers of lower-order derivatives. Equating the coefficients of all powers of the derivatives with zero, one obtains compatibility conditions connecting the functional coefficients of equations (4.2.2.1) and (4.2.2.2). In general, when analyzing the compatibility of two or more PDEs with a single unknown function, one has to address methods for investigating overdetermined systems of PDEs based on the Cartan algorithm or on the Riquier–Janet–Kuranishi algorithm. These algorithms are detailed, for example, in [184, 303, 333] (see also [74]), where some other information on the theory of overdetermined systems of PDEs can be found. 3◦ . One solves the system of differential equations obtained in Item 2◦ for the determining functions. Then these functions are substituted into the differential constraint (4.2.2.2) to obtain an equation of the form g(x, t, u, ux, ut , uxx , uxt , utt , . . . ) = 0.

(4.2.2.3)

A differential constraint (4.2.2.3) compatible with equation (4.2.2.1) in question is called an invariant manifold of equation (4.2.2.1). 4◦ . One should find the general solution of either equation (4.2.2.3) or some equation that follows from (4.2.2.1) and (4.2.2.3). The solution thus obtained will involve some arbitrary functions {ϕm }; these may depend on x and t as well as u. Notably, instead of the general solution, one may use, in certain cases, some particular solutions of equation (4.2.2.3) or equations that follow from (4.2.2.3). 5◦ . The solution obtained in Item 4◦ should be substituted into the original PDE (4.2.2.1). As a result, one arrives at an equation that serves to find the functions {ϕm }. Having found the {ϕm }, one should insert these functions into the solution from Item 4◦ . Thus, one obtains an exact solution of the original equation (4.2.2.1). Remark 4.11. Should the choice of a differential constraint be inadequate, equations (4.2.2.1) and (4.2.2.2) may happen to be incompatible (have no common solutions). Remark 4.12. Rather than one differential constraint, one can use several constraints of

the form (4.2.2.2).

4.3. First-Order Differential Constraints for Partial Differential Equations

321

Remark 4.13. In the last three stages of the method of differential constraints, one has to solve various equations (systems of equations). If no solution can be constructed in at least one of those steps, one fails to construct an exact solution of the original equation.

For greater clarity, a schematic of the method of differential constraints is displayed in Fig. 4.1.

Figure 4.1. Algorithm for constructing exact solutions by the method of differential constraints.

4.3. First-Order Differential Constraints for Partial Differential Equations 4.3.1. Second-Order Evolution Equations First-order differential constraint. Compatibility condition. Splitting procedure. Consider a general second-order evolution equation solved for the highestorder derivative: uxx = F (x, t, u, ux , ut ). (4.3.1.1) ∗ Usually,

this solution involves arbitrary functions and constants.

322

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Let us supplement this equation with a first-order differential constraint of the general form ut = G(x, t, u, ux ). (4.3.1.2) We obtain the compatibility condition for these equations by differentiating (4.3.1.1) with respect to t once and differentiating (4.3.1.2) with respect to x twice, and then equating the two resulting expressions of the third derivatives uxxt and utxx : Dt F = D2x G.

(4.3.1.3)

Here, Dt and Dx are the total differential operators with respect to t and x: ∂ ∂ ∂ ∂ + utt , + ut + uxt ∂t ∂u ∂ux ∂ut ∂ ∂ ∂ ∂ Dx = + uxt . + ux + uxx ∂x ∂u ∂ux ∂ut Dt =

(4.3.1.4)

The partial derivatives ut , uxx , uxt , and utt in formulas (4.3.1.4) must be expressed in terms of x, t, u, and ux using equations (4.3.1.1) and (4.3.1.2) as well as relations obtained by differentiating these equations. As a result, we find that ∂G ∂G ∂G , + ux +F ∂x ∂u ∂ux   ∂G ∂G ∂G ∂G ∂G ∂G = Gt + G . + uxt + Gx + ux +F utt = DtG = Gt + G ∂u ∂ux ∂u ∂u ∂ux ∂ux (4.3.1.5) In the expression of F , the derivative ut must be replaced with G according to (4.3.1.2). In view of (4.3.1.2), (4.3.1.4), and (4.3.1.5), the compatibility condition (4.3.1.3) will have the form R(x, t, u, ux ) = 0. The left-hand side of this relation can often be represented as a polynomial in the derivative ux : ut = G,

uxx = F,

uxt = Dx G =

M X

Rm (x, t, u)um x = 0.

m=1

The splitting procedure in the derivative leads to the system of determining equations Rm (x, t, u) = 0,

m = 1, . . . , M.

Illustrative examples. Below we consider a few examples that illustrate the application of the method of differential constraints for constructing exact solutions to nonlinear PDEs. For simplicity and clarity, we will not use the general formulas for computing the derivatives (4.3.1.3) and (4.3.1.4). ◮ Example 4.18. To look for exact solutions to nonlinear heat equation with a

source ut = [f (u)ux ]x + g(u),

(4.3.1.6)

4.3. First-Order Differential Constraints for Partial Differential Equations

323

we use the simplest differential constraint ut = ϕ(u).

(4.3.1.7)

The functions f (u), g(u), and ϕ(u) in equations (4.3.1.6) and (4.3.1.7) are unknown in advance and they need to be determined in the subsequent analysis. The procedure for constructing solutions consists of several stages detailed below. 1◦ . Transformation of the original PDE to a more convenient form. On solving equation (4.3.1.6) for the highest derivative, we obtain uxx =

ut − fu′ (u)u2x − g(u) ϕ(u) − g(u) − fu′ (u)u2x = . f (u) f (u)

Here, the derivative ut has been eliminated using the differential constraint (4.3.1.7). The resulting equation can be conveniently rewritten as uxx = h(u) −

fu′ (u) 2 u , f (u) x

h(u) =

ϕ(u) − g(u) . f (u)

(4.3.1.8)

For brevity, we will further omit the argument of the functions f = f (u), g = g(u), and ϕ = ϕ(u) in the intermediate results. 2◦ . Derivation of the determining system of equations. Differentiating constraint (4.3.1.7) twice with respect to x, we find the mixed third-order derivative utxx : ut = ϕ,

utx = ϕ′u ux ,

  f′ utxx = ϕ′u uxx + ϕ′′uu u2x = ϕ′u h − u u2x + ϕ′′uu u2x f   ′ f = hϕ′u + ϕ′′uu − ϕ′u u u2x . f

(4.3.1.9)

The second derivative uxx has been eliminated using relation (4.3.1.8). Differentiating (4.3.1.8) with respect to t, we find the third-order mixed derivative uxxt :  ′ ′ f′ fu ut u2x − 2 u ux uxt uxxt = h′u ut − f f (4.3.1.10)    ′ u′ ′ fu 2 ′ fu ′ ux . + 2ϕu = hu ϕ − ϕ f u f The derivatives ut and uxt have been eliminated using the differential constraint (4.3.1.7) and the second relation in (4.3.1.9). Equating the third-order derivatives uxxt and utxx , defined by (4.3.1.9) and (4.3.1.10), we obtain   ′ ′  ′ f ′′ ′ fu ϕuu + ϕu u2 + hϕ′u − h′u ϕ = 0. (4.3.1.11) +ϕ u f f u x

324

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Now equating the functional coefficients of different powers of ux with zero (which is the splitting procedure with respect to the derivative ux ), we arrive at the determining system of ODEs  ′ ′ f f′ = 0, ϕ′′uu + ϕ′u u + ϕ u f f u (4.3.1.12) ′ ′ hϕu − hu ϕ = 0. 3◦ . Solution of the determining system of equations. System (4.3.1.12) can be integrated, since the first equation is representable in the form [(f ϕ)′u /f ]′u = 0, while the second one is a separable equation. As a result, we get  Z  1 ϕ= A f du + B , f (4.3.1.13) g = (1 − Cf )ϕ, where A, B, and C are arbitrary constants. In deriving the formula for g in (4.3.1.13), the function h was replaced with other functions considering its definition in (4.3.1.8). We will now look at the nondegenerate case A 6= 0 in (4.3.1.13). While treating f = f (u) as an arbitrary function, we use (4.3.1.13) to find the functions g(u) and ϕ(u): Z Z   a a + cf f du + b , ϕ(u) = f du + b , (4.3.1.14) g(u) = f f where a = A, b = B/A, and c = −AC are arbitrary constants.

4◦ . Determining the general form of solution using the differential constraint. On substituting ϕ(u) from (4.3.1.14) into the differential constraint (4.3.1.7), we get the equation Z  a ut = f du + b . (4.3.1.15) f R The change of variable w = f du+b reduces (4.3.1.15) to a simple linear ODE: wt = aw. Integrating it gives the general solution to equation (4.3.1.15) in implicit form Z f (u) du = θ(x)eat − b. (4.3.1.16) Here, the function θ = θ(x) plays the role of an x-dependent constant of integration, since w depends on x and t, while equation wt = aw is independent of x explicitly; in the current stage, θ(x) can be treated as arbitrary. 5◦ . Obtaining a solution to the original PDE. Differentiating (4.3.1.16) with respect to x and t, we find that ut = aeat θ/f and ux = eat θx′ /f . Substituting these expressions into the original equation (4.3.1.6) and taking into account (4.3.1.14), we arrive at the second-order linear ODE ′′ θxx + cθ = 0.

(4.3.1.17)

4.3. First-Order Differential Constraints for Partial Differential Equations

Its general solution is expressed as √
√
 for c > 0,  C1 sin x c +
C2 cos x c √
√ θ = C1 sinh x −c + C2 cosh x −c for c < 0,  C1 x + C2 for c = 0,

325

(4.3.1.18)

where C1 and C2 are arbitrary constants. Formulas (4.3.1.16)–(4.3.1.18) describe exact solutions (in implicit form) to equation (4.3.1.6) with arbitrary f (u) and g(u) ◭ defined by (4.3.1.14). Remark 4.14. In the degenerate case, on setting A = 0, B = b, and C = −c/b in (4.3.1.13), we obtain the following expressions of the functions g(u) and ϕ(u) (as above, f is considered to be an arbitrary given function):

g(u) =

b + c, f

ϕ(u) =

b , f

where b and c are arbitrary constants. This solution can be derived from (4.3.1.14) by renaming the constants, b → b/a and c → ac/b, followed by going to the limit a → 0. After simple calculations, we obtain the corresponding solution to equation (4.3.1.6) in implicit form: Z 1 f (u) du = bt − cx2 + C1 x + C2 . 2 ◮ Example 4.19. Consider the nonlinear reaction–diffusion–convection equa-

tion ut = uxx + f1 (u)ux + f0 (u).

(4.3.1.19)

To seek its exact solutions, we will use the quasilinear first-order differential constraint ut = g1 (u)ux + g0 (u). (4.3.1.20) Equations (4.3.1.19) and (4.3.1.20) are special cases of equations (4.3.1.1) and (4.3.1.2) with F = ut − f1 (u)ux − f0 (u) and G = g1 (u)ux + g0 (u). The functions f1 (u), f0 (u), g1 (u), and g0 (u) are unknown in advance and they need to be determined in the subsequent analysis. Equating the right-hand sides of (4.3.1.19) and (4.3.1.20), we get uxx = h1 ux + h0 ,

where h1 = g1 − f1 ,

h0 = g 0 − f 0 .

(4.3.1.21)

Here and henceforth, the argument of the functions f1 , f0 , g1 , g0 , h1 , and h0 is omitted. Differentiating (4.3.1.20) with respect to x twice and taking into account the expression (4.3.1.21) for uxx , we find the mixed derivatives utx = g1 uxx + g1′ u2x + g0′ ux = g1′ u2x + (g1 h1 + g0′ )ux + g1 h0 , utxx = g1′′ u3x + (g1 h′1 + 3g1′ h1 + g0′′ )u2x +

(g1 h′0

+

3g1′ h0

+

g1 h21

+

g0′ h1 )ux

(4.3.1.22) + (g1 h1 +

g0′ )h0 ,

326

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

where the prime denotes a derivative with respect to u. Differentiating (4.3.1.21) with respect to t and using the expressions (4.3.1.20) and (4.3.1.22) for ut and uxt , we obtain uxxt = h1 uxt + h′1 ux ut + h′0 ut = (g1 h′1 + g1′ h1 )u2x + (g1 h21 + g0′ h1 + g0 h′1 + g1 h′0 )ux + g1 h0 h1 + g0 h′0 .

(4.3.1.23)

We equate the expressions for the third mixed derivatives from (4.3.1.22) and (4.3.1.23) and collect the terms with like powers of ux to obtain a polynomial of degree 3 in the derivative ux in the form g1′′ u3x + (2g1′ h1 + g0′′ )u2x + (3g1′ h0 − g0 h′1 )ux + g0′ h0 − g0 h′0 = 0.

(4.3.1.24)

For relation (4.3.1.24) to hold, we require that the functional coefficients of like powers of ux be zero: g1′′ = 0,

2g1′ h1 + g0′′ = 0,

3g1′ h0 − g0 h′1 = 0,

g0′ h0 − g0 h′0 = 0.

The general solution of this system of ordinary differential equations is given by the following formulas: g1 = C1 u + C2 , g0 = −C12 C3 u3 − C1 C4 u2 + C5 u + C6 , h1 = 3C1 C3 u + C4 , h0 = C3 g0 ,

(4.3.1.25)

where C1 , . . . , C6 are arbitrary constants. Using formulas (4.3.1.21) for h0 and h1 in conjunction with (4.3.1.25), we determine the unknown functions appearing in equations (4.3.1.19) and (4.3.1.20): f1 (u) = C1 (1 − 3C3 )u + C2 − C4 ,

f0 (u) = (−C12 C3 u3 − C1 C4 u2 + C5 u + C6 )(1 − C3 ), g1 (u) = C1 u + C2 ,

g0 (u) =

−C12 C3 u3

(4.3.1.26)

2

− C1 C4 u + C5 u + C6 .

Let us dwell on the special case of C1 = −k,

C2 = C4 = 0,

C3 = −1/k,

C5 = ak,

C6 = bk

in (4.3.1.26), where a, b, and k are arbitrary constants (k 6= 0). The corresponding equation (4.3.1.19) and the differential constraint (4.3.1.20) have the form ut = uxx − (k + 3)uux + (k + 1)(u3 + au + b), 3

ut = −kuux + k(u + au + b).

(4.3.1.27) (4.3.1.28)

The general solution of the first-order quasilinear equation R (4.3.1.28) can be written out in implicit form; it involves the integral I(u) = u(u3 + au + b)−1 du and its inverse. Due to its complicated structure, this solution is inconvenient for the construction of exact solutions to equation (4.3.1.27). In this case, instead of (4.3.1.28), we can use the corollary of equations (4.3.1.27) and (4.3.1.28) obtained by eliminating the derivative ut : uxx = 3uux − u3 − au − b.

(4.3.1.29)

327

4.3. First-Order Differential Constraints for Partial Differential Equations

This ordinary differential equation coincides with (4.3.1.21), where h1 and h0 are expressed by (4.3.1.25). The substitution u = −Ux /U transforms (4.3.1.29) into a third-order linear ODE with constant coefficients: Uxxx + aUx − b = 0.

(4.3.1.30)

Its solutions are determined by the roots of the cubic equation λ3 + aλ − b = 0. In particular, if all roots λn are real, then the general solutions of equations (4.3.1.29) and (4.3.1.30) are given by u = −Ux /U,

U = r1 (t) exp(λ1 x)+r2 (t) exp(λ2 x)+r3 (t) exp(λ3 x). (4.3.1.31)

The functions rn (t) are found by substituting (4.3.1.31) into equation (4.3.1.27) or (4.3.1.28). Note that equation (4.3.1.27) is a special case of equation (3.2.2.1), which was ◭ studied in detail in Subsection 3.2.2 using another method. ◮ Example 4.20. Consider the nonlinear diffusion-type equation

ut = uxx + u2x + u2 .

(4.3.1.32)

We use the first-order differential constraint ux = ϕ(x, t),

(4.3.1.33)

where ϕ is some (yet unknown) function of two arguments. With (4.3.1.33), we can rewrite the original equation (4.3.1.32) in the form ut = ϕx + ϕ2 + u2 .

(4.3.1.34)

Let us find the compatibility condition for equations (4.3.1.33) and (4.3.1.34). To this end, we differentiate (4.3.1.33) with respect to t and (4.3.1.34) with respect to x and eliminate the mixed derivative from the resulting relations bearing in mind that uxt = utx . Using (4.3.1.33) to replace ux with ϕ, we get ϕt = ϕxx + 2ϕϕx + 2uϕ. By expressing u,

ϕt − ϕxx − 2ϕϕx . (4.3.1.35) 2ϕ and substituting (4.3.1.35) into (4.3.1.33) and (4.3.1.34), we arrive at the overdetermined system of nonlinear PDEs [90, 227, 357]: ∂  ϕt − ϕxx − 2ϕϕx  = ϕ, ∂x 2ϕ (4.3.1.36)    ϕ − ϕ − 2ϕϕ 2 ∂ ϕt − ϕxx − 2ϕϕx t xx x = ϕx + ϕ2 + . ∂t 2ϕ 2ϕ u=

It is not difficult to verify that the first equation of (4.3.1.36) has a multiplicative separable solution of the form ϕ(x, t) = ψ(t) sin(x + C),

(4.3.1.37)

328

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

where ψ(t) is an arbitrary function and C is an arbitrary constant. Substituting (4.3.1.37) into the second equation in (4.3.1.36), we arrive at an ODE for ψ(t):    ′ 2 d ψt′ + ψ ψt + ψ = + ψ2 . (4.3.1.38) dt 2ψ 2ψ Once a solution to the second-order ordinary differential equation (4.3.1.38) is found, a solution to the original equation (4.3.1.32) can be obtained as [90, 227]: u(x, t) =

ψt′ + ψ − ψ cos(x + C); 2ψ ◭

this formula results from substituting (4.3.1.37) into (4.3.1.35).

Remark 4.15. In general, solving the first-order partial differential equation (4.3.1.2) reduces to solving a system of ordinary differential equations [161, 277].

4.3.2. Second-Order Hyperbolic Equations In a similar way, one can consider second-order hyperbolic equations of the form uxt = F (x, t, u, ux , ut ),

(4.3.2.1)

supplemented by a first-order differential constraint (4.3.1.2). We will assume that ∂G/∂ux 6= 0. One can obtain a compatibility condition for these equations by differentiating (4.3.2.1) with respect to t and (4.3.1.2) with respect to t and x, and then equating the resulting expressions of the third derivatives uxtt and uttx with each other: Dt F = Dx [Dt G].

(4.3.2.2)

Here, Dt and Dx are the total differential operators of (4.3.1.4) in which the partial derivatives ut , uxx , uxt , and utt must be expressed in terms of x, t, u, and ux with the help of relations (4.3.2.1) and (4.3.1.2) and those obtained by differentiating (4.3.2.1) and (4.3.1.2). Let us show how the second derivatives can be calculated. We differentiate (4.3.1.2) with respect to x and replace the mixed derivative with the right-hand side of (4.3.2.1) to obtain the following expression for the second derivative with respect to x: ∂G ∂G ∂G + ux + uxx = F (x, t, u, ux , ut ) ∂x ∂u ∂ux

=⇒

uxx = H1 (x, t, u, ux ).

(4.3.2.3) Here and henceforth, we bear in mind that (4.3.1.2) allows us to express the derivative with respect to t through the derivative with respect to x. Further, differentiating (4.3.1.2) with respect to t yields utt =

∂G ∂G ∂G ∂G ∂G ∂G +ut +uxt = +G +F =⇒ utt = H2 (x, t, u, ux ). ∂t ∂u ∂ux ∂t ∂u ∂ux (4.3.2.4)

329

4.3. First-Order Differential Constraints for Partial Differential Equations

Replacing the derivatives ut , uxt , uxx , and utt in (4.3.1.4) with their expressions from (4.3.1.2), (4.3.2.1), (4.3.2.3), and (4.3.2.4), we find the total differential operators Dt and Dx , which appear in the compatibility condition (4.3.2.2). ◮ Example 4.21. Consider the nonlinear equation

uxt = f (u).

(4.3.2.5)

Let us supplement (4.3.2.5) with the quasilinear differential constraint ux = ϕ(t)g(u).

(4.3.2.6)

Differentiating (4.3.2.5) with respect to x and then replacing the first derivative with respect to x with the right-hand side of (4.3.2.6), we get uxxt = ϕgfu′ .

(4.3.2.7)

Differentiating further (4.3.2.6) with respect to x and t, we obtain two relations uxx = ϕgu′ ux = ϕ2 ggu′ , uxt = ϕ′t g + ϕgu′ ut .

(4.3.2.8) (4.3.2.9)

Eliminating the mixed derivative from (4.3.2.9) using equation (4.3.2.5), we find the first derivative with respect to t: ut =

f − ϕ′t g . ϕgu′

(4.3.2.10)

Differentiating (4.3.2.8) with respect to t and replacing ut with the right-hand side of (4.3.2.10), we get uxxt = 2ϕϕ′t ggu′ + ϕ2 (ggu′ )′u ut = 2ϕϕ′t ggu′ + ϕ(ggu′ )′u

f − ϕ′t g . gu′

(4.3.2.11)

Equating now the third derivatives (4.3.2.7) and (4.3.2.11) with each other, canceling them by ϕ, and rearranging, we obtain the determining equation ′′ ϕ′t g[(gu′ )2 − gguu ] = ggu′ fu′ − f (ggu′ )′u ,

(4.3.2.12)

which has two different solutions. Solution 1. Equation (4.3.2.12) holds identically for any ϕ = ϕ(t) if ′′ (gu′ )2 − gguu = 0, ′ ′ ′ ′ ggu fu − f (ggu )u = 0.

The general solution of this system is f (u) = aeλu ,

g(u) = beλu/2 ,

(4.3.2.13)

where a, b, and λ are arbitrary constants. For simplicity, we will consider the case a = b = 1,

λ = −2.

(4.3.2.14)

330

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

We substitute g(u) defined by (4.3.2.13)–(4.3.2.14) into the differential constraint (4.3.2.6) and integrate the resulting equation to obtain u = ln[ϕ(t)x + ψ(t)],

(4.3.2.15)

where ψ(t) is an arbitrary function. Substituting (4.3.2.15) into the original equation (4.3.2.5) with the right-hand side (4.3.2.13)–(4.3.2.14), we arrive at a linear ordinary differential equation for ψ(t): ψϕ′t − ϕψt′ = 1. The general solution of this equation is expressed as Z dt ψ(t) = Cϕ(t) − ϕ(t) , 2 ϕ (t)

(4.3.2.16)

where C is an arbitrary constant. Thus, formulas (4.3.2.15) and (4.3.2.16), where ϕ(t) is an arbitrary function, define an exact solution to the nonlinear PDE with an exponential nonlinearity uxt = e−2u . Solution 2. The second solution is determined by the linear relation ϕ(t) = at + b,

(4.3.2.17)

where a and b are arbitrary constants. In this case, the functions f (u) and g(u) are related by (4.3.2.12), with ϕ′t = a. Integrating (4.3.2.6) taking into account constraint (4.3.2.17) yields the solution structure u = u(z),

z = (at + b)x + ψ(t),

(4.3.2.18)

where ψ(t) is an arbitrary function. Inserting it into the original equation (4.3.2.5) and changing x to z with the help of (4.3.2.18), we obtain [az + (at + b)ψt′ − aψ]u′′zz + au′z = f (u).

(4.3.2.19)

For this relation to be an ordinary differential equation for u = u(z), one should set (at + b)ψt′ − aψ = const. Integrating yields ψ(t) in the form ψ(t) = ct + d,

(4.3.2.20)

where c and d are arbitrary constants. Formulas (4.3.2.18) and (4.3.2.20) define a solution to equation (4.3.2.5) for arbitrary f (u). The function u(z) is described by the equation (az + bc − ad)u′′zz + au′z = f (u), which follows from (4.3.2.19) and (4.3.2.20). To the special case a = d = 0 there corresponds a traveling wave solution, while to b = c = d = 0 there ◭ corresponds a self-similar solution.

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

331

4.3.3. Second-Order Equations of General Form Consider the second-order equation of general form F (x, t, u, ux , ut , uxx , uxt , utt ) = 0

(4.3.3.1)

with a first-order differential constraint G(x, t, u, ux , ut ) = 0.

(4.3.3.2)

Let us successively differentiate equations (4.3.3.1) and (4.3.3.2) with respect to both variables so as to obtain differential relations involving second and third derivatives. We get Dx F = 0,

Dt F = 0,

Dx [Dx G] = 0,

Dx G = 0,

Dx [Dt G] = 0,

Dt G = 0,

Dt [Dt G] = 0.

(4.3.3.3)

The compatibility condition for (4.3.3.1) and (4.3.3.2) can be found by eliminating the derivatives ut , uxx , uxt , utt , uxxx , uxxt , uxtt , and uttt from the nine equations of (4.3.3.1)–(4.3.3.3). In doing so, we obtain an expression of the form H(x, t, u, ux ) = 0.

(4.3.3.4)

If the left-hand side of (4.3.3.4) is a polynomial in ux , then the compatibility conditions result from equating the functional coefficients of the polynomial with zero.

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations 4.4.1. Second-Order Differential Constraints Constructing exact solutions of nonlinear partial differential equations with the help of second- and higher-order differential constraints requires finding exact solutions of these differential constraints. The latter is generally rather difficult or even impossible. For this reason, one employs some special differential constraints that involve derivatives with respect to only one variable. In practice, one considers secondorder ordinary differential equations in, say, x and the other variable, t, is involved implicitly or is regarded as a parameter, so that the integration constants will depend on t. Below we discuss the problem of compatibility between a second-order evolution equation ut = F1 (x, t, u, ux , uxx ) (4.4.1.1) and a similar differential constraint ut = F2 (x, t, u, ux , uxx ).

(4.4.1.2)

332

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

First of all, this problem can be reduced to a simpler problem with a first-order differential constraint, which was considered in Subsection 4.2.1. To this end, we need to eliminate the second derivative uxx from equations (4.4.1.1) and (4.4.1.2). As a result, we obtain a first-order equation like (4.3.1.2), which can be treated as a simpler differential constraint than the original one. Then this equation should be analyzed for compatibility with the original equation (4.4.1.1). Secondly, on eliminating the derivative with respect to t from (4.4.1.1) and (4.4.1.2), we arrive at an equation of the form H(x, t, u, ux , uxx) = 0,

(4.4.1.3)

which can be treated as an ordinary differential equation with independent variable x involving t as a parameter. The constants of integration arising in solving equation (4.4.1.3) will be arbitrary functions of t: C1 = C1 (t) and C2 = C2 (t). Hence, an evolution second-order differential constraint of the general form (4.4.1.1) is equivalent, on the one hand, to a suitable first-order constraint (4.3.1.2) and, on the other hand, to a second-order differential constraint in the form of the ordinary differential equation (4.4.1.3). These equivalent constraints are easier to use than the original differential constraint (4.4.1.2). Remark 4.16. The left-hand sides of equation (4.4.1.1) and the differential constraint (4.4.1.2) can have the second derivative utt (or utx ) instead of ut . Then, the elimination of utt (or utx ) also results in a second-order ordinary differential equation (4.4.1.3), which is more reasonable to use rather than the original differential constraint.

◮ Example 4.22. Let us revisit the class of nonlinear heat equations with a

source ut = [f1 (u)ux ]x + f2 (u).

(4.4.1.4)

To seek its exact solutions, we will now use an autonomous second-order differential constraint with a quadratic nonlinearity in the second derivative: uxx = g1 (u)u2x + g2 (u).

(4.4.1.5)

The functions f2 (u), f1 (u), g2 (u), and g1 (u) appearing in (4.4.1.4) and (4.4.1.5) are to be determined in the further analysis. Eliminating the second derivative from (4.4.1.4) and (4.4.1.5), we obtain the first-order PDE ut = ϕ(u)u2x + ψ(u), (4.4.1.6) where ϕ(u) = f1 (u)g1 (u) + f1′ (u),

ψ(u) = f1 (u)g2 (u) + f2 (u).

Here and henceforth, the prime stands for a derivative with respect to u. On differentiating (4.4.1.5) with respect to t, we get uxxt = 2g1 ux uxt + g1′ u2x ut + g2′ ut .

(4.4.1.7)

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

333

We eliminate the derivatives uxxt , uxt , and ut from this relation with the help of equations (4.4.1.5) and (4.4.1.6) as well as their corollaries derived by differentiation. As a result, we obtain a polynomial of degree 4 in the derivative ux : (2ϕg12 + 3ϕ′ g1 + ϕg1′ + ϕ′′ )u4x + (4ϕg1 g2 + 5ϕ′ g2 + ϕg2′ − g1 ψ ′ − ψg1′ + ψ ′′ )u2x + 2ϕg22 + ψ ′ g2 − ψg2′ = 0. Equating the functional coefficients of the different powers of ux with zero, one obtains three equations, which, for convenience, can be written in the form (ϕ′ + ϕg1 )′ + 2g1 (ϕ′ + ϕg1 ) = 0, 4g2 (ϕ′ + ϕg1 ) + (ϕg2 − ψg1 )′ + ψ ′′ = 0,

(4.4.1.8)

ϕ = − 21 (ψ/g2 )′ .

The first equation can be satisfied by putting ϕ′ + ϕg1 = 0. The corresponding particular solution of system (4.4.1.8) is   1 1 C2 µ′′ ϕ = − µ′ , ψ = µg2 , g1 = − ′ , g2 = 2C1 + p , (4.4.1.9) 2 µ |µ| µ′

where µ = µ(u) is an arbitrary function. Taking into account (4.4.1.7), we find the functional coefficients of the original equation (4.4.1.4) and differential constraint (4.4.1.5):   1 f1 = C3 − u µ′ , f2 = (µ − f1 )g2 , 2   (4.4.1.10) C2 1 µ′′ . g1 = − ′ , g2 = 2C1 + p µ |µ| µ′ In view of (4.4.1.10), equation (4.4.1.5) admits the first integral p   1 , u2x = 4C1 µ + 4C2 |µ| + 2σt′ (t) (µ′ )2

(4.4.1.11)

where σ(t) is an arbitrary function. Let us eliminate u2x from (4.4.1.6) by means of (4.4.1.11) and substitute the functions ϕ and ψ from (4.4.1.9) to obtain the equation p µ′ ut = −C2 |µ| − σt′ (t). (4.4.1.12)

Let us dwell on the special case C2 = C3 = 0. Integrating equation (4.4.1.12) and taking into account that µt = µ′ ut yields µ = −σ(t) + θ(x),

(4.4.1.13)

where θ(x) is an arbitrary function. Substituting (4.4.1.13) into (4.4.1.11) and taking into account the relation µx = µ′ ux , we obtain θx2 − 4C1 θ = 2σt − 4C1 σ. Equating both sides of this equation with zero and integrating the resulting ordinary differential equations, we find the functions on the right-hand side of (4.4.1.13): σ(t) = A exp(2C1 t),

θ(x) = C1 (x + B)2 ,

(4.4.1.14)

334

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

where A and B are arbitrary constants. Thus, an exact solution of equation (4.4.1.4) with the functions f1 and f2 from (4.4.1.10) with C2 = C3 = 0 can be represented in implicit form as follows: µ(u) = −A exp(2C1 t) + C1 (x + B)2 . The function µ(u) in the original equation, its solution, and determining relations ◭ (4.4.1.10) can be chosen arbitrarily. ◮ Example 4.23. Consider the class of nonlinear second-order PDEs

ut = f2 (u)uxx + f1 (u)ux + f0 (u). We choose a differential constraint of the form uxx = g1 (u)ux + g0 (u). The compatibility analysis of these equations leads us to the following relations for the determining functions: f2 (u) is an arbitrary function, f1 (u) = C1 u + C2 − (3C1 C3 u + C4 )f2 (u),

f0 (u) = (−C12 C3 u3 − C1 C4 u2 + C5 u + C6 )[1 − C3 f2 (u)], g1 (u) = 3C1 C3 u + C4 , g0 (u) = C3 (−C12 C3 u3 − C1 C4 u2 + C5 u + C6 ), ◭

where C1 , . . . , C6 are arbitrary constants.

4.4.2. Higher-Order Differential Constraints. Determining Equations 1◦ . Third- and higher-order differential constraints involving derivatives in two or more independent variables are used very rarely, since they lead to cumbersome computations and rather complex equations (the original equations are often simpler). As a rule, a higher-order differential constraint represents an ordinary differential equation, only involving derivatives with respect to a single independent variable; the other independent variables appear as free parameters. An example of such a differential constraint of the second order is given by equation (4.4.1.3). For simplicity of the presentation, we first look at the second-order evolution equation ut = f (t, x, u, ux , uxx ) (4.4.2.1) and an n-order differential constraint of the form (n−1) h ≡ u(n) ) = 0. x + g(t, x, u, ux , . . . , ux

(4.4.2.2)

Let [f ] denote equation (4.4.2.1) and its derivatives with respect to x and let [h] denote constraint (4.4.2.2) and its derivatives with respect to x.

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

335

Equation (4.4.2.1) and the differential constraint (4.4.2.2) satisfy compatibility conditions if and only if Dt (h) [f ]∩[h] = 0. (4.4.2.3)

If condition (4.4.2.3) holds, the differential constraint (4.4.2.2) represents an invariant manifold of equation (4.4.2.1). The study [164] showed that for n ≥ 4, condition (4.4.2.3) can be represented as the equivalent determining equation Dt (h) [f ] = f2 D2x (h) + [f1 + nDx (f2 )]Dx (h) + [f0 + nDx (f1 ) − hn−1 Dx (f2 ) +

1 2

n(n − 1)D2x (f2 ) + f2 hhn−1,n−1 − 2f2 Dx (hn−1 )]h.

(4.4.2.4)

Here and henceforth, Dt and Dx are total differential operators with respect to t and x, respectively. In addition, we use the following short notation: f0 =

∂f ∂f ∂f ∂h ∂2h , f1 = , f2 = , hn−1 = , hn−1,n−1 = , w = u(n−1) . x ∂u ∂ux ∂uxx ∂w ∂w2

Equation (4.4.2.4) is a complicated nonlinear equation for h. 2◦ . Instead of the nonlinear equation (4.4.2.4), we can use the simpler, linear equation Dt (h) [f ] = f2 D2x (h) + [c1 f1 + c2 Dx (f2 )]Dx (h) + [c3 f0 + c4 Dx (f1 ) + c5 D2x (f2 )]h (4.4.2.5) as a determining invariant manifold of the evolution equation (4.4.2.1). The constants c1 , . . . , c5 are to be determined. Equation (4.4.2.5) has been obtained by discarding the nonlinear terms in equation (4.4.2.4) and replacing the numeric constants with undetermined coefficients. If, for some values of the constants c1 , . . . , c5 , the function h satisfies (4.4.2.5), then the equation h = 0 is an invariant manifold of equation (4.4.2.1). ◮ Example 4.24. For the equation

ut = uxx + u2x + u2 , the linear determining equation (4.4.2.5) is written as Dt (h) [f ] = D2x (h) + 2c1 ux Dx (h) + 2(c2 w + c3 uxx )h.

(4.4.2.6)

(4.4.2.7)

A solution to equation (4.4.2.7) is sought in the form h = uxxx + aux ,

(4.4.2.8)

where a is some constant. Successively differentiating (4.4.2.6) with respect to x, we get uxt = uxxx + 2ux uxx + 2uux, 2 2 uxxt = u(4) x + 2ux uxxx + 2uxx + 2uuxx + 2ux ,

uxxxt =

u(5) x

+

2ux u(4) x

+ 6uxx uxxx + 2uuxxx + 6ux uxx .

(4.4.2.9)

336

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

In view of (4.4.2.8) and (4.4.2.9), we calculate the left- and right-hand sides of the determining equation (4.4.2.7) to obtain (4) Dt (h) [f ] = u(5) x + 2ux ux + 6uxx uxxx + 2uuxxx + (6 + 2a)ux uxx + auxxx + 2auux,

D2x (h)

(4) + 2c1 ux Dx (h) + 2(c2 u + c3 uxx )h = u(5) x + 2c1 ux ux + 2c3 uxx uxxx + 2c2 uuxxx + 2(c1 + ac3 )ux uxx + auxxx + 2ac2 uux .

It follows that the linear determining equation (4.4.2.7) can be satisfied by setting a = 1,

c1 = c2 = 1,

c3 = 3

in (4.4.2.7) and (4.4.2.8). Putting a = 1 in (4.4.2.8) and solving the equation h = 0, we find that u = ϕ0 (t) + ϕ2 (t) cos x + ϕ3 (t) sin x. (4.4.2.10) By substituting this expression into the original equation (4.4.2.6), one can obtain a ◭ system of ordinary differential equations for ϕk (t). ◮ Example 4.25. For the equation

ut = uuxx −

2 2 3 ux

the linear determining equation (4.4.2.5) becomes Dt (h) [f ] = uD2x (h) + b1 ux Dx (h) + b2 uxx h,

(4.4.2.11)

(4.4.2.12)

where the coefficients b1 and b2 can be expressed in terms of c1 , . . . , c5 . It can be verified by direct substitution that for n = 4 and with a suitable selection of the coefficients b1 and b2 , equation (4.4.2.12) can be satisfied by setting h = uxxxx. It follows that the equation uxxxx = 0 represents an invariant manifold. Hence, the original equation (4.4.2.11) admits a solution in the form of a cubic polynomial in the space variable x: u = ϕ3 (t)x3 + ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t). By substituting this expression into (4.4.2.11), one can obtain a system of ordinary ◭ differential equations for ϕk (t). Remark 4.17. Examples 4.24 and 4.25 as well as some other examples of using linear determining equations can be found in [163–165]. The last article investigated nonlinear heat equations of the form ut = (uk ux )x + f (u) using second- and third-order linear determining equations.

3◦ . Consider the mth-order evolution equation ut = f (t, x, u, ux , . . . , u(m) x ) in conjunction with the differential constraint (4.4.2.2).

(4.4.2.13)

337

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

To find the function h, one can use a linear determining equation of the form [164, 165] m X i X m−i Dt (h) [f ] = cij Di−j (h), x (fm−j )Dx

(4.4.2.14)

i=0 j=0

where cij are some constants to be found. Equation (4.4.2.14) is a generalization of (4.4.2.5) to the case of m > 2. Remark 4.18. Section 4.5 contains examples of second- and third-order differential constraints that are equivalent to most common types of exact solutions.

4.4.3. Utilizing Several Differential Constraints. Systems of Nonlinear Equations Remark 4.12 from Subsection 4.2.2 (see also Subsection 4.5.3) noted that several differential constraints of the form (4.2.2.2) can be used instead of a single constraint. In general, all differential constraints must be analyzed for consistency with the original equation. The method of differential constraints can also be used to construct exact solutions to nonlinear systems of equations. We will illustrate this with a few specific examples. ◮ Example 4.26. Consider the nonlinear heat equation with a source

ut = [f (u)ux ]x + g(u).

(4.4.3.1)

Following [227, 357], we set two first-order differential constraints of the form ut = ϕ(x, t, u), ux = ψ(x, t, u),

(4.4.3.2)

where ϕ and ψ are some (yet arbitrary) functions of three arguments. First, let us find the consistency condition for the differential constraints (4.4.3.2). To this end, we differentiate the first relation in (4.4.3.2) with respect to x and the second relation with respect to t. In the resulting relations, we replace the first derivatives with the right-hand sides of (4.4.3.2) to obtain utx = ϕx + ϕu ux = ϕx + ψϕu , uxt = ψt + ψu ut = ψt + ϕψu . Equating the mixed derivatives, utx = uxt , we get the consistency condition ϕx + ψϕu − ψt − ϕψu = 0.

(4.4.3.3)

Now substituting (4.4.3.2) into the original equation (4.4.3.1), we obtain ϕ = (ψx + ψψu )f + ψ 2 fu′ + g.

(4.4.3.4)

338

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Eliminating ϕ from the consistency condition (4.4.3.3) using (4.4.3.4) and taking into account (4.4.3.2), we arrive at the following equation for ψ: ′′ ψt = (ψxx + 2ψψxu + ψ 2 ψuu )f + (3ψψx + 2ψ 2 ψu )fu′ + ψ 3 fuu + ψgu′ − gψu . (4.4.3.5) Equation (4.4.3.5) has three independent variables, x, t, and u, and looks more complicated than the original equation (4.4.3.1), which contains only two independent variables, x and t. However, the presence of the extra variable u provides a wider selection of solutions, which can be sought by prescribing the structure of the function ψ. It will be shown below how to find some classes of exact solutions to the nonlinear heat equation with a source (4.4.3.1) on the basis of equation (4.4.3.5).

Case 1. First, we look for x-independent particular solutions to equation (4.4.3.5) in the product form [227, 357]: ψ = α(t)h(u),

(4.4.3.6)

where the functions α = α(t) and h = h(u) are to be determined. Formula (4.4.3.6) defines the solution structure; the functions α(t) and h(u) are yet unknown and will be determined in the subsequent analysis. Substituting (4.4.3.6) into (4.4.3.5) yields α′t = α3 h(f h)′′uu + αh(g/h)′u . This equation has a nontrivial solution if the relations h(f h)′′uu = A, h(g/h)′u = B,

(4.4.3.7)

hold, where A and B are arbitrary constants. Equations (4.4.3.7) involve three unknown functions; one of them can be treated as preset arbitrarily, while the others need to be determined. The function (4.4.3.6) generates a solution to the original equation (4.4.3.1). By virtue of the second equation in (4.4.3.2), this solution can be represented in implicit form: Z du = α(t)x + β(t), (4.4.3.8) h(u) where the function α = α(t) satisfies the Bernoulli equation α′t = Aα3 + Bα, which is easy to integrate. The function β(t) is determined from the ordinary differential equation resulting from substituting solution (4.4.3.8) into the original equation (4.4.3.1). Assuming that the function h = h(u) is prescribed (it can be defined arbitrarily), we integrate equations (4.4.3.7) to obtain the forms of the functions that determine the equation in question (4.4.3.1): Z A C1 u + C2 f (u) = Q(u) du + , h(u) h(u) Z   du g(u) = h(u) BQ(u) + C3 , Q(u) = , h(u)

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

339

where C1 , C2 , and C3 are arbitrary constants. Case 2. Now we look for t-independent particular solutions to equation (4.4.3.5) in the product form ψ = θ(x)p(u). (4.4.3.9) Substituting (4.4.3.9) into (4.4.3.5) and rearranging, we obtain ′′ θxx f p + θθx′ p(2f p′u + 3fu′ p) + θ3 p2 (f p)′′uu + θ(pgu′ − p′u g) = 0.

(4.4.3.10)

Such functional differential equations are discussed in detail in Chapter 1. Solutions to equation (4.4.3.10)— there are several solutions — can be obtained by the splitting method using the results of Section 1.5; see the functional equation (1.5.2.6) and its solutions (1.5.2.7) and (1.5.2.8). Without performing a complete analysis of equation (4.4.3.10), we write out one exact solution: Z au + b (au + b)p′u (u) du θ(x) = x, f (u) = , g(u) = −3(au + b) − 2p(u) , p(u) p2 (u) (4.4.3.11) where p = p(u) is an arbitrary function; a and b are arbitrary constants. Substituting (4.4.3.9) into the second differential constraint in (4.4.3.2) and taking into account that θ(x) = x [see (4.4.3.11)], we find that ux = xp(u). Integrating the last relation yields Z 1 du = x2 + ξ(t). (4.4.3.12) p(u) 2

Differentiating (4.4.3.12) with respect to t and taking into account the form of the first differential constraint in (4.4.3.2), we find that ϕ = ξt′ p(u). Substituting the expressions of ϕ and ψ into (4.4.3.4) and bearing in mind relations (4.4.3.11) and (4.4.3.12), we obtain a linear ordinary differential equation for ξ(t). The solution of this equation results in the exponential dependence ξ(t) = Ce−2at ,

(4.4.3.13)

where C is an arbitrary constant. Formulas (4.4.3.12) and (4.4.3.13) define a solution to the nonlinear heat equation (4.4.3.1) in implicit form; the determining functions f (u) and g(u) are given by (4.4.3.11), where p(u) is an arbitrary function. ◭ ◮ Example 4.27. In the three-dimensional case, unsteady motion of a viscous incompressible fluid is described by the Navier–Stokes equations and the continuity equation; in short notation, these are written as [192, 329]:

Vt + (V · ∇)V = −∇p + ν∆V, ∇ · V = 0,

(4.4.3.14)

where V = (V1 , V2 , V3 ) is the fluid velocity vector, t is time, p is pressure divided by fluid density, ν is kinematic viscosity, and ∆ and ∇ are the Laplace and gradient operators with respect to the space coordinates x, y, z.

340

4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Equations (4.4.3.14) admit an exact solution of the form [15, 16] V1 = x(− 12 Fz + w) + yv,

V2 = xu − y( 12 Fz + w), V3 = F, Z 2 2 1 1 1 2 p = p0 − 2 αx − 2 βy − γxy − 2 F + νFz − Ft dz,

(4.4.3.15)

where p0 = p0 (t), α = α(t), β = β(t), and γ = γ(t) are arbitrary functions, while F = F (t, x), u = u(t, x), v = v(t, x), and w = w(t, x) are unknown functions that are described by the nonlinear system of four coupled PDEs Ftx + F Fxx − 21 Fx2 = νFxxx + 2(uv + w2 ) − α − β, ut + F ux − uFx = νuxx + γ, vt + F vx − vFx = νvxx + γ, wt + F wx − wFx = νwxx +

1 2 (α

− β).

(4.4.3.16) (4.4.3.17) (4.4.3.18) (4.4.3.19)

Following [15, 16], we will analyze equations (4.4.3.16)–(4.4.3.19) by supplementing them with three first-order differential constraints u = mFx + A,

v = nFx + B,

w = kFx + C,

(4.4.3.20)

where m, n, k, A, B, and C are unknown functions of t. By requiring that the four equations (4.4.3.16)–(4.4.3.19) must coincide after inserting (4.4.3.20), one arrives at a nonlinear system for the unknowns, consisting of one algebraic and six ordinary differential equations: 1 , 4 ′ ′ A − mt B − nt C − kt′ = = = 2(An + Bm + 2Ck), m n k ′ ′ γ − At γ − Bt α − β − 2Ct′ = = = −α − β + 2AB + 2C 2 . m n 2k mn + k 2 =

(4.4.3.21) (4.4.3.22) (4.4.3.23)

This system has seven equations for nine unknowns: six functions m, n, k, A, B, and C from (4.4.3.20) and three functions α, β, and γ from (4.4.3.16)–(4.4.3.19); in this case, the last three functions are also treated as unknown. It can be shown that the last equation in (4.4.3.22) is a corollary of the other three equations in (4.4.3.21) and (4.4.3.22). Therefore, three unknown functions in system (4.4.3.21)–(4.4.3.23) can, in general, be taken arbitrarily. In view of (4.4.3.20)–(4.4.3.23), system (4.4.3.16)–(4.4.3.19) is reduced to a single partial differential equation Ftx + F Fxx − Fx2 = νFxxx + qFx + p,

(4.4.3.24)

where the functions p = p(t) and q = q(t) are determined by p=

γ − A′t , m

q=

A − m′t . m

(4.4.3.25)

4.4. Second- and Higher-Order Differential Constraints. Some Generalizations

341

For m = n, the general solution to system (4.4.3.21)–(4.4.3.23) can be represented in the form 1 1 sin ϕ, k = cos ϕ, 2 2 1 1 A = B = (q sin ϕ + ϕ′t cos ϕ), C = (q cos ϕ − ϕ′t sin ϕ), 2 2 1 2 1 ′ 2 1 α = q + (ϕt ) − p(1 − cos ϕ) + Ct′ , 4 4 2 1 2 1 ′ 2 1 β = q + (ϕt ) − p(1 + cos ϕ) − Ct′ , 4 4 2 1 ′ γ = p sin ϕ + At , 2 m=n=

(4.4.3.26)

where p = p(t), q = q(t), and ϕ = ϕ(t) are arbitrary functions. For convenience, the free functions p and q in (4.4.3.26) are chosen so that system (4.4.3.16)–(4.4.3.19) in conjunction with the differential constraints (4.4.3.20) and (4.4.3.26) is reduced to a single equation (4.4.3.24) with the same functions p = p(t) and q = q(t). Thus, we have proved the following important statement: any solution to equation (4.4.3.24) for any p = p(t) and q = q(t) generates a solution to the system of PDEs (4.4.3.16)–(4.4.3.19). The case m 6= n and a large number of exact solutions to equation (4.4.3.24) can ◭ be found in [15–17, 258]. ◮ Example 4.28. Let us look at the nonlinear hydrodynamic-type system of two

equations Ftx + F Fxx − Fx2 = νFxxx + qFx + p, Gt + F Gx − GFx = νGxx ,

(4.4.3.27)

the first of which is independent of G and coincides with (4.4.3.24) and the second is linear in G. The functions p = p(t) and q = q(t) in the first equation in (4.4.3.27) can be chosen arbitrarily. It can be shown that exact solutions to system (4.4.3.27) generate some exact solutions to system (4.4.3.16)–(4.4.3.19) and, hence, exact solutions to the unsteady three-dimensional Navier–Stokes equations [15, 17]. Let us supplement system (4.4.3.27) with the second-order differential constraint G = a(t) + b(t)Fx + c(t)Fxx .

(4.4.3.28)

In the second equation of system (4.4.3.27), we eliminate G using (4.4.3.28) and compare the resulting equation with the first equation in (4.4.3.27) (as well as with the equations obtained by differentiating the first equation in (4.4.3.27) with respect to x). This results in the compatibility conditions of system (4.4.3.27) and the differential constraint (4.4.3.28): b′t

a′t + bp = 0, + bq − a = 0, c′t + qc = 0.

(4.4.3.29)

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R The last equation in (4.4.3.29) is easy to integrate: c = C1 exp(− q dt), where C1 is an arbitrary constant. From the first two equations in (4.4.3.29), we obtain a linear second-order ODE for b: b′′tt + qb′t + (p + qt′ )b = 0.

(4.4.3.30)

On solving this equation, one can find a = a(t) from the second equation in (4.4.3.29) without integration. The result obtained can be rephrased as follows. Suppose an exact solution F = F (t, x) to the first equation in (4.4.3.27) is known. Then the corresponding exact solution to the second equation in (4.4.3.27) can be obtained by formula (4.4.3.28), where a = a(t), b = b(t), and c = c(t) are determined by solving the system of ordinary differential equations (4.4.3.29). It is noteworthy that formula (4.4.3.28) can be used for the analysis of nonlinear ◭ stability (instability) of solutions to system (4.4.3.27) [246, 247]. Remark 4.19. For the use of differential constraints to construct exact solutions in gas dynamics, see [206, 333].

4.5. Connection Between the Method of Differential Constraints and Other Methods 4.5.1. Preliminary Remarks The method of differential constraints, based on the compatibility theory of two or more PDEs, is one of the most (if not the most) common methods for constructing exact solutions to nonlinear partial differential equations. Many other methods can be treated as its special cases. The most informal and challenging problem in utilizing the method is finding a suitable differential constraint for a given nonlinear partial differential equation. If, for example, a differential constraint contains insufficient functional arbitrariness, then no solutions may be obtained. If it is too general, then the compatibility analysis of the nonlinear partial differential equations in question may be so complex that one may fail to find any exact solutions at all. The successful solution to the above problem lies beyond the formal description of the method and, in each specific case, is usually determined by the researcher’s experience and intuition. Furthermore, in several stages of applying the method of differential constraints, one has to solve various differential equations or systems of such equations that involve arbitrary functions. This can significantly complicate the analysis. If one cannot find a solution in at least one of these stages, one fails to construct an exact solution to the original equation. Therefore, the method of differential constraints is usually more complicated to utilize than other methods described in the present book. These circumstances explain why simpler though less common methods for constructing exact solutions to nonlinear PDEs are often preferable in practical use. Notably, less common methods can frequently allow one to find important relations constructively that can be treated as differential constraints; such relations

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cannot be obtained a priori directly from the method of differential constraints. The ideas and algorithms underlying these methods often make them more effective∗ than the method of differential constraints for constructing exact solutions to individual classes of nonlinear partial differential equations. Thus, it seems that the methods of generalized and functional separation of variables, the direct method of symmetry reductions, and the method of differential constraints in the aggregate complement each other. Each method has its advantages and disadvantages and may be more effective than others for appropriate classes nonlinear PDEs. Below we will discuss the connection between the method of differential constraints and the other methods described in the preceding chapters.

4.5.2. Generalized Separation of Variables and Differential Constraints The methods of generalized separation of variables can be restated in terms of the method of differential constraints, in the sense that any generalized separable solution can be matched up with an equivalent differential constraint. Generalized separable solutions of a given form can be reduced to differential constraints by eliminating arbitrary functions through differentiation (a similar procedure is used in Section 1.4 and also Subsection 4.2.1). Table 4.2 lists examples of some first- and second-order differential constraints that are equivalent to the most common forms of separable solutions. The functions f and g can be expressed in terms of the original functions ϕ, ψ, and χ. It is apparent that each separable or generalized separable solution can be matched up with several equivalent differential constraints. Conversely, by integrating the differential constraints given in the last column of Table 4.2, one can obtain the general form of separable solutions and generalized separable solutions, specified in the penultimate column of the table. The generalized separable solutions specified in rows 3 and 4 of Table 4.2, which involve three free functions, can be matched up with the equivalent third-order differential constraints uxxx = 0 ux uxxt − uxx uxt = 0

(solution structure: u = ϕ(t)x2 + ψ(t)x + χ(t)), (solution structure: u = ϕ(t)ψ(x) + χ(t)),

which already do not have functional arbitrariness. Seeking a generalized separable solution in the form u(x, t) = ϕ1 (x)ψ1 (t) + · · · + ϕn (x)ψn (t) with 2n unknown free functions is equivalent to prescribing a differential constraint of order 2n that already does not involve free functions. In general, the number of ∗ By this, we mean that less common methods may lead to the desired results much quicker with fewer intermediate calculations and, besides, they are much easier to use.

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Table 4.2. First- and second-order differential constraints corresponding to some classes of separable solutions. No.

Type of solution

Structure of solution

Differential constraints

1

Additive separable solution

u = ϕ(x) + ψ(t)

ux = f (x) ut = g(t) uxt = 0

2

Multiplicative separable solution

u = ϕ(x)ψ(t)

ux = f (x)u ut = g(t)u uuxt − ux ut = 0

3

4

Generalized separable solution

Generalized separable solution

u = ϕ(t)x2 + ψ(t)x + χ(t)

ux = f (t)x + g(t) xux − u = f (t)x2 + g(t) xux − 2u = f (t)x + g(t) uxx = f (t) xuxx − ux = f (t)

u = ϕ(t)ψ(x) + χ(t)

ux = f (t)g(x) ut = f (t)u + g(t) ux = f (x)[u + g(t)] uxx − f (x)ux = 0 uxt − g(t)ux = 0

the unknown functions ϕi (x) and ψi (t) equals the maximum possible order of the differential constraint. For the types of solution listed in Table 4.2, using the methods of generalized separation of variables is preferable, since these methods require fewer steps in which solving intermediate differential equations is necessary. Furthermore, in constructing exact solutions to third- or higher-order PDEs, the method of differential constraints leads to very cumbersome calculations and highly complicated equations; the original equations usually look much simpler. On the other hand, the method of generalized separation of variables often allows one to construct exact solutions to higher-order nonlinear PDEs without much effort; the books [274, 275] present a large number of similar solutions to various equations. Remark 4.20. The differential relations that result from utilizing the splitting method (see Section 1.5) in studying bilinear functional differential equations, which arise after substituting generalized separable solutions into the PDEs in question, can be treated as differential constraints. Thus, the splitting method allows one to find differential constraints constructively during the solution process. Importantly, these constraints are not known in advance, nor can they be prescribed at the beginning of the analysis from a priori considerations.

4.5.3. Functional Separation of Variables and Differential Constraints Functional separable solutions and equivalent differential constraints. The methods of functional separation of variables can be restated in terms of the method of

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differential constraints, in the sense that any functional separable solution can be matched up with an equivalent differential constraint. Functional separable solutions of a given form can be reduced to differential constraints by eliminating arbitrary functions through differentiation. Table 4.3 gives a few examples of first- and second-order differential constraints that are equivalent to the two most common forms of functional separable solutions. The functions f and g can be expressed in terms of the original functions U , ϕ, and ψ. It is apparent that each generalized separable solution can be matched up with several equivalent differential constraints. Conversely, by integrating the differential constraints given in the last column of Table 4.3, one can obtain the general form of functional separable solutions, specified in the penultimate column of the table. Table 4.3. First- and second-order differential constraints corresponding to two classes of functional separable solutions. No.

1

2

Type of solution

Structure of solution

Functional separable solution u = U (z), z = ϕ(t)x + ψ(t) (generalized traveling-wave solution)

Functional separable solution

Differential constraints ux = f (t)g(u) ut = [f (t)x + g(t)]ux uxx − f (u)u2x = 0 ux uxt − utuxx = f (t)u2x

ux = f (x)g(u) ut = f (t)g(u) ut = f (x)g(t)ux u = U (z), z = ϕ(x) + ψ(t) uuxt − f (u)uxut = 0 ux uxt − utuxx = f (x)ux ut ux utt − utuxt = g(t)uxut

The functional separable solutions specified in Table 4.3, which involve three free functions, can also be matched up with the equivalent third-order differential constraints ux(utuxxx − uxuxxt) = 2uxx(utuxx − uxuxt)

uxut(utuxxt − uxuxtt) = uxt(u2t uxx − u2xutt)

(solution: u = U (ϕ(t)x + ψ(t))), (solution: u = U (ϕ(x) + ψ(t))),

which already do not have functional arbitrariness. Remark 4.21. The differential relations that derive from the PDEs resulting from applying transformation (2.7.1.2) and utilizing the method of functional separation of variables in conjunction with the generalized splitting principle, can be interpreted as differential constraints. Thus, the generalized splitting principle allows one to find differential constraints constructively during the solution process. Importantly, these constraints are not known in advance, nor can they be prescribed at the beginning of the analysis from a priori considerations.

Comparison of the effectiveness of the methods of differential constraints and functional separation of variables. Although any functional separable solution of a given form can be replaced with an equivalent differential constraint, the procedure for finding exact solutions by the method of differential constraints, based on a

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compatibility analysis of PDEs, and the procedure of direct functional separation of variables differ significantly. In the generic case, these procedures lead to different results. Let us compare the effectiveness of these methods by looking at the reaction– diffusion type equation (2.7.3.1) (its functional separable solutions were obtained above in Subsection 2.7.3). In order to use the method of differential constraints, we differentiate formula (2.7.1.2) with respect to t to obtain ut = Θ(x, t)φ(u),

(4.5.3.1)

where Θ(x, t) = ϑt and φ(u) = 1/ζ(u). Relation (4.5.3.1) can be treated as a first-order differential constraint of a special form. We will use this constraint to find exact solutions of equation (2.7.3.1) through a compatibility analysis of the overdetermined pair of differential equations (2.7.3.1) and (4.5.3.1) with the single unknown u. The differential constraint (4.5.3.1) is equivalent to relation (2.7.1.2). In the initial stage, both functions Q(x, t) and φ(u) included on the right-hand side of (4.5.3.1) are considered arbitrary, and the specific form of these functions will be determined in the subsequent analysis. We solve equation (2.7.3.1) for the highest derivative uxx and eliminate ut with the help of (4.5.3.1) to obtain  ′  f′ ax b g Θφ − ch uxx = − u u2x − + . (4.5.3.2) ux + f a a f af Differentiating (4.5.3.1) twice with respect to x and taking into account relation (4.5.3.2), we get utx = Θφ′u ux + Θx φ, utxx = Θφ′u uxx + Θφ′′uu u2x + 2Θx φ′u ux + Θxx φ   fu′ ′ ′′ φ u2 + A1 (x, t, u)ux + A0 (x, t, u), = Θ φu − f u x    ′ b g a φ′u , A1 (x, t, u) = 2Θx − Θ x + a a f cΘ h ′ Θ2 φ ′ φu + φ . A0 (x, t, u) = Θxx φ − a f a f u

(4.5.3.3)

where the functions A1 (x, t, u) and A0 (x, t, u) are independent of the derivative ux and are expressed in terms of the functions appearing in the original PDE (2.7.3.1) and differential constraint (4.5.3.1). Differentiating (4.5.3.2) with respect to t and using relation (4.5.3.1) and the first formula of (4.5.3.3), we find the mixed derivative in a different way:    ′ ′ f f′ uxxt = −Θ φ u +2 u φ′u u2x +B1(x, t, u)ux +B0(x, t, u), f u f  ′ ′ f ax Θ ′ b gφ (4.5.3.4) B1(x, t, u) = −2Θxφ u − φu − Θ , f a a f u  ′  ′ ax Θ x bΘx g cΘ h Θt φ Θ2 φ B0(x, t, u) = − φ− φ− φ + + φ , a a f a f u a f a f u

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where the functions B1 (x, t, u) and B0 (x, t, u) are independent of ux . Equating the third-order mixed derivatives (4.5.3.3) and (4.5.3.4), we get the following relation, quadratic in ux : Ku2x + M ux + N = 0,

(4.5.3.5)

where   ′ ′  f′ f K = Θ φ′′u + φ′u u + φ u , f f u  ′   bΘ g f′ φ, M = 2Θx φ′u + u φ + f a f u    ′   ′ Θt φ h b g cΘ Θ2 φ2 fu′ h ′ ax − φ . + + − φu + N = Θxx φ + Θx φ a a f a f a f u f a f2 (4.5.3.6) The functional coefficients K, M , and N depend on a, b, c, f , g, h, Θ, and φ and their derivatives, but are independent of ux . Equating the functional coefficients in (4.5.3.5) with zero (this is splitting with respect to ux ), we obtain the determining system of equations K = 0, M = 0, and N = 0. For what follows, it will only suffice to consider the first equation of this system (K = 0). After having been divided by Θ, it becomes  ′ ′ ′ f ′′ ′ fu φu + φu +φ u = 0. (4.5.3.7) f f u This equation admits the first integral φ′u + φ

fu′ = C1 . f

(4.5.3.8)

Considering f to be an arbitrary function and φ to be an unknown function and integrating (4.5.3.8), we find the general solution of equation (4.5.3.7):   Z 1 φ= (4.5.3.9) C1 f du + C2 , f where C1 and C2 are arbitrary constants. Thus, the method of differential constraints based on constraint (4.5.3.1) leads to exact solutions in which the functions f and φ, involved in the original equation and the differential constraint, are related by (4.5.3.9). Using the differential constraint (4.5.3.1) is equivalent to representing the solution in the form (2.7.1.2). Since φ = 1/ζ, solution (4.5.3.9) can be rewritten in terms of f and ζ as −1  Z ζ = f C1 f du + C2 .

(4.5.3.10)

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Now we look at some solutions obtained in Subsection 2.7.3 by the method of functional separation of variables. Solution (2.7.3.48) to equation (2.7.3.47) and solution (2.7.3.128) to equation (2.7.3.127) are special cases of solutions (2.7.1.2) with ζ = f /u. These solutions differ from (4.5.3.10); consequently, they cannot be obtained by the method of differential constraints with constraint (4.5.3.1). Also, the more complex solutions (2.7.3.10), (2.7.3.16), (2.7.3.33), (2.7.3.39), (2.7.3.57), and (2.7.3.88), in which ζ depends not only on f (u) but also on other functional coefficients, g(u) or/and h(u), to the class of equations (2.7.3.1) in question, cannot be obtained by the method of differential constraints with constraint (4.5.3.1). Remark 4.22. Furthermore, the above exact solutions cannot be obtained by the method of differential constraints even with a differential constraint of the form ut = U (x, t, u), which is more general than (4.5.3.1).

Some remarks on weak symmetries. In applying the method of differential constraints to equation (2.7.3.1), the loss of some exact solutions occurred when the splitting procedure in powers of ux was applied to relation (4.5.3.5)–(4.5.3.6). Theoretically, in order to avoid such losses, we can further look for weak symmetries [90, 309, 357]. Consider two possible algorithms for finding weak symmetries by looking at the example of the nonlinear equation (2.7.3.1). The first algorithm. This algorithm consists of two stages. 1◦ . The first (composite) stage suggests that relation (4.5.3.5) should be derived. Hence, it leads to the same results as with the method of differential constraints up until the splitting procedure in powers of ux . 2◦ . The second stage suggests a compatibility analysis of three partial differential equations (4.5.3.1), (4.5.3.2), and (4.5.3.5)–(4.5.3.6) in order to derive the determining equation, which then needs to be integrated. The compatibility analysis of these PDEs is carried out as follows. Equation (4.5.3.5) is differentiated with respect to t, after which the derivatives ut and uxt are eliminated from the resulting expression using relation (4.5.3.1) and the first formula of (4.5.3.3). As a result, we obtain P u2x + Qux + R = 0,

(4.5.3.11)

where P = Kt + U Ku + 2Uu K, Q = Mt + U Mu + Uu M + 2Ux K, R = Nt + U Nu + Ux M, U = Θ(x, t)φ(u).

(4.5.3.12)

The functions K, M , and N are defined by formulas (4.5.3.6). Next, on eliminating the derivative ux from equations (4.5.3.5) and (4.5.3.11), we obtain the determining equation. In the nondegenerate case (M P − KQ 6≡ 0), it has the form K(N P − KR)2 − M (M P − KQ)(N P − KR) + N (M P − KQ)2 = 0. (4.5.3.13)

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Equation (4.5.3.13) is a rather complicated and cumbersome nonlinear PDE. It involves the third-order derivatives Θxxt and φ′′′ uuu (recall that Θ and φ are unknown functions); in expanded form, it almost fills, considering relations (4.5.3.5) and (4.5.3.12), the entire page. Furthermore, equation (4.5.3.13), which involves one or more arbitrary functions, f (u), g(u), etc., must be solved in conjunction with equations (4.5.3.1) and (4.5.3.2) (or the original equation). As a result, instead of one equation (2.7.3.1) (or equation (2.7.3.2) together with (2.7.1.2)), one has to deal here with a much more complex system of nonlinear PDEs. In other words, the method in question, which requires an analysis of three PDEs, (4.5.3.1), (4.5.3.2), and (4.5.3.5), is extremely difficult for practical use. ◮ Example 4.29. For greater clarity, let us look at the linear heat equation ut = uxx , which is obtained from (2.7.3.1) by setting

a(x) = 1,

b(x) = 0,

c(x) = 0,

f (u) = 1.

In this case, one has to substitute the following functions into equation (4.5.3.13): K = Uuu , M = 2Uxu , N = Uxx − Ut , U = Θφ; P = Utuu + U Uuuu + 2Uu Uuu , Q = 2(Uxtu + U Uxuu + Uu Uxu + Ux Uuu , R = Uxxt − Utt + U (Uxxu − Utu ) + 2Ux Uxu .

(4.5.3.14)

One can see that the third-order nonlinear PDE (4.5.3.13)–(4.5.3.14) becomes isolated (can be solved independently of the original equation); it is far more compli◭ cated than the second-order linear heat equation in question. The degenerate case of M P − KQ ≡ 0 can be treated likewise. The second algorithm. In this case, we differentiate formula (2.7.1.2) with respect to t and x. As a result, we obtain two relations ut = ϑt φ(u),

ux = ϑx φ(u),

(4.5.3.15)

which can be interpreted as two compatible differential constraints, where the functions ϑ = ϑ(x, t) and φ(u) = 1/ζ(u) are to be determined. Differentiating the second relation (4.5.3.15) with respect to x, we find the second derivative uxx = ϑxx φ + ϑx φ′u ux = ϑxx φ + ϑ2x φφ′u ,

φ = φ(u).

(4.5.3.16)

We insert the derivatives (4.5.3.15) and (4.5.3.16) into (2.7.3.1) to arrive at an equation that, up to renaming, coincides with equation (2.7.3.2). Further, by employing the generalized splitting principle described in Subsection 2.7.1, one can construct the exact solutions obtained in the first part of Subsection 2.7.3 without using equivalent equations. However, constructing the solution from the second part of this subsection with equivalent equations will be impossible. To find these solutions, one has first to integrate the differential relations (4.5.3.15) and return to the original relation (2.7.1.2) and then consider the equivalent equations described in Subsection 2.7.3.

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Thus, we have shown that utilizing transformation (2.7.1.2) in conjunction with the generalized splitting principle can in some cases be more effective for the construction of exact solutions than using one or two equivalent differential constraints.

4.5.4. Direct Method of Symmetry Reductions and Differential Constraints Consider a symmetry reduction based on seeking solutions in the form
u(x, t) = F x, t, w(z) , z = z(x, t),

(4.5.4.1)

where F (x, t, w) and z(x, t) should be selected so as to obtain ultimately a single ordinary differential equation for w(z); see Subsection 3.1.3. Following [224], let us show that employing the solution structure (4.5.4.1) is equivalent to looking for a solution with the help of a first-order quasilinear differential constraint ξ(x, t)ut + η(x, t)ux = ζ(x, t, u). (4.5.4.2) Indeed, the characteristic system of ordinary differential equations dt dx du = = , ξ(x, t) η(x, t) ζ(x, t, u) which is associated with the first-order PDE (4.5.4.2), has the first integrals z(x, t) = C1 ,

ϕ(x, t, u) = C2 ,

(4.5.4.3)

where C1 and C2 are arbitrary constants. Therefore, the general solution of equation (4.5.4.2) can be written as follows [277]:
ϕ(x, t, u) = w z(x, t) , (4.5.4.4) where w(z) is an arbitrary function. On solving (4.5.4.4) for u, we obtain a representation of solutions in the form (4.5.4.1). Recall that the representation of the desired function u in the form (4.5.4.1) is redundant and, depending on the imposed conditions on the functions F (x, t, w) and z(x, t) as well as further actions, can ultimately lead to different solutions (compare the method descriptions and examples in Sections 3.1 and 3.2).

4.5.5. Nonclassical Method of Symmetry Reductions Brief description. The nonclassical method of symmetry reductions [33] is an important special case of the method of differential constraints. It is based on supplementing the nonlinear equation in question, (4.2.2.1), with two differential constraints. One of the constraints is a first-order quasilinear PDE of the general form ξ(x, t, u)ux + η(x, t, u)ut = ζ(x, t, u).

(4.5.5.1)

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It is referred to as an invariant surface condition. The other constraint represents an auxiliary equation, ξFx + ηFt + ζFu + ζ1 Fux + ζ2 Fut + ζ11 Fuxx + ζ12 Fuxt + ζ22 Futt + · · · = 0, (4.5.5.2) which coincides with the invariance conditions underlying the classical method of symmetry reductions [34, 150, 225, 230, 231]. The functions ξ = ξ(x, t, u), η = η(x, t, u), and ζ = ζ(x, t, u) in equations (4.5.5.1) and (4.5.5.2) are unknown, while ζi and ζij are the coordinates of the first and second prolongations, whose expressions can be found in the cited books. The nonclassical method of symmetry reductions leads to a nonlinear determining system of PDEs for the unknown functions. For the discussion of this method as well as a number of specific examples of its use, see [33, 70, 71, 73, 74, 187, 223, 227, 275, 278, 308, 310, 323]. For first-order differential quasilinear constraints, the results of applying the method of differential constraint and the nonclassical method of symmetry reduction coincide (provided that the differential constraint coincides with the invariant surface condition). Remark 4.23. The study [223] showed, using the Fitzhugh–Nagumo equation as an example, that the nonclassical method of symmetry reductions is more general than the direct method of symmetry reductions. Remark 4.24. The studies [256, 257] established that, in looking for exact solutions to nonlinear PDEs, the methods of functional separation of variables can be more effective than the nonclassical method of symmetry reductions based on an invariant surface condition. This fact is illustrated with examples of nonlinear reaction–diffusion and convection–diffusion equations with variable coefficients as well as nonlinear Klein–Gordon-type equations.

Classical method of symmetry reductions. With the classical method of symmetry reductions, based on the Lie group analysis of PDEs [34, 150, 225, 230, 231, 275], one first deals with two equations: (4.2.2.1) and (4.5.5.2). One eliminates one of the highest derivatives (for example, utt for second-order PDEs), while the other derivatives (ux , ut , uxx , and uxt ) are considered ‘independent’. The resulting expression is then split into powers of the independent variables, which means that the functional coefficients of the different powers of the independent variables are all equated with zero. As a result, one arrives at an overdetermined system of equations and determines the functions ξ, η, and ζ from this system. One then inserts these functions into the first-order quasilinear equation (4.5.5.1) and solves it to determine the general form of solutions with arbitrary functions. Next, using (4.2.2.1), one refines the solution structure obtained in the preceding step. The study [204] showed that the classical method of symmetry reductions is a special case of the method of differential constraints. Remark 4.25. The classical method of symmetry reductions leads to a linear determining system of PDEs for the unknown functions ξ , η , and ζ . This method may result in the loss of some solutions (which can however be found by the nonclassical method of symmetry reductions), since in the first step of splitting, it is assumed that the first derivatives ux and ut are independent, whereas these are actually linearly dependent by virtue of equation (4.5.5.1).

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4. M ETHOD OF D IFFERENTIAL C ONSTRAINTS

Remark 4.26. As a rule, the classical method of symmetry reductions fails to find exact solutions that can be obtained with the methods of generalized and functional separation of variables, the direct method of symmetry reductions, and the method of different constraints.

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Index A Abel equation of second kind, 185, 205, 225 additive separable solutions, xvi, 1, 3–5, 12, 18, 55, 73, 318, 344 adiabatic index, 15, 64 analysis and solutions of determining system, 180, 191, 268, 278, 289 anisotropic heat equation in two space variables, 5, 6, 123, 125 anisotropic heat equations in three or more independent variables, 208, 209 anisotropic wave equation in two space variables, 210, 256 anisotropic wave equations in three or more independent variables, 208, 211 axisymmetric boundary layer equations, 260, 265, 266 for extended body of revolution, 260, 276, 278 for generalized Sisko model, 299 for non-Newtonian fluids, 285–287, 299 generalization of solutions, 287 three-parameter rheological model, 298 two-parameter rheological model, 300

B Barenblatt–Zel’dovich dipole solution, 20 Bernoulli equation, 10, 59, 96, 101, 294, 338 bilinear equation, 98, 106, 121, 132, 158, 214, 260, 267, 277 reduction to, 216, 231, 235, 242, 247 bilinear functional equation, 46, 48, 52, 55, 91, 98, 119, 146, 213, 215, 344, see also bilinear equation solutions, 46, 47, 49 method of differentiation, 49 splitting principle, 46, 128 Blasius problem, 274 Blasius type problem for unevenly heated flat plate, 273 boundary layer equation, 6, 25, 41, 270 generalized Sisko model non-Newtonian fluid, 299

plane, 269 non-Newtonian fluid, general model, 296 unsteady, 265 variable viscosity, 266, 273, 286 three-parameter model, 297 variable viscosity physical interpretation, 273 boundary layer equations axisymmetric extended, 278 for extended body of revolution, 276 for generalized non-Newtonian fluids, 296 for non-Newtonian fluids, 286 for power-law fluids, 287 laminar, 266 properties, 287 steady-state, 260 unsteady, 266 hydrodynamic, x, 260, 265 plane for non-Newtonian fluids, 285 solutions to determining system, 290 unsteady, 265 steady-state, 260 system, reduction to single PDE, 276 Boussinesq equation, 44, 45, 252, 258 special form of reductions, 254 unnormalized, 44 Burgers equation, 262 generalized, 126 potential, 6 Burgers–Huxley equation, 262 Burgers–Korteweg–de Vries equation, generalized, 251, 256

C Caputo fractional derivative, 84 Cattaneo–Vernotte differential model, 78 Clarkson–Kruskal direct method (of symmetry reductions), xi, 251 classical method of symmetry reductions, xi, 129, 155, 351, 352

375

376

I NDEX

classical Newtonian fluid model, 265, 285 comb model, 84 comparison of effectiveness of methods, x, 93, 260, 345 compatibility analysis, xi, 93, 278, 303, 320, 334, 342, 348 compatibility condition, xi, 318–322, 328, 331, 341 compatibility method for two equations, 309 complicated one-dimensional diffusion type equations, 198 constant of separation, 1, 116 convection–diffusion equation nonlinear, 145–149 convection–diffusion equations, nonlinear, with variable coefficients, 145, 190 convection–diffusion type equations, 145, 190 Crocco type transformation, 287

D derivation of determining system of equations, 323 derivation of functional differential equation, 130, 146, 157 determining equation, 6, 287, 329, 336, 348 equivalent, 335 linear, 335–337 determining equations higher-order differential constraints, 337 determining system algebraic, 305 analysis and solutions, 180, 191, 208, 278, 289, 324 axisymmetric boundary layer, solutions, 268 derivation, 323 modified, 271, 280, 291 ODEs, 48, 98, 215 PDEs, 180, 191, 201, 268, 278, 281, 288– 291, 294, 300, 351 plane boundary layer, solutions, 269 solutions, 281, 283, 290, 295, 296 determining system of equations, see determining system differential constraints, xi, 93, 129, 303, 313, 318, 320, 331 arbitrary order, 309 connection with other methods, 342 equivalent, 344 first-order, 303 for evolution equations, 321

for hyperbolic equations, 328 for ODEs, 303–309 for PDEs, 318, 321, 337 for second-order equations of general form, 331 linear, 313 nonlinear, 314 of general form, 322 quadratic, 315 quasilinear, 325, 329, 350 simple, 323, 327 for evolution equations, 321, 331, 336 for higher-order PDEs, 331 for ODEs, 303–342 for second-order PDEs, 331 general description of method, 319, 321 higher-order, 334 nonclassical method of symmetry reductions, 350 nth-order, 334 second-order for ODEs, 311 for PDEs, 334 using several ones, 316, 320, 337, 340 using with weak symmetries, 348 vs. direct method of symmetry reductions, 350 vs. functional separation of variables, 344, 345 vs. generalized separation of variables, 343, 344 differential-difference equations with finite relaxation time, 78 differentiation method, 36, 42, 46 diffusion equation, 84, see also heat equation, reaction–diffusion equation, convection– diffusion equation linear fractional, 84 nonlinear, 75 fractional, 84 steady-state, 116 with volume reaction, 21 diffusion type equations, 6, 139, see also reaction–diffusion type equations, convection–diffusion type equations one-dimensional, 198 with cubic nonlinearity, 262 with n spatial variables, 170–172 with several spatial variables, 200 with variable coefficients, 216 dilation transformation, 8 direct construction of exact solutions, 187, 197

I NDEX direct functional separation of variables, 93, 346 direct method of symmetry reductions, xi, 251–260 direct method of symmetry reductions and differential constraints, 350 direct method of weak symmetry reductions, 260–302 direct procedure for seeking exact solutions, 180, 191, 201 direct reduction method, see direct method of symmetry reductions

E Emden–Fowler type equation, 142, 238 equation, see also equations Abel, second kind, 185, 205, 225 Bernoulli, 10, 59, 96, 101, 294, 338 boundary layer, see boundary layer equation(s) Boussinesq, 44, 45, 252, 258 special form of reductions, 254 unnormalized, 44 Burgers, 262 generalized, 126 potential, 6 Burgers–Huxley, 262 Burgers–Korteweg–de Vries, generalized, 251, 256 determining, 6, 287, 329, 336, 348 equivalent, 335 higher-order differential constraints, 337 linear, 335–337 diffusion type, with cubic nonlinearity, 262 Emden–Fowler type, 142, 238 functional differential derivation, 130, 146, 157 Ginzburg–Landau type, 178 Guderley, 15, 17 generalized, 64 Harry Dym, 258, 259 general form of reductions, 259 Helmholtz, 171 hyperbolic, 50, 89, 125, 210 delay, 77 second-order, 328 thermal conductivity, 78 with variable coefficients, 202 hyperbolic type, 125, 156, 190 Klein–Gordon type in anisotropic medium, 210 nonlinear, 160, 164 with n spatial variables, 173

377

Kolmogorov–Petrovskii–Piskunov, 304 Korteweg–de Vries generalized, nth-order, 103, 251, 256 potential, 6 Korteweg–de Vries type, 173 linear, diffusion, fractional, 84 nonlinear diffusion, 75 diffusion, fractional, 84 diffusion, steady-state, 116 parabolic, 36, 60, 86, 198, 200, 284 Poisson, 171 porous medium, 19, 247 generalized, 242, 246 reaction–diffusion, see reaction–diffusion equation delay, 72 reaction–diffusion type, 6, 139 with n variables, 170–172 with variable coefficients, 130 Riccati, 183, 193, 203, 284 Richards, 146, 216 Schr¨odinger type nonlinear, general form, 177 nonlinear, with variable coefficients, 178 several nonlinear operators, 65 von Mises, 23 equation involving several nonlinear operators, 65 equations, see also equation admitting simple separable solutions, 6 axisymmetric boundary layer, see axisymmetric boundary layer equations boundary layer, see boundary layer equation(s) convection–diffusion nonlinear, 145, 146, 149, 154, 179 convection–diffusion type nonlinear, reduction to ODEs, 191 nonlinear, variable coefficients, 145, 190 delay reaction–diffusion, 75, 77 delay, admitting simple separable solutions, 72 determining, see determining equation(s) differential-difference, finite relaxation time, 78 diffusion type, see diffusion type equations diffusion, steady-state, in anisotropic media, 209 equivalent, 213–215, 230, 247 evolution, 58, 98 evolution, mth-order, 336 evolution, second-order, 321, 331, 334

378

I NDEX

equations (continued) first-order, 268, 278, 288 nonlinear, 2, 123, 173 functional differential, see functional differential equation(s) generalized porous medium, with nonlinear source, 242 Ginzburg–Landau type, 178 heat, see heat equations hyperbolic, delay, 77 hyperbolic, second-order, 328 integro-differential, nonlinear, 80 Klein–Gordon type nonlinear, 157, 200, 351 nonlinear, overview of solutions, 155 nonlinear, with variable coefficients, 155, 156 Korteweg–de Vries type, 173 Navier–Stokes, 26, 39 solutions, 27, 34, 47, 341 three-dimensional, 339 nonlinear diffusion-type, overview of solutions, 129 Klein–Gordon type, overview of solutions, 155 systems of, 5, 337 third- and higher-order, 173 with fractional derivative, 81, 84 with three independent variables, 5, 26, 254 with three or more independent variables, 5, 26, 170, 208 with time-dependent delay, 71, 73–75, 77 parabolic, 78, 89 partial differential delay, nonlinear, 71 first-order, nonlinear, 2, 123, 173, 313 linear, 1, 13 nonlinear, 13 plane boundary layer for non-Newtonian fluids, 285 solutions to determining system, 290 unsteady, 265 pseudo-differential, 85, 86 reaction–convection–diffusion nonlinear, 179, 216 reduction to ODE, 179 several spatial variables, 189 reaction–diffusion nonlinear, 129, 130, 231, 235 nonlinear, with variable coefficients, 129, 130, 179

with delay, 71, 77 with variable coefficients, 129 reaction–diffusion type, nonlinear, 6, 129, 130, 179, 185, 189 second-order, general form, 331 telegraph type, nonlinear, reduction to ODE, 201 telegraph, nonlinear, 200 telegraph, nonlinear, with variable coefficients, 200 wave anisotropic, in three or more independent variables, 208 nonlinear, in anisotropic medium, 210, 211 with fractional derivative, nonlinear, 81, 84 equations of mathematical physics, linear, 1, 13 equivalent differential constraints, 93, 343– 345 equivalent differential equations, 230 generalizations, 214 equivalent equations, 213–215, 230, 247, 249 exact solutions, see solutions exact solutions to Navier–Stokes equations, x, 26, 27, 34, 47, 253, 341

F first-order differential constraints, 303, 316, 337, 340 for ordinary differential equations, 303, 310, 316 for partial differential equations, 321 first-order partial differential equations determining system of, 268, 278, 288 nonlinear, 2, 181, 328 quasilinear, 350 fluid, see Newtonian fluid, non-Newtonian fluid fractional derivative, 81 Caputo, 84 definition, 82, 84 main properties, 83 of constant, 83 of order µ, 82 Riemann–Liouville, 82 sufficient conditions for existence, 83 fractional integral, 81, 82 definition, 82 main properties, 83 Riemann–Liouville, 82 function inner, 91, 96

I NDEX function (continued) outer, 91, 94, 96, 179 stream, 24, 27, 30, 39, 99, 260, 265, 266, 273, 286, 296 stream, modified, 277 functional differential equation, 2, 10–12, 36–38, 52, 57, 98, 114, 170, 215 bilinear, 128, 146, 213, 344 derivation, 130, 146, 157 functional differential equations, 2, 14, 58 solution by method of differentiation, 35, 110 solution by splitting method, 46, 48 splitting principle, 128 functional separable solutions, xvi, 77, 91, 121, 123, 136, 138, 144, 150, 152, 161, 163, 168, 170, 173, 177, 206 direct construction, 193, 197 direct method of weak symmetry reductions, 260 equivalent differential constraints, 344, 345 examples of construction, 94, 111, 187 general form, 179 heat equations in anisotropic medium, 125 heat-type equations, 117 implicit form, 92, 125 direct method for construction, 127 generalization of traveling wave solutions, 125 method of differentiation, 110, 111 schematic of construction, 215 simple, 96 simplified method for construction, 93 special form, 96, 99, 105 structure, 91 wave-type equations, 117 functional separation of variables, x, xi, 91, 117 direct, 93 general explicit representation of solutions, 179 implicit representation of solutions, 211 indirect, 93 methods, 91–250 functional separation of variables and differential constraints, 344 functions, see function

G general compatibility method for two equations, 309 general form of functional separable solutions, 179

379

general functional separation of variables explicit representation of solutions, 179 implicit representation of solutions, 211 general non-Newtonian fluid model, 296 generalization of exact solutions, 8, 125, 208, 287 generalizations based on equivalent equations, 214 generalized Burgers equation, 126 generalized Burgers–Korteweg–de Vries equation, 251 generalized Guderley equation, 64 generalized porous medium equations with nonlinear source, 242 generalized separable solutions, xvi, 13, 16, 38, 50, 52, 66, 74, 94, 283, 343 Barenblatt–Zel’dovich dipole solution, 20 boundary layer equation, 42 delay PDEs, 71 differentiation method, 36 Guderley equation, 15 hydrodynamic type equation, 29 integro-differential equations, 80 method of invariant subspaces, 58 Monge–Amp`ere type equation, 52 most common structure, 14 nonhomogeneous Monge–Amp`ere equation, 18 PDEs in three or more independent variables, 26 porous medium equation, 19 schematic of construction, 48 simplified method for construction, 15 splitting method, 50 structure, 12 generalized separation of variables methods, xi, 1, 14, 26, 28, 41, 262, 344 generalized separation of variables and differential constraints, 343 generalized Sisko model, non-Newtonian fluid, 299 generalized splitting principle, 211, 212, 214, 215, 260, 345, 350 generalized traveling wave solutions, xvi, 96, 103, 131, 136, 140, 146, 166, 203 algorithm for construction, 98 examples of construction, 99 generalized traveling wave solutions of special form, 96 Ginzburg–Landau type equations, 178 Guderley equation, 15, 17 generalized, 64

380

I NDEX

H Harry Dym equation, 258, 259 general form of reductions, 259 heat equations, 13, 336 anisotropic, in three or more independent variables, 208 in three or more independent variables, 5 reducible to equations with quadratic or cubic nonlinearity, 97 with source, 78, 106, 332 with source, in anisotropic medium, 125 heat type equations, 9, 42, 94, 95, 117 Helmholtz equation, 171 higher-order differential constraints, 331, 334

I index, adiabatic, 15, 64 indirect functional separation of variables, 93 inner functions, 91, 96 invariant manifold, 320, 321, 335 invariant surface condition, xi, 351

K Klein–Gordon type equations nonlinear, 157, 200, 351 in anisotropic medium, 210 overview of solutions, 155 with n spatial variables, 173 with variable coefficients, 155, 156 Kolmogorov–Petrovskii–Piskunov equation, 304 Korteweg–de Vries equation generalized, nth-order, 103, 251, 256 potential, 6 Korteweg–de Vries type equations, 173

L linear determining equation, 335–337 linear equation, 4, 91, 121, 169, 171, 199, 227 determining invariant manifold, 335 diffusion, fractional, 84 heat, 1, 4, 13 linear equations of mathematical physics, 1, 13 linear fractional diffusion equation, 84 linear partial differential equations, 4, 13 linear subspace invariant under nonlinear operator, 58–68, 75–77 linear superposition principle, 1

M method Clarkson–Kruskal direct, xi, 251 reduction direct, see method of symmetry reductions, direct symmetry reductions classical, xi, 351, 352 direct, xi, 260, 273, 283, 343, 350 direct, modification, 266 nonclassical, xi, 350 tanh-function, 127, 317 Titov–Galaktionov, 58 weak symmetry reductions construction of exact solutions, 270, 280, 291 direct, 260, 273, 283 method of differential constraints, xi, 77, 129, 303 algorithm, 321 application, 322, 337 description by example, 317 for ordinary differential equations, 303 for partial differential equations, 317 general description, 319 vs. direct method of symmetry reductions, 350 vs. functional separation of variables, 344, 345 vs. generalized separation of variables, 343, 344 vs. other methods, 342 method of differentiation, 35, 110, see also differentiation method construction of functional separable solutions, 111 nonlinear functional equations, 118 method of invariant subspaces, 58, 71 application for PDEs with fractional derivatives, 84 finding linear subspaces, 67 method of symmetry reductions classical, xi, 351, 352 direct, x, xi, 251, 260, 273, 283, 343, 350 direct, modification, 266 nonclassical, xi, 350 method of undetermined coefficients, 15, 127 method of undetermined functions, 15 method of weak symmetry reductions construction of exact solutions, 270, 280, 291 direct, 260, 273, 283

I NDEX methods, see also method comparison of effectiveness, 93, 260, 345, 346 functional separation of variables, xi, 91, 344, 351 generalized separation of variables, xi, 1, 14, 26, 28, 343, 344 model Cattaneo–Vernotte, differential, 78 diffusion, 84 Newtonian fluid, classical, 265, 285 non-Newtonian fluid, 285 non-Newtonian fluid, general, 296 percolation (comb), 84 propagation of electric pulse along nerve fiber, 262 seepage of water through unsaturated soils, 146, 216 Sisko, 298 generalized, non-Newtonian fluid, 299 three-parameter, 299 two-parameter, special, 300 thermal conductivity, 78 three-parameter polynomial boundary layer equation, 297 modified stream function, 277 multiplicative separable solutions, xvi, 1, 5, 8, 22, 74, 79, 98, 181, 192

N Navier–Stokes equations, 27 generalized self-similar solutions, 47 solutions, x, 26, 27, 34, 253, 341 three-dimensional, 253, 339 unsteady, 34 Newtonian fluid, 99, 265, 276, 286, 291 non-Newtonian fluid, 285, 292 general model, 296 generalized Sisko model, 299 power-law, 285, 291 special two-parameter Sisko model, 300 with viscoelastic properties, 319 non-Newtonian fluid model, 285 nonclassical method of symmetry reductions, xi, 350, 351 noninvariant self-similar solution, 145, 238 noninvariant traveling wave solution, 149, 184, 194, 204 nonlinear equations, see equation(s) notations, xv nth-order autonomous ODE, 316 nth-order generalized Burgers–Korteweg–de Vries equation, 251, 256

381

nth-order generalized Korteweg–de Vries equation, 103 nth-order linear differential operator, 175– 177 nth-order nonlinear differential operator, 75, 76 nth-order ODE, 308 nth-order PDE, 25, 26, 41, 103, 174

O operator biharmonic, xv fractional differential, 84 fractional differentiation, 83 fractional integration, 82 fractional, inversion, 83 fractional, linearity, 83 gradient, 200, 339 Laplace, xv, 12, 27, 190, 200, 339 linear differential, 62, 65, 77, 171 in t, 80 nth-order, 175–177 linear fractional differential, 84 linear integral, 81 linear integro-differential, 80 nonlinear differential, 61–69 nth-order, 58, 75, 76 parametrically dependent on t, 62 second-order, 59–61, 80 pseudo-differential, 86 total differential, 322, 328, 335 outer function, 91, 94, 96, 179 overdetermined systems of ODEs and PDEs, 92, 179, 260 overview of exact solutions to nonlinear diffusion type PDEs, 129 overview of exact solutions to nonlinear Klein–Gordon type equations, 155

P partial differential equations, see equations PDE, see equation(s) percolation model, 84 plane boundary layer equation, 265, 287, 290 general model of non-Newtonian fluids, 296 non-Newtonian fluids, 285 solutions to determining system, 269 variable viscosity, 266 physical interpretation, 273 reduction to, 286

382

I NDEX

point transformation, 313, 317 Poisson equation, 171, 172 porous medium equation, 19 preface, ix problem Blasius, 274 Blasius type, for unevenly heated flat plate, 273 pseudo-differential equations, 85 nonlinear, 86

R reaction–convection–diffusion equations nonlinear, 179, 216 reduction to ODE, 179 with several spatial variables, 189 reaction–diffusion equations nonlinear, 129, 130, 231, 235 nonlinear, with variable coefficients, 129, 130, 179 with delay, 71, 77 with variable coefficients, 129 reaction–diffusion type equations nonlinear, 6, 129, 130, 179, 185, 189 reduction of nonlinear convection–diffusion type equations to ODEs, 191 reduction of nonlinear reaction–convection– diffusion equations to ODE, 179 reduction of nonlinear telegraph type equations to ODE, 201 reduction to plane boundary layer equation with variable viscosity, 266, 286 Riccati equation, 183, 193, 203, 284 Richards equation, 146, 216

S Schr¨odinger type equation nonlinear, general form, 177 nonlinear, with variable coefficients, 178 second-order differential constraints for ODEs, 311 for PDEs, 334 second-order equations of general form, 331 second-order evolution equations, 321, 331, 334 second-order hyperbolic equations, 328 seeking exact solutions using equivalent equations, 214, 247, 349 self-similar solution, ix, xvi, 111, 139, 154, 165, 188, 198, 233, 249, 277 generalized, xvi, 47, 98

noninvariant, 145, 238 special form, 156 separable solutions additive, see additive separable solutions functional, see functional separable solutions generalized, see generalized separable solutions multiplicative, see multiplicative separable solutions simple, see simple separable solutions separation of variables, 3 functional, see functional separation of variables generalized, see generalized separation of variables nontrivial, 10 simple cases, 3 simple separable solutions, 1, 3, 5, 6, 10, 12, 19 delay PDEs, 72, 74 special form, 8 Sisko model, 298 generalized, non-Newtonian fluid, 299 three-parameter, 299 two-parameter, special, 300 solution, see also solutions Barenblatt–Zel’dovich, dipole, 20 functional separable, see functional separable solutions self-similar, see self-similar solution Titov, 16 traveling wave, see traveling wave solution solution of functional differential equations by method of differentiation, 35 solution of functional differential equations by splitting method, 46 solutions additive separable, see additive separable solutions bilinear functional equation, 46, 47, 49 method of differentiation, 49 splitting principle, 46, 128 direct construction, 187, 197 direct procedure for construction, 180, 191, 201 functional separable, see functional separable solutions generalized separable, see generalized separable solutions multiplicative separable, see multiplicative separable solutions Navier–Stokes equations, 27, 34, 47, 341

I NDEX solutions (continued) nonlinear equations three independent variables, 5, 26, 29, 170, 208, 338 three or more independent variables, 5, 26, 170, 208 two independent variables, xvi, 15, 92, 179, 251, 254, 258 seeking using equivalent equations, 213– 215, 230, 247, 249 self-similar, see self-similar solution separable, see separable solutions simple separable, see simple separable solutions traveling wave, see traveling wave solutions using transformations of independent variables, 28 solutions of bilinear functional equations, 47, 50 solutions to determining system, 180, 191, 208, 278, 281, 283, 289, 290, 295, 296, 324 axisymmetric boundary layer, 268 plane boundary layer, 269 special two-parameter Sisko model, 300 splitting principle, 46–48, 127, 158, 227 generalized, 211, 212, 214, 215, 260, 345, 350 splitting procedure, 321, 324, 348 steady-state boundary layer equations, 260 stream function, 24, 27, 30, 39, 99, 260, 265, 266, 273, 286, 296 modified, 277 structure of functional separable solutions, 91 structure of generalized separable solutions, 12, 14 subspaces invariant under nonlinear differential operator, 58, 63, 67 systems determining, see determining system overdetermined ODEs and PDEs, 92, 179, 260 systems of nonlinear equations, 5, 337

T tanh-function method, 317 thermal conductivity model, 78 three-parameter polynomial model, boundary layer equation, 297 three-parameter Sisko model, 299

383

Titov solution, 16 Titov–Galaktionov method, 58, see also method of invariant subspaces total differential operator, 322, 328, 335 transformation, 212, 239 continuous, xi Crocco type, 287 dilation, 8 scaling, 9 translation, 14 transformations, 129, 265, 277, 286, 297, see also transformation nonlinear, 181, 192, 201, 212, 225 special type, 211 of independent variables to construct exact solutions, 28 of unknown function, 93 point, 313, 317 traveling wave solution, 92, 111, 113, 116, 126, 155, 158, 167 noninvariant, 149, 184, 194, 204 traveling wave solutions, ix, xvi, 92, 98, 121, 127, 304 generalized, see generalized traveling wave solutions implicit form, 125 two-parameter Sisko model, special, 300

U unnormalized Boussinesq equation, 44 unsteady axisymmetric boundary layer equations, 266, 276, 287 unsteady plane boundary layer equations, 265, 266, 285, 287, 296 using several differential constraints, 316, 337

V von Mises equation, 23

W wave equations anisotropic, in three or more independent variables, 208 nonlinear, in anisotropic medium, 210, 211 nonlinear, in inhomogeneous medium, 210 weak symmetries, 251, 348