Asymptotic Methods for Engineers 9781032725420, 9781032740584, 9781003467465

Citation preview

Asymptotic Methods for Engineers Asymptotic Methods for Engineers is based on the authors’ many years of practical experience in the application of asymptotic methods to solve engineering problems. This book is devoted to modern asymptotic methods (AM), which is widely used in engineering, applied sciences, physics, and applied mathematics. Avoiding complex formal calculations and justifications, the book’s main goal is to describe the main ideas and algorithms. Moreover, not only there is a presentation of the main AM, but there is also a focus on demonstrating their unity and inseparable connection with the methods of summation and asymptotic interpolation. The book will be useful for students and researchers from applied mathematics and physics and of interest to doctoral and graduate students, university and industry professors from various branches of engineering (mechanical, civil, electro-mechanical, etc.).

Asymptotic Methods for Engineers

Igor Andrianov Jan Awrejcewicz

Designed cover image: Fomenko Anatoly Timofeevich First edition published 2024 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Igor Andrianov and Jan Awrejcewicz 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 [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-72542-0 (hbk) ISBN: 978-1-032-74058-4 (pbk) ISBN: 978-1-003-46746-5 (ebk) DOI: 10.1201/9781003467465 Typeset in Nimbus Roman by KnowledgeWorks Global Ltd.

Contents Preface......................................................................................................................ix Notations and Abbreviations..................................................................................xi Used special functions and constants.............................................xii Authors...................................................................................................................xiii Introduction: From the principle of idealization to asymptotology .................. xv Chapter 1

ASYMPTOTIC APPROXIMATIONS AND SERIES .................... 1 1.1 1.2 1.3

Chapter 2

Asymptotic Series...................................................................1 Asymptotic Symbols and Operations on Asymptotic Expansions..............................................................................5 Choice of Asymptotic Integration Parameters........................ 8

REGULAR ASYMPTOTIC EXPANSIONS ................................ 13 2.1 2.2 2.3 2.4 2.5

Perturbation and Iteration Methods ......................................13 Eigenvalue Problems ............................................................ 18 Adjoint Equations Method....................................................27 Non-Power Asymptotics....................................................... 28 Elimination of Non-Uniformities of Asymptotic Expansions............................................................................29 2.6 Choice of Zero Approximation............................................. 35 2.7 Homotopy Perturbation Method ...........................................37 2.8 Small Delta Method..............................................................41 2.9 Large Delta Method .............................................................. 46 2.10 Lyapunov-Schmidt Method .................................................. 50 2.11 Boundary Shape Perturbation Method.................................. 52 2.12 Asymptotic and Real Errors.................................................. 54

Chapter 3

SINGULAR ASYMPTOTIC EXPANSIONS ............................... 56 3.1 3.2 3.3 3.4 3.5 3.6

Gol’denveizer-Vishik-Lyusternik Method ............................ 56 Newton’s Polygon and its Generalizations ...........................63 An Example of an Asymptotic Splitting of a Partial Differential Equation ............................................................ 70 Setting of Boundary Conditions ........................................... 72 Papkovich-Fadle Method (Method of Homogeneous Solutions).............................................................................. 74 Corner Boundary Layers....................................................... 76 v

vi

Contents

3.7 3.8 3.9 3.10

The Applying of Generalized Functions............................... 78 Multiple Scales Method........................................................81 Method of Matched Asymptotic Expansions .......................84 On Boundary Value Problems of Non-Classical Theories of Beams, Plates, and Shells .................................. 87 3.11 Kirchhoff’s and Bolotin’s Approximations in the Theory of Non-linear Beams Oscillations ............................ 88 3.12 Idealization Avenges Itself.................................................... 92 Chapter 4

DYNAMIC EDGE EFFECT METHOD ....................................... 93 4.1 4.2 4.3 4.4 4.5

Chapter 5

CONTINUALIZATION ..............................................................107 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Chapter 6

Linear Vibrations of the Beam.............................................. 93 Non-linear Vibrations of the Beam....................................... 96 Non-linear Vibrations of the Rectangular Plate.................... 99 Combination of Asymptotic and Variational Methods ....... 103 Non-linear Normal Modes of a Continuous Systems ......... 105

Discrete and Continuous Models of Mechanics .................107 Chain of Elastically Connected Masses.............................. 108 Classic Continuous Approximation.................................... 111 ”Splashes”........................................................................... 112 Envelope Continualization..................................................113 Refined Continuous Approximations for Natural Oscillations .........................................................................115 Forced Oscillations .............................................................117 Logistic Equations: Discretization and Continualization ...119

AVERAGING AND HOMOGENIZATION METHODS .......... 123 6.1 6.2 6.3 6.4 6.5 6.6

Averaging ........................................................................... 123 Freezing Method for Viscoelastic Problems....................... 126 WKB Method......................................................................127 Kuzmak-Whitham Method ................................................. 130 Differential Equations with Rapidly Varying Coefficients . 133 Differential Equations with Periodically Discontinuous Coefficients ......................................................................... 138 6.7 Periodically Perforated Media ............................................ 142 6.8 Waves in a Periodically Inhomogeneous Medium.............. 147 6.9 Higher Order Asymptotic Homogenization for Dynamical Problems...........................................................150 6.10 Hilbert Transform ............................................................... 153

vii

Contents

Chapter 7

SUMMATION OF ASYMPTOTIC SERIES .............................. 156 7.1 7.2 7.3

Chapter 8

APPLICATIONS OF PADE´ APPROXIMATIONS .................... 168 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Chapter 9

9.4 9.5 9.6 9.7

Method of Asymptotically Equivalent Functions ............... 186 Two-Point Pad´e Approximants ........................................... 188 Other Methods for Constructing Asymptotically Equivalent Functions .......................................................... 192 Method of Composite Equations ........................................194 Example: Schroedinger Equation ....................................... 195 Example: Asymptotically Equivalent Functions in the Theory of Composite Materials .......................................... 196 Example: Thomas-Fermi Equation..................................... 197

USING OF SPECIAL NONSMOOTH TRANSFORMATIONS .............................................................. 201 10.1 10.2 10.3 10.4 10.5

Chapter 11

Acceleration of the Convergence of Iterative Processes..... 168 Elimination of Nonuniformities..........................................170 Localized Solutions ............................................................171 Hermite-Pad´e Approximation and Bifurcation Problems... 172 Bounds of the Effective Characteristics of Composite Materials ............................................................................. 173 Method of Perturbation of the Type of Boundary Conditions...........................................................................173 Reduction of the Gibbs-Wilbraham Phenomenon .............. 181 And That’s Not All! ............................................................ 183

JOINING OF ASYMPTOTIC EXPANSIONS ........................... 186 9.1 9.2 9.3

Chapter 10

Power Series Analysis ........................................................ 156 Pad´e Approximants and Continued Fractions .................... 160 Borel Summation ................................................................ 166

Governing equations and zero-order homogenization........ 201 Structure attached sawtooth coordinate .............................. 203 Homogenization procedure................................................. 207 Higher-order asymptotics ................................................... 208 Physical meaning and technical details of nonsmooth transformations ................................................................... 210

MATHEMATICAL MODELS IN PURE AND APPLIED MATHEMATICS......................................................................... 214 11.1 11.2 11.3 11.4

Pure and Applied Mathematics - Definitions...................... 214 Poincar´e, Lyapunov, and Lord Rayleigh.............................215 Mathematical Model - Definition .......................................215 Simple Example..................................................................216

viii

Contents

11.5 Problem of Truncation ........................................................218 11.6 Highly Likely: Rigor of Pure Mathematics? ...................... 218 11.7 Don’t Go Beyond Asymptotic Restrictions ........................ 219 Chapter 12

CONCLUSION ........................................................................... 220

Chapter 13

RECOMMENDED LITERATURE.............................................223

References .............................................................................................................225 Index...................................................................................................................... 241

Preface How well is Nature simulated by the varied asymptotic models that imaginative scientists have invented? G. Birkhoff [58]

The importance of asymptotic methods in mechanics was emphasized by the D. Crighton: “Design of computational or experimental schemes without the guidance of asymptotic information is wasteful at best, dangerous at worst, because of possible failure to identify crucial (stiff) features of the process and their localization in coordinate and parameter space. Moreover, all experience suggests that asymptotic solutions are useful numerically far beyond their nominal range of validity, and can often be used directly, at least at a preliminary product design stage, for example, saving the need for accurate computation until the final design stage, where many variables have been restricted to narrow ranges” [94]. Let us define the features of the book, which can be considered as an elementary introduction to modern asymptotic methods. This book differs from the monograph [38, 143, 226, 228] by a significant expansion of the topics considered, and from the monograph [11, 12], in addition to adding new material, by a significant simplification of the presentation. Wherever possible, we tried to get rid of unnecessary complications and pseudo-generalizations. Our book differs from well-known remarkable books on perturbation methods [143, 226, 228, 232, 310] by the description of summation methods and the construction of asymptotically equivalent functions. It is focused on the solution of differential equations; therefore, it almost does not consider methods for the asymptotic calculation of integrals, since extensive literature is devoted to this issue [38, 75, 143, 226, 228]. The book can be used by all persons who have taken standard courses in higher mathematics and theoretical mechanics at a technical university (ideally, at least some special courses in mechanics of deformable solids and oscillations). We proceeded from the fact that “examples are more useful than rules” (a phrase attributed to Newton). As examples, we took problems from non-linear dynamics, the theory of plates and shells, fluid mechanics, and quantum mechanics. Philosophical questions of asymptotology are reflected in books and articles [10, 18, 118, 165, 178, 270]. Our book is intended, primarily, for engineers, mechanics and physicists who use or are going to use asymptotic methods in their practical activities. Therefore, when determining the level of rigor of presentation, we focused on the Rayleigh words [287]: “In the mathematical investigations I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of deficient rigour. But to this question there are two sides. For, however important it may be to maintain a uniformly

ix

x

Preface

high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a different order of ideas, the more severe procedure oft he pure mathematician may appear not more but less demonstrative. And further, in many cases of difficulty to insist upon the highest standard would mean the exclusion of the subject altogether in view of the space that would be required”. Andrianov would like to note that he was introduced to asymptotic methods in 1968 by his teacher Leonid Isaakovich Manevitch [10]. He is indebted to him for education and friendship. Many of the ideas presented below appeared as a result of our long-term cooperation with R.G. Barantsev, V.V. Danishevskyy, A.Yu. Evkin, E.G. Kholod, S.G. Koblik, G.A. Krizhevsky, A.I. Manevich, Yu.V. Mikhlin, B.V. Nerubaylo, V.I. Olevskyi, A.V. Pavlenko, V.N. Pilipchuk, A.D. Shamrovskii, G.A. Starushenko, S. Tokarzewski, H. Topol, A. Vakakis, W.T. van Horssen, D. Weichert. The authors express their deep gratitude to Professor, Academician of the Russian Academy of Sciences A.T. Fomenko for kind permission to use his painting “Morse functions and the theorem about the Euler characteristics” for the design of the book cover.

Notations and Abbreviations g F(x, y) A BAM O, o AEF BCCL x, y ξ,η F D ρ V, v U, u W, w FcL T Ω, ω L, l a, b Q, q M, m I PA ν r θ Φ R, R1 , R2 RBM RRM c G SCL ε σ SSS h t, τ TPPA E

- acceleration of gravity - Airy’s function - amplitude - Bolotin’s asymptotic method - asymptotic symbols - asymptotically equivalent function - body-centered cubic lattice - Cartesian coordinates (dimensional) - Cartesian coordinates (dimensionless) - cross-sectional area - cylindrical stiffness - density - displacement (circumferential of shell) - displacement (longitudinal of rod or shell) - displacement (normal beam, membrane, plate, shell) - face-centered lattice - force - frequency - length (dimensional, dimensionless) - lengths (membrane or plate sides) - load (dimensional, dimensionless) - mass - moment of inertia - Pad´e approximations - Poisson’s ratio - polar coordinate - polar coordinate - potential function - radii of curvature - Rayleigh-Bolotin method - Rayleigh-Ritz method - rigidity - shear modulus - simple cubic lattice - small parameter - stress - stress-strain state - thickness - time (dimensional, dimensionless) - two-point Pad´e approximations - Young’s modulus

xi

xii

Notations and Abbreviations

USED SPECIAL FUNCTIONS AND CONSTANTS B(..., ...) Bi K(...) ψ(...) δ (...) γ E1 (...), Ei(...) Γ(...) H(...) γ(..., ...) sn(..., ...), cn(..., ...) In (...) Kn (...) er f(...) ζ (x) Si(...) Hn (...)

- beta function ([1], ch. 6) - Bernoulli numbers ([1], ch. 23) - complete elliptic integral of the first kind ([1], ch. 16) - digamma function ([1], ch. 6) - Dirac delta function [320] - Euler’s constant ([1], ch. 6) - exponential integral ([1], ch. 5) - gamma function ([1], ch. 6) - Heaviside function [320] - incomplete gamma function ([1], ch. 6) - Jacobi elliptic sine and cosine ([1], ch. 16) - modified Bessel function of the first kind ([1], ch. 9) - modified Bessel function of the second kind ([1], ch. 9) - error function ([1], ch. 7) - Riemann zeta function ([1], ch. 23) - sine integral function ([1], ch. 5) - Struve function ([1], ch. 12)

Authors ORCID: 0000-0002-2762-8565 Website: andrianov.freeservers.com Scopus: ID 7004511057; h-index 24 Google Scholar: h-index 33; i10-index 127 WoS: ID G-8772-2016; h-index 20

Prof. Igor Andrianov: The academic history of Prof. Igor Andrianov consists of the following steps: 1971 - MSc, Mechanics, Dnepropetrovsk State University (DSU); 1975 - PhD, Physics and Mathematics, DSU; 1990 - DSc Thesis, Mechanics of Solids, Moscow Institute of Electronic Engineering; 1991 - Professor; 1996 -Soros Professor. He worked at DSU, Dnepropetrovsk Civil Engineering Institute, RWTH Aachen University. Co-author of 19 monographs published by various leading publishers. Editor of three books. Author or co-author of more than 300 articles in leading international journals. Speaker, invited speaker at numerous international conferences, organizer a large number of mini-symposiums. Recipient of scientific grants from the USA, Germany, the Netherlands, England. He supervised 25 PhD students. Scientific interests: Asymptotology, Nonlinear Dynamics, Composite Materials, Theory of Thin-Walled Structures, Mechanics of Solids, Science popularization.

xiii

xiv

Authors

ORCID: 0000-0003-0387-921X Website: abm.p.lodz.pl/en/jan-awrejcewicz Scopus: ID 7007114678; h-index 38 Google Scholar: h-index 51; i10-index 377 WoS: ID DXG-6904-2022; h-index 32 Prof. Jan Awrejcewicz: The academic history of Prof. Jan Awrejcewicz consists of the following steps: 1977 - MSc, Mechanics, Lodz University of Technology (LUT); 1978 - MSc, Philosophy, University of Lodz; 1981 - PhD, Mechanics, LUT; 1990 - DSc Thesis, Mechanics, LUT; 1994 - Title Professor, President of Poland; 1997 - Ordinary Professor, Ministry of Education, Poland. His subject areas include dynamical systems, mechanics, biomechanics, mechatronics, control, applied mathematics. He published: journal papers (687), conference papers (363), monographs (English - 28, Polish - 26, Russian - 1, Slovenian - 1), text books (2), chapters in books (332), edited books (36), edited conference proceedings (23), edited journal special issues (44), other (10), prefaces and guest editorials (47), short comm. (518), tech. reports (14), other papers and reports (12). He is co-holder of 4 patents and served as a principal investigator of 25 research grants. His scientific work was awarded by the following honorary positions: real member of the Polish Academy of Sciences (PAS), (2021–), Member of the European Academy of Sciences and Arts (2021–), Member of the International Advisory Board of the Institute of Thermomechanics of the Czech Academy of Sciences (2019– 2023); Member of the Scientific Council of the Institute of Fundamental Technological Research of PAS (2019–2022); Corresponding Member of PAS (2016–2020); Member of the Central Commission of Scientific Degrees and Titles, Poland (2013– 2016, 2017–2020); Vice President of the Polish Society of Biomechanics (2012– 2016); Member of the Academy of Engineering, Poland (2005–); Chairperson of the Technical Committee for Nonlinear Oscillations of the Interational Federation of Mechanisms and Machines (FToMM), (1995, 2000–2005, 2006–2009); Member of the Committee of Mechanics of PAS (1993–). He delivered 69 international and 18 national invited/keynote addresses and supervised 33 PhD students (including two from Russia, three from China, one from Ukraine, one from Germany).

Introduction From the principle of idealization to asymptotology My friend Walter Meyerstein has pointed out, that was the Greek notion of beauty; beauty was symmetry was mathematics! Plato connects all of these things. So then that notion of beauty is sort of like the notion of simplicity, the absence of complexity, a simple structure G.I. Chaitin [80]

The key concept of Western science is the principle of idealization. Any real system or phenomenon consists of a large (often infinite) number of subsystems or less significant phenomena. To describe a real system or phenomenon, one has to build idealized models, neglecting details that are insignificant for further purposes. The principle of idealization was apparently first explicitly formulated by Galileo. This is how he described the essence of the science that we now call mechanics: “White or red, bitter or sweet, sounding or silent, sweet-smelling or evil-smelling are names for certain effects upon the sense-organs; they can no more be ascribed to the external objects than can the tickling or the pain caused sometimes by touching these objects. If ears, tongues, and noses are removed, I am of the opinion that shape, quantity, and motion would remain, but there would be an end of smells, tastes, and sounds, which abstractedly from the living creature, I take to be mere words” [160]. And here is how Galileo was able to investigate the motion of bodies: “Hence, in order to handle this matter in a scientific way, it is necessary to cut loose from these difficulties [air resistance, friction, etc.] and having discovered and demonstrated the theorems, in the case of no resistance, to use them and apply them with such limitations as experience will teach” [160]. The following speaks about the nontriviality of such a step. Science began when “The ancient Greek philosophers showed an amazing, completely original ability to see the world divided into objects, and objects divided into qualities. It is difficult for us to appreciate the non-triviality of this vision now that our upbringing in our civilization has largely taught us this. ... Theoretical thinking originated, of course, not from those who were able to see connections (religio) in the world of things and interpenetration in the world of concepts, but from those who were able to disembowel, dissect the world in order to find out what was inside it, and explore current concepts as unchanging things. The brilliant Greeks invented length without width, space without time, atoms without qualities” [324]. And, of course, Western science is closely connected with the dominant religion in Europe, since “In accordance with the monotheistic principle, Christianity (regardless of the desire of the educated person) lays in the consciousness the principle of the hierarchical construction of nature, which requires the continuous completion of

xv

xvi

Introduction

a system of interconnected parts of knowledge around a certain semantic center” [324]. From here, by the way, it is clear that the principle of idealization is not one of the innate concepts of a person, it needs to be taught. In our opinion, both in secondary and in higher schools, sufficient attention is not paid to really basic things, to which the principle of idealization belongs. “According to Andronov, Mandelstam believed that questions of idealization should occupy a fundamental place in any teaching of physics, both at school and at the university. Already a schoolboy should be aware that in any physical theory we work with ideal models of real things and processes” [265]. Once again, let us be surprised at the genius of Galileo, because “The mathematical method of abstraction is indeed a step away from reality, but paradoxically, it leads back to reality with greater power than if all the factors actually present are taken into account at once” [160]. However, the principle of idealization is far from omnipotent. Let’s reread Galileo: “... apply with the limitations that experience tells us”, “discard ... air resistance, friction, etc.” And if we want to take into account these perturbing factors? The works of many great scientists shown how an idealized solution can be refined, considering it as the first approximation of some iterative process. The term “asymptotics” appeared. Recall that in the school mathematics course we met with “asymptotes” - straight lines approaching infinitely close to some curves when the independent variable x tends to infinity. In other words, starting from some values of x, the curves can be replaced by asymptotes, and such an approximation will be the more accurate, the larger x. Similarly, asymptotic formulas describe phenomenon the more accurately, the smaller (or larger) some parameters become. The development of asymptotic mathematics led to the understanding that “Asymptotic description is not only a convenient tool in the mathematical analysis of nature, it has a more fundamental significance” [117]. As a result, Kruskal suggested talking about a new science - asymptotology [165]. It is not easy to give a definition of asymptotology, to outline its scope and limits of applicability. Barantsev suggested using the accuracy-locality-simplicity triad to describe the asymptotology [15, 18]. In other words, the antagonistic conflict “precision-simplicity” is resolved through “locality” - simple asymptotic models are correct only in a certain narrow range of parameters. Asymptotic methodology, in our opinion, is based on the following assumptions: 1. Every system has a small parameter, preferably with a physical meaning. The solution can be presented as a series expansion in powers of this parameter. 2. Every reasonable method or theory in physics, mechanics, or engineering has asymptotic nature. It is often hard to discover, but this has to be done to understand the essence of the problem and improve the model. 3. To evaluate the accuracy of an approximate model, we need to find its place in the asymptotics’ hierarchy. 4. The principle of asymptotic complementarity holds: if a system has meaningful asymptotics when some parameter tends to zero, then there exists meaningful asymptotics when this parameter tends to infinity as well.

Introduction

xvii

5. In practice, an asymptotic solution usually has a wider scope of applicability than is determined by a rigorous mathematical analysis. Description of the technical methods of asymptotology is the subject of this book. Here is a description of the sections of the book with brief comments. Chapter 1 contains a brief description of the basic concepts of asymptotic methods, as well as examples of choosing or introducing to the system asymptotic integration parameters. We recommend that the reader be sure to read this section. Here we presented the used terminology and notation, and also described the permissible (and, more importantly, unpermissible!) actions with asymptotic expansions. We also justified to use for analysis of obtained truncated series summation methods. The choice of the parameter of the asymptotic expansion and the asymptotic ansatz are the most crucial and creative moment in the application of asymptotic methods. All available information should be used here, starting from the analysis of known solutions in particular cases. Using of the method of successive approximations and numerical experiments can suggest a lot. Non-dimensionalization of the system make it possible to identify dimensionless combinations of parameters characteristic for a given problem. A hybrid strategy is possible: start with a few numerical explorations, to identify small or large parameters and use asymptotic expansions to obtain more general insight. The traditional terms “small” or “large” parameters (which are used in our book also) may be misleading. For example, expansion in a small parameter after using summation methods may yield a solution for non-small values of it. Unobvious choices of an expansion parameter can greatly extend the range and usefulness of perturbation series. Engineering intuition and experience allow to construct many successful approximate theories. As a rule, they are of an asymptotic nature. Detect the basic asymptotic parameter and choice of suitable asymptotic technique makes it possible to determine the range of applicability of such approximations and refine them. Many examples of this feature can be found in our book. An important non-trivial step in the application of asymptotic ideology is the choice of a specifc technique. As a rule, the zero approximations of various asymptotic methods coincide, but the construction of subsequent approximations can differ greatly. Therefore, before constructing an approximate solution, it is useful to analyze the arsenal of available asymptotic techniques. Chapter 2 and Chapter 3 are devoted to regular and singular asymptotic expansions, respectively. Before using any asymptotic method, it is useful to guess the zeroth approximation (idealized model) and estimate what effect the discarded terms of the equations (unaccounted factors) might have. For example, you assume that a small changing of analyzed system will lead to small consequences. Mathematically, such problems are described by ODEs or PDEs, in which small parameters are the coefficients standing by the lower derivatives. In this case, Chapter 2 is at your service. This chapter contains recommendations for choosing the zero approximations, estimating the error of asymptotic solutions, and it includes examples of non-trivial choice of parameters of the asymptotic expansion. Eigenvalue problems containing

xviii

Introduction

small parameters are analyzed in detail. Lyapunov-Schmidt method for bifurcation problems is also described. Problems in which small disturbances lead to large, but localized changes in time or space, are studied in Chapter 3. Mathematically they deal with ODEs or PDEs having small parameter near highest derivatives. Typical for this equations localized solutions are boundary layers, which also named as edge effects, skin effects, internal or corner boundary layers. Such problems are characterized by dividing the spatial or temporal region under study into zones with fast and slow varying of the solution. To determine these zones (create an “asymptotic portrait” of the system) and obtain variation index of solutions, Newton’s polygon is employed. In Chapter 3, signifcant attention is paid to the issues of setting boundary conditions, that ensure the correctness of truncated boundary value problems. Technically, the solving of the singular perturbated problems can be based on methods of multiple scale or matched asymptotic expansions. In the frst case, instead of the original variable, a sequence of scaled ones is introduced, i.e. “stretched” in accordance with the powers of the original small parameter (variables). Next, the solution is represented as a function depending on these new variables, which allows us to formally regularize the problem (in the new PDEs, the small parameter is not the coefficient at the highest derivative). In the second case, solutions are constructed separately in the areas of their slow and rapid change, followed by matching according to certain rules. A detailed description of the technique of singular perturbations is carried out using the example of the static deformation of a circular cylindrical shell. An asymptotic portrait of this system is compiled, all possible truncated equations are obtained, as well as the correct boundary value problems for them are formulated. Chapter 4 is dedicated to the dynamical edge effect method (DEEM), which uses the inverse of the vibration frequency as a small parameter. It allows analytically investigate eigenvalue problems described by ODEs or PDEs with complicated boundary conditions. Unlike the problems described in Chapter 3, these solutions in inner and edge areas change rapidly and with the same rate, that is why we described DEEM in the separate chapter. DEEM can be used to a wide range of problems, linear and non-linear, for thin-walled structures of various shapes and structures. Chapter 5 is devoted to the relationship between discrete and continuous models in engineering practice. Continuous mathematics has accumulated a huge arsenal of tools for engineer. The situation began to change with the advent of modern computers, and now discrete approximations are one of the main engineering methods. However, continuous models have not lost their importance, since they very convenient for qualitative analysis. They advantage also is in the possibility of using the powerful arsenal of continuous mathematics. At the same time, in applying the continuous models to describe a discrete structure (or vice versa), one must bear in mind the natural limitations of these approximations in order to avoid various “paradoxes” and artifacts. Some of these pitfalls are described in Chapter 5. The connection between discrete and continuous approximations is provided by the discretization and continualization procedures. In the first case, local (differential) operators are replaced by non-local, difference ones. In the second case, nonlocal (discrete,

Introduction

xix

pseudodifferential, integral) operators are replaced by the local (differential) ones. Discretization and continualization often lead to a dramatic change of the mathematical model. For example, discretization of a deterministic continuous dynamical system can lead to a discrete system with chaotic behavior. The opposite is also possible - continualization can eliminate the chaotic behavior inherent to a discrete system. “Naive” continualization is based on the Taylor series expansion of pseudodifferential or integral nonlocal operators. It is more effective to use one- and two-point Pad´e approximants for this purpose, as shown in Chapter 5. Chapter 6 presents averaging of nonlinear ODEs and homogenization of PDEs with rapidly varying or periodically discontinuous coefficients. We are talking here about the separation of fast and slow periodic components of the solution according to time or spatial variables. Multiple scales method or Hilbert transform can be used for these aims. Averaging is one of the main and natural tools for analyzing nonlinear dynamic problems, described by ODEs. Homogenization approach for PDEs allows to study periodically inhomogeneous structures (eg, grid stiffened, lattice, corrugated, perforated, etc., plates and shells) and media (eg, composite materials). This section also describes the WKB and Kuzmak-Whithem methods. The first method can be considered as a variant of the averaging while the second is based on the multiple scale approach. The features of described approaches are clarified based on examples from non-linear dynamics and mechanics of solids. Chapters 7–9 cover summation and interpolation methods. As the technique of asymptotic integration is well developed and widely used, elimination of the locality of expansion, evaluation of the convergence domain, construction of uniformly suitable solutions are very important. The solution to these problems is usually based on the construction of asymptotically equivalent functions (AEFs). AEF, having the same asymptotic behavior as the original approximation, describes the desired solution in a wider range of values of the expansion parameter used. Ideally, a formula can be obtained that is suitable for all values of the parameter. There exist many approaches to construction of AEF, like the method of analytic continuation based on the Domb-Sykes diagram, Euler and Shanks transformations, various summation and interpolation methods. Among the latter, the leading place belongs to one- and multipoint Pad´e approximants and continued fractions. Chapters 7-9 presents various methods for constructing AEFs, as well as numerous specific examples showing their effectiveness, are given. Chapter 10 is devoted to the method of nonsmooth transformations (NSTs). The main objective of NSTs is to simplify solving of problems with nonsmooth or discontinuous elements. Using of NSTs allows to reduce the problems to PDEs or ODEs with smooth coefficients and then asymptotic approaches can be employed. In Chapter 11, the concepts of the mathematical models in pure and applied mathematics are analyzed. The role of asymptotic ideology as basis for mathematical modeling is discussed. The main conclusion can be formulated as follows. Any generalization or refinement of an idealized model should be approached with caution. One cannot simply add some terms to the equations describing such a model without comparison of their orders with the original asymptotic estimations.

1 ASYMPTOTIC APPROXIMATIONS AND SERIES 1.1

ASYMPTOTIC SERIES

As Dingle notes [98], the theory of asymptotic series, dormant for decades, has made great strides in recent years. This is due to the understanding of the fact that the successful application of asymptotic methods is inextricably connected with the use of a certain method of their summation. There is nothing surprising in this: when writing out any series, you need to be aware of how to calculate it. Very rarely does the naive procedure of adding successive terms succeed. When calculating even convergent series, one often has to resort to various tricks, to say nothing of divergent ones [71]! To clarify the situation, we write out a rather general form of an asymptotic series typical for problems in physics and mechanics [70, 71]:  n ∞ ε (1.1) ∑ Mn ε0 Γ(n + a), n=1 where a is usually an integer; Γ(...) is the gamma function [1]. The quantity ε0 is often called a singulant, Mn is called a modifying factor. The sequence Mn tends to a constant as n → ∞ and gives information about the slowly changing part of the function; the constant ε0 is related to the first singular point of the series (1.1). Let us give the definition: a power series is an asymptotic one of the function f (ε) if for a fixed N and sufficiently small ε > 0 the inequality N
j f (ε) − ∑ a j ε ∼ O ε N+1 , j=0
where the symbol O ε N+1 means “of order ε N+1 ” (see 1.2). In other words, it is about the limit ε → 0, N = N0 . Series (1.1) can diverge at ε = 6 0, but its first terms at ε 0 is the constant, ε is the “small parameter” of the asymptotic expansion. Thus, terminating the asymptotic series at the smallest term, we introduce an exponentially small error. At the same time, taking into account such exponentially small terms is often important from a DOI: 10.1201/9781003467465-1

1

2

Asymptotic Methods for Engineers

computational point of view, since it increases the actual accuracy of the asymptotic solution [231, 70, 54, 55]. To illustrate the point, we use the Stieltjes function [71], Z∞

S(ε) = 0

exp(−t) dt. 1 + εt

(1.2)

Using equality 1 = 1 + εt

N

∑ (−εt) j +

j=0

(−εt)N+1 , 1 + εt

(1.3)

after substituting expression (1.3) into integral (1.2), we have N

S(ε) =

∑ (−ε)

j=0

where

j

Z∞

t j exp(−t)dt + EN (ε),

(1.4)

0

Z∞

EN (ε) = 0

exp(−t)(−εt)N+1 dt. 1 + εt

(1.5)

Calculating the integrals in (1.4) by integrating by parts, we find N

S(ε) =

∑ (−1) j j!ε j + EN (ε).

j=0

If we let N tend to infinity, we get a divergent series. This is understandable: the integrand has a simple pole at the point t = −1/ε, so expansion (1.3) is valid only for |t| < 1/ε. Using the obtained results in the entire range 0 ≤ t < ∞ is illegal. Let us estimate the order of the divergent part by representing the function S(ε) in the form 1/ε Z

S(ε) = S1 (ε) + S2 (ε) = 0

exp(−t) dt + 1 + εt

Z∞

exp(−t) dt. 1 + εt

1/ε

Since 1/(1 + εt) ≤ 1/2 for t > 1/ε, we get the estimation S2 (ε) < 0.5 exp(−1/ε ). Thus, there is an exponential decrease in the error with decreasing ε, a situation typical for asymptotic series. Let us estimate the optimal number of terms in the series for S(ε). To do this, we find when the term t N+1 exp(−t) in formula (1.4) is minimal. It is easy to obtain the estimation t = 1/(N + 1). For t ≥ 1/ε we have divergence, so the estimation is as follows: N = [ 1/ε ], where [...] means the integer part of the number. The asymptotic series with optimal number of terms is called superasymptotic series [54]. Hyperasymptotics [55] means overcoming the accuracy barrier that arises

3

ASYMPTOTIC APPROXIMATIONS AND SERIES

when using an asymptotic series. Indeed, after the optimal termination of the original asymptotic series, some other ideas are needed to improve the accuracy of the solution. As a rule, we are talking about some method of summing a divergent series (these issues are considered in more detail in this book in Chapters 7–9). It is possible, for example, to rearrange the segment of the series 2N

S(ε) ≈

∑ (−1) j j!ε j

(1.6)

j=0

in the Pad´e approximant, i.e. fractional-rational function of the form N

1 + ∑ α jε j S(ε) ≈

j=1 N

,

(1.7)

1 + ∑ βi ε i i=1

where the constants α j , βi are selected from the condition that the first 2N + 1 terms of the Maclaurin expansion of function (1.7) coincided with the coefficients of the series (1.6). It is shown [131], that the sequence of Pad´e approximants (1.7) for the Stieltjes function and the error in determining S(ε) decreases in propor pconverges,  tion to exp −4 N/ε . Along with asymptotic series, there are also antiasymptotic series. For example, Laurent series (−1) j exp(−1/z) = ∑ j j=1 j!z is convergent, but antiasymptotic: the terns of the expansion get larger and larger as z → 0 [93]. The definition of an asymptotic series points the way to a numerical verification of the asymptoticity of expansions [67]. Let, for example, there is reason to assume that the solution Ua (ε) is an asymptotic of the exact solution Ue (ε) , so E = Ue (ε) −Ua (ε) = Kε α . Ue can be a numerical solution. To determine α, graphs of the dependence of ln E on ln ε are plotted for various values of ε. The corresponding dependencies should be close to linear, and the constant α can be determined using the least squares method. The choice of ε is quite difficult: for large ε it is difficult to notice the asymptotic nature of the solution, for small ones it is problematic to obtain a numerical solution; however, as a rule, this difficulty can be overcome. As an example, consider the function Z∞ −t e dt I(ε) = εeε t ε

4

Asymptotic Methods for Engineers

Figure 1.1 Verification of the asymptotic nature of partial sums (1.8).

for large values of ε. Series (−1)n n! εn n=0 ∞

I(ε) =

∑

diverges for all values ε, but partial sums (−1)n n! ∑ εn n=0 M

IM (ε) =

(1.8)

are asymptotically exact in order O(ε −M ) with error O(ε −M−1 ) at ε → ∞. Fig. 1.1 shows the dependence of lg EM (ε) on lg ε, where EM (ε) = Inum (ε) − IM (ε), and curves from top to bottom correspond to values from M = 1 to M = 5.

Table 1.1 Slope lg EM (ε) as a function of lg ε, determined by the least squares method. EM (ε) 1 2 3 4 5

ε ∈ [5, 50] -1.861 -2.823 -3.789 -4.758 -5.729

ε ∈ [50, 200] -1.972 -2.963 -3.954 -4.945 -5.937

ε ∈ [200, 500] -1.991 -2.988 -3.985 -4.981 -5.999

Exact slope -2.0 -3.0 -4.0 -5.0 -6.0

It can be seen that the slopes of the curves differ; however, given in Table 1.1, the results of applying the method of least squares show quite satisfactory accuracy of the method. We also mention the method for determining asymptotic expansions in terms of known values of a function at several points [35, 235]. Let there be a numerical

5

ASYMPTOTIC APPROXIMATIONS AND SERIES

solution for some values of the parameter ε: f (ε1 ), f (ε2 ), f (ε3 ). If it is known a priori that function has asymptotic expansion and its form is known (for example, the expansion occurs in integer powers of ε), then we can write the system of equations 3

f (ε) = ∑ ε i ai ,

j = 1, 2, 3,

i=1

and determine the coefficients ai . This technique can be applied in the following case. It is often difficult to obtain a numerical solution for small values of the parameter ε, but it is quite simple for values of ε of the order of 1. Let us know from a priori considerations that the asymptotics for ε → 0 exist and have a power character; however, it is impossibly difficult to determine it analytically. Then the asymptotic extrapolation of the numerical solution described above can be applied.

1.2

ASYMPTOTIC SYMBOLS AND OPERATIONS ON ASYMPTOTIC EXPANSIONS

Let us introduce the main symbols and concepts of asymptotic analysis, considering the function f (x) at x → x0 . In the asymptotic method, the behavior of the function f (x) at the vicinity of the point x = x0 is of interest. The aim is to find another, simpler function ϕ(x) that describes f (x) at x → x0 . Quantitative comparisons are based on the notion of the order of a variable. Let us introduce the main order symbols and gauge functions. For describing the rate at which some function tends to the limit value, one can use a set of gauge functions. These are functions that are so familiar that their limiting behavior can be regarded as known intuitivly. The simplest and the most useful of these are xn , n = ..., −2, −1, 0, 1, 2, ... Instead of integer numbers n can be some real numbers. Examples of others gauge functions can be found in our book, as well as in other monographs devoted to asymptotic methods. When choosing gauge functions, be guided by the essence of your problem. We say that f (x) is a quantity of order ϕ(x) at x → x0 , writing f (x) = O(ϕ(x)) at x → x0 , if there exists a number A such that in some neighborhood ∆ of the point x0 the following inequality holds: | f (x) | ≤ A | ϕ(x) |. In addition, we say that f (x) is a quantity of order less than ϕ(x) at x → x0 , writing f (x) = o(ϕ(x))

at

x → x0 ,

if for any ε > 0 there exists a neighborhood ∆ of the point x0 where | f (x) | ≤ ε | ϕ(x) |.

6

Asymptotic Methods for Engineers

In the first case, the relation | f (x) |/| ϕ(x) | is bounded in ∆, while in the second it tends to zero at x → x0 . For example, sin x = O(1) when x → ∞; ln x = o(xα ) for any α > 0 at x → ∞. The symbols O(...) and o(...) are often called Edmund Landau symbols. They tell us how fast a function grows or declines. Sometimes it is advisable to use additional symbols that describe order of function. Namely, if f (x) = O(ϕ(x)), but f (x) 6= o(ϕ(x)) at x → x0 , then the notation ˜ f (x) = O(ϕ(x)) at

x → x0

can be used. ˜ The symbol O(ϕ(x)) is called the exact order symbol (note that the symbol Oe(ϕ(x)) is sometimes used). If f (x) = O(ϕ(x)), ϕ(x) = O( f (x)) at x → x0 , then we say that f (x) is asymptotically equal to ϕ(x) at x → x0 , writing f (x)  (x) at

x → x0 .

(Note that the symbol ≈ is sometimes used). The asymptotic equality implies the existence of such numbers a > 0 and A > 0 that in some neighborhood of the point x0 the inequality a | ϕ(x) | ≤ | f (x) | ≤ A | ϕ(x) | takes place. (The symbols O˜ and  can be expressed through O, o and are used only for brevity). It is useful to distinguish the following stages of asymptotic approximation. First, upper (or lower) estimates of the type f (x) = O(ϕ(x)) are constructed. Usually such an estimate turns out to be overestimated, i.e. actually f (x) = o(ϕ(x)). In ˜ 0 (x)) is found. Then the course of its improvement, the exact order f (x) = O(ϕ the asymptotic equality f (x) ∼ a0 ϕ0 (x) is reached. Having completed such a cycle, one can investigate the remainder in the same way, obtain the asymptotic equality f (x) − a0 ϕ0 (x) ∼ a1 ϕ1 (x) and move on. The sequence { ϕn (x)}, n = 0, 1, ... at x → x0 is called asymptotic if ϕn+1 (x) = o(ϕn (x)). For example, the sequence { xn } is asymptotic at x → 0. ∞

A series ∑ an ϕn (x) with constant coefficients is called asymptotic if { ϕn (x)} is n=0

an asymptotic sequence. We say that f (x) has an asymptotic expansion in the sequence { ϕn (x)}, writing N

f (x) ∼

∑ an ϕn (x),

N = 0, 1, 2, ...,

(1.9)

n=0

if

m

f (x) =

∑ an ϕn (x) + o(ϕm (x)),

m = 0, 1, 2, ..., N.

(1.10)

n=0

We study the uniqueness of asymptotic expansions. Let the function f (x) expand ∞

at x → x0 into a series in the asymptotic sequence { ϕn (x)}, f (x) ∼ ∑ an ϕn (x). Then n=0

7

ASYMPTOTIC APPROXIMATIONS AND SERIES

the expansion coefficients an are uniquely determined by the formula " # n−1

an = lim

x→x0

f (x) − ∑ ak ϕk (x) ϕn−1 (x). k=0

The same function f (x) can be expanded in various asymptotic sequences, for example, ∞ 1 ∼ ∑ xn 1 − x n=0

∞ 1 ∼ ∑ (1 + x)x2n 1 − x n=0

at x → 0;

at

x → 0.

On the other hand, the same asymptotic expansion can correspond to several functions, for example: ∞ 1 1 + e−1/x ∼ ∼ ∑ xn 1−x 1−x n=0

at x → 0.

In other words, an asymptotic series represents not one, but a whole class of asymptotically equivalent functions. This property can often be successfully used (see Section 9). Asymptotic expansions of the functions f (x) and g(x) at x → x0 in the sequence { ϕn (x)}, ∞

f (x) ∼

∞

∑ an ϕn (x),

g(x) ∼

n=0

∑ bn ϕn (x),

n=0

one can add and multiply by constants: ∞

α f (x) + β g(x) ∼

∑ (αan + β bn )ϕn (x).

n=0

where α and β are the constants. Generally speaking, it is impossible to multiply asymptotic series, since the products { ϕn (x) · ϕm (x)} (m, n = 0, 1, . . . ) cannot always be ordered into an asymptotic sequence. However, if this can be done, then term-by-term multiplication is possible. Power asymptotic series can be divided if in the denominator b0 6= 0. The logarithm of the asymptotics does not cause difficulties, and exponentiating it is necessary to take special care of √ the correct estimate of the remainder. Consider, for example, the function f (x) = ( x ln x + 2x)ex , for which the estimation f (x) = [2x + o(x)]ex

at x → ∞

(1.11)

takes place. Let g(x) = ln[ f (x)], then, in accordance with equality (1.11), g(x) = x + ln[2x + o(x)] = x + ln x + ln 2 + o(1) ∼ x + o(x) at x → ∞. Exponentiating the expansion for g(x), we find f (x) ∼ ex at x → ∞. In this case, the multiplier 2x is lost in the main term. The point is that the potentiation does

8

Asymptotic Methods for Engineers

not take into account the terms ln x and ln 2, which affect the leading term of the asymptotics f (x), and only the quantities o(1) do not change the coefficient, because exp { o(1)} ∼ 1. Power asymptotic expansion ∞

f (x) ∼

∑ an x−n

at x → ∞,

n=2

can be integrated term by term. It is impossible to differentiate asymptotic expansions in the general case. For example, the function   f (x) = e−1/x sin e−1/x has a degenerate power expansion f (x) ∼ 0 · 1 + 0 · x + 0 · x2 + ..., at the same time, the derivative of the function f (x) does not admit a power expansion. If the derivative f 0 (x), which is continuous as x ≥ d > 0, has, like the function f (x), a power asymptotic expansion at x → ∞, then it is obtained by term-by-term differentiation of the expansion of the function f (x). We emphasize that the vast majority of errors in the use of asymptotic methods are associated with an incorrect change in the order of passages to the limit and operation on series. Be carefull! We give nontrivial example. Let the Bubnov-Galerkin method be used to solve the some problem that contains a small parameter ε. How many terms of the expansion N should be left in the solution? In the general case lim lim (...) 6= lim lim (...).

N→∞ ε→0

ε→0 N→∞

The optimal choice of N is N ∼ ε −1 . The reader can get additional information about asymptotic expansions and operations on them from book on asymptotic methods, see, for example, [75, 310, 228, 226, 143].

1.3

CHOICE OF ASYMPTOTIC INTEGRATION PARAMETERS When a nondimensional physical parameter ε is small, a perturbation series in ε is an obvious strategy. However, unobvious choices of an expansion parameter can greatly extend the range and usefulness of perturbation series. J. Boyd [72]

As Vishik and Lyusternik noted [319], “There are a large number of problems “for perturbation” when, together with an “unperturbed” problem, one is investigated in a certain sense close to it. The most natural thing is to introduce the numerical

9

ASYMPTOTIC APPROXIMATIONS AND SERIES

parameter ε and consider a family of problems that depend continuously in a certain sense on ε”. At the same time, nothing is said about the “smallness” of the parameter ε. This “smallness” itself is often taken too literally. The most nontrivial issue of asymptotic analysis is the choice of the “small” or “large” parameters. The main ways to search for the natural parameters of asymptotic integration are nondimensionalization and dimensional analysis [161, 226]. Dimensional analysis is a method for reducing complex physical problems to their simplest forms prior to quantitative analysis. Dimensional analysis reduces a number of problem’s parameters to the minimum and thus suggests the most economical scaling laws. Pi theorem states: if a dependent variable F is completely determined by the values of a set of n independent parameters, of which a k parameters form a complete, dimensionally independent subset, then a suitable dimensionless F is completely determined by n − k dimensionless similarity parameters. In other words, the number of independent parameters may be reduced by the number k [161]. Well-chosen dimensionless combinations of parameters, such as Mach and Reynolds numbers in gas and hydrodynamics, Batdorf parameter in the theory of shells, Bergman contrast parameter in the theory of composites, often serve as asymptotic integration parameters. However, sometimes it is more advantageous to use an initial approximation that is far from obvious and perhaps even strange at first glance. To illustrate the above, consider a simple example: the algebraic equation [47] x5 + x = 1.

(1.12)

We look for the real root of this equation, the exact value of which can be determined numerically: x = 0.75487767 . . . The “small” parameter does not appear explicitly in equation (1.12). Let us consider various possibilities of introducing the parameter ε into equation (1.12). 1. Let us introduce a parameter ε at the non-linear term of equation (1.12) εx5 + x = 1.

(1.13)

x = a0 + a1 ε + a2 ε 2 + ...,

(1.14)

Representing x as substituting the expansion (1.14) into equation (1.13) and equating the terms ε at the same powers, we obtain a0 = 1,

a1 = -1,

a2 = 5,

a3 = −35,

a4 = 285,

a5 = −2530,

a6 = 23751 .

For an one obtains: an = [(−1)n (5n)!]/[n!(4n + 1)!], 5

and the convergence radius R of series (1.14), R = 44 /5 = 0.08192, is defined. Therefore, for ε = 1, the series (1.14) diverges very quickly, so that the sum of the first six terms is 21476. The situation can be corrected using the Pad´e approximant

10

Asymptotic Methods for Engineers

method (see Chapter 7 for more details). Building the Pad´e approximant with three terms in the numerator and denominator and calculating it with ε = 1, we obtain the value of the root x = 0.76369 (difference from the exact value of 1.2%). 2. Let us now introduce a parameter ε at the linear term of equation (1.12) x5 + εx = 1.

(1.15)

Representing the solution of equation (1.15) in the form x = b0 + b1 ε + b2 ε 2 + ...,

(1.16)

after applying the standard procedure of the perturbation method, we have b0 = 1,

b1 = −1,

b4 = 0,

b2 = −1/25,

b5 = 21/15625,

b3 = −1/125,

b6 = 78/78125.

One can obtain a general expression for the coefficients: bn = −(Γ[(4n − 1)/5]/(5Γ[(4 − n)/5]n!) and determine the convergence radius of series (1.16), R = 5/44/5 = 1.64938 . . . The value x (1) when taking into account the first six terms of the series (1.16) x = 0.75434 differs from the exact one by 0.07%. 3. Let us to introduce the small parameter δ into the exponent x1+δ + x = 1,

(1.17)

x = c0 + c1 δ + c2 δ 2 + . . .

(1.18)

and represent x as We also use the expansion: x1+δ = x(1 + δ ln |x| + ...). The coefficients of series (1.18) are easily determined: c0 = 0.5, c1 = 0.25 ln 2, c2 = −0.125 ln 2, ... The radius of convergence in this case is equal to one, and it must be calculated at δ = 4. Using the Pad´e approximant with three terms in the numerator and denominator, for ε = 1 we find x = 0.75448, which is only 0.05% different from the exact result, and by calculating ci to i = 12 and constructing the Pad´e approximant with six terms in the numerator and denominator, find x = 0.75487654 (error 0.00015%). The described method is called the “small delta method” (for details, see Section 2.8).

11

ASYMPTOTIC APPROXIMATIONS AND SERIES

4. We now consider the exponent as a large parameter. Consider the equation xn + x = 1.

(1.19)

Setting n → ∞ (“large δ method”, see section 2.9 for details), we represent the desired solution in the form  1/n 1 x = (1 + x1 + x2 + ...) , (1.20) n where 1 > x1 > x2 > ... Substituting (1.20) into equation (1.19) and taking into account that 1 n1/n = 1 + ln n + ..., n 1 x1/n = 1 + ln(1 + x1 + x2 + ...) + ... n we obtain in order of increasing accuracy  x≈  x≈

ln n n

1/n

ln n − ln ln n n

,

(1.21)

1/n .

(1.22)

For n =√2 formula (1.21) gives x = 0.58871; the error compared to the exact solution (0.5( 5 − 1) ≈ 0.618034) is 4.7%. For n = 5 we get x = 0.79745 (4.4% error). Formula (1.22) for n = 5 gives x = 0.74318 (error 2.7%). Thus, even the first terms of the asymptotics give good results. Consequently, in the case under consideration, the “large delta method” wins in terms of the number of approximations, providing good accuracy even in low orders of the perturbation method. Approximations (1.21), (1.22) give an example of a nonpower asymptotics (see section 2.3 for details). The obtained solution can be improved using the Lambert functions W (z) [92] (proposed by A.V. Pichugin), which is governed by the following equation z = W (z)exp(W (z)). Then, the solution to our problem has the form  x≈ 1 where C = 2n ln (error 0.06%).



W (n) n

1/n 1 +C n W( ) , n 1 +C

 . Note that for n = 5, the above formula yields x = 0.75443

12

Asymptotic Methods for Engineers

Let us also note a rather trivial, but often forgotten true: the asymptotic analysis should begin with the nondimensionalization of the initial relations and the choice of the most natural dimensionless parameters for a given problem [161, 226, 228]. This makes it possible to most simply take into account the symmetries inherent in the system [18]. The choice of the asymptotic method, and the parameters of asymptotic integration, is the most essential and informal part of the study. Experience and intuition, analysis of the physical essence of the problem, experimental and numerical results should help here. One should not forget Manevitch’s criterion [10]: the true simplification is closely connected with the increase in the symmetry of the original system. After the parameters of asymptotic integration have already been introduced and the asymptotic method has been chosen, there is no need for “reinventing the wheel”: you can use the well-known and well-developed techniques, some of which are described in this book.

ASYMPTOTIC 2 REGULAR EXPANSIONS 2.1

PERTURBATION AND ITERATION METHODS

Next, we need some facts from linear algebra and the theory of differential equations [89]. A matrix A is called self-adjoint if it is equal to its conjugate A∗ . The conjugate matrix is obtained from the original matrix by transposition followed by complex conjugation. Self-adjoint matrices play a role among all matrices similar to the role of real numbers among all complex ones, since any matrix can be represented as A = A1 + iA2 , where A1 , A2 are self-adjoint matrices, i is the imaginary unit. If A is a self-adjoint matrix, then for any vectors x, y of order n, the equality holds (Ax, y) = (x, Ay),

(2.1)

where the symbol (..., ...) means the scalar product. Self-adjoint matrices have only real eigenvalues. Equality (2.1) serves as the basis for defining the self-adjointness of an operator. Let, for example, an ordinary differential equation (for partial differential equations, the definition is similar) L(y) − λ M(y) = 0,

(2.2)

and homogeneous boundary conditions Qi (y)|x=xi = 0, where

i = 1, 2,

(2.3)

  d n−1 dn L(y) = an n + an−1 n−1 + . . . + a0 y, dx dx   dm d m−1 M(y) = bm m + bm−1 m−1 + . . . + b0 y, dx dx   n−2 n−1 (i) d (i) d (i) (i) d Qi (y) = cn−1 n−1 + cn−2 n−2 + . . . + c1 + c0 y. dx dx dx

Let us introduce the so-called test function u(x) satisfying the given boundary conditions (2.3). The boundary value problem (2.2),(2.3) is called self-adjoint if Zx2

[uL(v) − vL(u)]dx = 0,

x1

Zx2

[uM(v) − vM(u)]dx = 0

(2.4)

x1

for any test functions u(x), v(x). DOI: 10.1201/9781003467465-2

13

14

Asymptotic Methods for Engineers

Figure 2.1 Compressed simply supported beam.

The self-adjointness of a linear differential operator can be easily verified by integrating by parts. For example, let us prove the self-adjointness of a boundary value problem describing the buckling of a compressed simply supported beam (Fig. 2.1): EIwxxxx + T wxx = 0, at x = 0, L

w = wxx = 0,

(2.5) (2.6)

where E is the Young’s modulus; I is the moment of inertia of the cross section of the beam; T is the compressive force. It is easy to show that for this case ZL 

u

 d4u d4v − v dx = 0. dx4 dx4

0

Similarly, one can show that the operator d 2 /dx2 is self-adjoint. Let us proceed to the construction of asymptotic solutions. As the simplest problem, consider the system of linear algebraic equations (A0 + εA1 ) x = b,

(2.7)

where A0 , A1 are matrices of order n and x, b are n-dimensional vectors; 0 < ε n) ∪ (l > n) ∪ (s > n). Thus, at µ = 0 system (2.137) takes a “triangular” form, and at µ = 1 it returns to its original form. Next, we look for a solution in the form of expansions: ω1 = ω (0) + µω (1) + µ 2 ω (2) + ..., (0)

(1)

(2)

Am,n = Am,n + µAm,n + µ 2 Am,n + ...,

m, n = 1, 2, 3...,

(2.138) (m, n) 6= (1, 1) , (2.139)

and then set µ = 1. This approach makes it possible to keep all equations in system (2.137). In the expansions (2.138), (2.139), we restrict ourselves to the first two terms. Let us analyze the solution. Note that in the problem under consideration the parameter c plays the role of a bifurcation parameter. In the general case, when c 6= 0 and has order 1, system (2.137) admits the following solution: Ai, j = 0, ω1 =

i, j = 1, 2, 3..., 27 A2m,n ω0
2 , 128 ω m,n

(i, j) 6= (m, n) ,

m, n = 1, 2, 3...

40

Asymptotic Methods for Engineers

The amplitude-frequency characteristic is determined by the following expression Ωm,n = ωm,n + 0.2109375

A2m,n ε + ... ωm,n

Of particular interest is the case when the linear component of the restoring force is equal to zero (c = 0) and the phenomenon of internal resonance between the eigenmodes occurs. Solving system (2.137) by the described method, we find Am,n = 0, (m, n) 6= (1, 1),

m, n = 1, 2, 3, ...,

(m, n) = 6 (2i − 1, 2i − 1),

A3,3 = −4.5662 · 10−3 A1,1

i = 1, 2, 3, ...,

(2.140)

A5,5 = 2.1139 · 10−5 A1,1 , ...,

ω1 = 0.211048A21,1 /ω0 . In the zeroth approximation, all odd “diagonal” modes are excited. If oscillations are excited by one of the higher modes, then as a result of the redistribution of energy, modes of lower orders appear. Let us take a look at the next question. At present, various versions of the so-called non-linear Galerkin method have become popular [171, 196, 285]. Perhaps it would be more correct to call these methods the non-linear Kantorovich methods, since we are talking about the reduction of the original partial differential equations to infinite systems of ordinary differential equations. Recall that the Kantorovich method is a variant of the Bubnov-Galerkin method for solving partial differential equations, when the desired functions are approximated with respect to only one variable [151]. Then the coefficients of such an expansion depend on another variable, and an infinite system of ordinary differential equations is obtained to determine them. Let us show that the non-linear Galerkin methods can be interpreted as variant of the homotopy perturbation method. Let us analyze an infinite system of ordinary differential equations ut + f (u, w) = 0, (2.141) wt + F(u, w) = 0,

(2.142)

where u is the n-dimensional vector, w is the vector of infinite dimension. The usual truncation method reduces the infinite system of equations (2.141), (2.142) to a finite one u0t + f (u0 , 0) = 0. (2.143) Let us introduce the parameter µ into the system (2.142) as follows [196, 285] µwt + F(u, w) = 0. If we suppose u = u0 + µu1 + ..., w = w0 + µw1 + ...,

41

REGULAR ASYMPTOTIC EXPANSIONS

at µ = 0 we obtain system of n ordinary differential equations and an infinite system of non-linear algebraic equations u0t + f (u0 , w0 ) = 0,

(2.144)

F(u0 , w0 ) = 0.

(2.145)

The vector w0 (u0 ) found from equations (2.145) is substituted into system (2.144). The zero approximation of the homotopy perturbation method coincides with the non-linear Galerkin approach. The parameter µ can also be introduced as follows: ut + f (u, µw) = 0, µwt + F(u, w) = 0. For µ = 0, we obtain system (2.143). Next from the system F(u0 , w0 ) = 0 one can define w0 and to solve the first approximation system u1t + f (u1 , w0 ) = 0. The zero and first approximation of the homotopy perturbation method in this case coincide with post-processed Galerkin method [171].

2.8

SMALL DELTA METHOD

In papers [45, 47], small δ method was proposed, which we explain with examples. Let us construct a periodic solution of the problem xtt + x3 = 0, at t = 0 x = 1,

(2.146) xt = 0.

(2.147)

We introduce a parameter δ into equation (2.146) as follows: xtt + x1+2δ = 0.

(2.148)

In the final expression, we should put δ = 1, but in the process of solving we consider δ 1, both boundary conditions must be set to equation (3.6), for p < 1 to equations (3.13). Hence, the only consistent value p = 1. Thus, the order of the solution of the boundary layer is estimated in terms of the solution of the degenerate equation. Taking p = 1, equating in conditions (3.15) the terms of the same order of ε to zero and taking into account that from equations (3.6) follows z0 0 = −z0 , we get at x = 0 z0 = 1, z1 = −zb0 , ..., (3.16)

60

Asymptotic Methods for Engineers

εz0 b0 = z0 ,

εz0 b1 = z1 ,

...

(3.17)

Boundary conditions (3.16), (3.17) are given for equations (3.6), (3.7) and (3.13), (3.14), respectively. The splitting of boundary conditions is an operation that leads to the greatest error in solving specific problems. In this connection, an asymptotically inconsistent procedure was proposed in [307], which, however, leads to an increase in the real accuracy of the solution. Let us show it on the described example. We not split the boundary conditions, writing them in the zero approximation in the form z0 + zb0 = 1, z0 0 + z0 b0 = 0

at x = 0.

The solution of the boundary layer equation (3.13) has the form zb0 = C exp(−ε −1 x). Substituting this expression into the boundary conditions leads to the system z0 +C = 1, z0 0 − ε −1C = 0

at x = 0.

Eliminating the constant C, we obtain the boundary conditions for the degenerate equation (3.6) z0 + εz0 0 = 1 at x = 0. As already noted, this condition is asymptotically inconsistent, but increases the real accuracy of the solution. The boundary layer is often described by ordinary differential equations with constant coefficients, since the variable coefficients in the boundary layer equations can be “frozen”. Let us explain the essence of this method using the example of a pendulum with a small variable mass. Physically, this mean that the pendulum is in a media which gives strong damping, e.g., in viscous fluid [237]. The original equation can be written as: εϕ(t)x¨ + x˙ + x = 0,

at t = 0 x = x0 ,

x˙ = 0,

(3.18)

˙ where 0 < ϕ(t) < 1, ϕ(t) ∼ ϕ(t). The degenerate equation has the form x˙ + x = 0.

(3.19)

The boundary layer equation should be written as εϕ(t)x¨b + x˙b = 0.

(3.20)

Equation (3.20) contains a coefficient that is variable in t. This coefficient varies slowly in time, boundary layer decays rapidly, it gives possibility to put in the first

61

SINGULAR ASYMPTOTIC EXPANSIONS

approximation ϕ(t) = ϕ(0). Then, to determine the boundary layer, we arrive at an equation with constant coefficients εϕ(0)x˙b + xb = 0. We also note the following point. The components of the boundary layer type in this case, as can be seen from the above solution, are of the order of ε in comparison with the main solution. So, the zeroth approximation of the boundary layer can be ignored when determining the displacement, since x = x0 +O(ε). But boundary layer component should be taken into account when determining the velocity x, ˙ since the component of the boundary layer increases by a factor of ε −1 upon differentiation and, consequently, x˙ = x˙0 + x˙b0 + O(ε). Since derivatives determine stresses in mechanics, it is impossible to correctly determine the stress-strain state without taking into account components of the boundary layer type. An important condition for the applicability of the boundary layer method is the requirement that the degeneracy be regular. The degeneration is said to be regular if the number of roots with negative real parts of the characteristic equation for the boundary layer equation coincides with the number of boundary conditions that drop out in the transition to the degenerate problem [319]. Let us now demonstrate the application of the boundary layer method to the solution of a non-linear problem with a small parameter at the highest derivative. Let us take as a model equation a van der Pol pendulum with a small mass: ε x¨ + (1 − x2 )x˙ + x = 0, at t = 0 x = a,

x˙ = 0,

(3.21) a = const.

(3.22)

Here we set a < 1 and a ∼ 1, so that the coefficient at the second term of equation (3.21) is not small. For ε = 0 we get
ln |x0 | = 0.5x02 − t +C. 1 − x02 x˙0 = −x0 , The construction of the boundary layer is now much more complicated and is inseparable from the analysis of the initial conditions. In addition, in non-linear problems, one must be careful about the concept of the variation index introduced earlier. The point is that this concept is essentially oriented toward the exponential boundary layer. In non-linear problems, generally speaking, boundary layers of another type are possible, for example, power-law ones, so each non-linear problem requires careful analysis. In the case under consideration, focusing on the solution of the linearized problem, it is natural to accept the following variability of the boundary layer solution: ∂ xb ∼ ε −1 xb . ∂t

(3.23)

62

Asymptotic Methods for Engineers

Then xb ∼ εx0 , and the decomposed initial conditions are at t = 0 x0 = a,

(3.24)

x˙b = −x˙0 .

(3.25)

Relations (3.23)–(3.25) imply that xb is small compared to x0 : xb ∼ εx0 .

(3.26)

Now writing the solution of the original problem in the form x = x0 + xb , substituting this expression into equation (3.21) and taking into account relations (3.23), (3.26), with respect to xb we obtain a linear equation with a variable coefficient ε x¨b + (1 − x02 )x˙b = 0. The slowly varying coefficient x02 does not change significantly comparison with the boundary layer, so it can be “frozen” at the boundary, i.e. set equal to the value at zero (see condition (3.24)). Finally, we arrive at a linear equation with constant coefficients ε x¨b + (1 − a2 )x˙b = 0. (3.27) The general solution of equation (3.27) is xb = C1 exp[−(1 − a2 )ε −1t]. From the boundary conditions (3.24), (3.25), we determine the constants C and C1 : a2 εa C = ln | a | − , C1 = − . 2 (1 − a2 )2 Reduction of the ODE order can lead not only to the impossibility of satisfying the given boundary (initial) conditions, but also to the appearance of discontinuities in solutions (or their derivatives) inside the domain. In this case, solutions appear that are localized in the vicinity of points or (in the case of partial differential equations) lines, which are called internal boundary layers [38, 313]. Consider, for example, the boundary value problem [38] ε

 2 d2y dy + = 1, 2 dx dx

(3.28)

y(0) = y(1) = 0.

(3.29)

Degenerate equation 

dy0 dx

2 =1

63

SINGULAR ASYMPTOTIC EXPANSIONS (1)

(2)

has solutions y0 = −x +C1 , y0 = x +C2 . From the boundary conditions (3.29), we find C1 = 0, C2 = −1; so the limit solution can be written as y0 = −x[H(x) − H(x − 0.5)] + (x − 1)[H(x − 0.5) − H(x − 1)], where H(x) is the Heaviside function. Let us proceed to the construction of the internal boundary layer. Consider first the solution in the region 0 ≤ x ≤ 0.5. Assuming (1)

(1)

(2)

y0 = y0 + yb + yb ,

(3.30)

substituting expression (3.30) into equation (3.28) and restricting ourselves to the main terms, we find (i) (i) d 2 yb dyb − 2 = 0, i = 1, 2. (3.31) ε dx2 dx The solutions of equations (3.31) can be represented as (i)

yb = C3 exp[2(−1)i+1 ε −1 (x − 0.5)],

i = 1, 2.

The constant C3 is determined from the conditions (i)

(i)

dyb dy + 0 =0 dx dx

at x = 0.5.

Hence, C3 = 0.5ε. The situation is shown schematically in Fig. 3.2.

Figure 3.2 Example of an internal boundary layer

The solutions of the internal boundary layer introduce exponentially small discrepancy into the boundary conditions (3.29), which can be compensated in the following linear approximations.

3.2

NEWTON’S POLYGON AND ITS GENERALIZATIONS

Let an implicit function be given F(ε, x) = 0,

F(0, x0 ) = 0,

64

Asymptotic Methods for Engineers

and we need to define an explicit dependence x on ε x = x0 + b1 ε β1 + b2 ε β2 + . . . ,

β1 < β2 < β3 < . . .

Determining the exponents βi is a problem solved using Newton’s polygon (the term Newton’s diagram is also used) [38, 298]. Following [38], consider an example n

∑ ak (ε)xk = 0,

F(ε, x) =

(3.32)

k=0

where

mk

ak (ε) =

( j) α ( j) k

∑ αk

ε

,

k = 0, 1, . . .

(3.33)

j=0

The asymptotic expansion of the roots xi of equation (3.32) is sought in the form ∞

xi ∼

( j) βi( j)

∑ xi

at ε → 0,

ε

( j+1)

βi

( j)

> βi ,

i = 1, 2, . . . ,

(3.34)

j=0

i.e., xi ∼ yit βi + O(t βi ), (0)

(3.35)

(0)

where yi = xi , βi = βi . To determine the possible values yi and βi , we substitute relation (3.35) into (3.32), collect the terms with the lowest degree, and equate zero coefficient at the specified degree. Until the values of βi are determined, it is not known which of the terms have the smallest order in ε. It is only clear that these terms are among the following: (0)

(0)

a0 ε α0 ,

(0)

(0)

a1 yi ε α1

+βi

,

(0)

(0)

a2 y2i ε α2

+2βi

,

...,

(0)

(0)

a2n yni ε αn

+nβi

.

For the mutual annihilation of terms of the smallest order, it is necessary that two of the exponents (0)

α0 ,

(0)

α1 + βi ,

(0)

α2 + 2βi ,

...,

(0)

αn + nβi

would be equals, and the rest would be no less than them. Equating the exponents, we find all possible values βi , and n then we o determine yi . Namely, on the plane (k, α) we (0) construct n + 1 point Mk = k, αk with integer abscissas (Fig. 3.3). Let us draw a line segment M0 M1 . The tangent of the angle of inclination of the line segment to the k axis is equal to the value βi , at which the orders of the first and second terms coincide. It is easy to check that the points lying above the line passing through M0 and M1 correspond to terms of a higher order of smallness. Only the main terms are of interest, so we connect the points of Mk by line segments in such a way that the points that do not lie on these segments are above the resulting broken line. To build it, we turn counterclockwise the ray drawn vertically downward from the point

65

SINGULAR ASYMPTOTIC EXPANSIONS

M0∗ = M0 . Denote M1∗ the first of the points Mk that the ray intersects. Then the vertical ray from the point M1∗ will be rotated in the same direction until it passes through the next point M2∗ . The process continues until the next ray passes through the point Ms∗ = Mn . If there are several points Mk on the ray, then the rightmost point (with the maximum k) is taken as Ms∗ . The broken line connecting the points Mi∗ is ∗ to the k axis gives called Newton’s diagram. The slope of the line segment Mi∗ Mi+1 the order βi of the root xi , the length of the projection of the line segment onto the k axis is equal to the number of roots xi of this order, and the number of points Mk , through which the line segment passes, is equal to the number of terms of the equation for calculating yi .

Figure 3.3 Newton’s diagram.

Consider, for example, the algebraic equation ε 3 x5 + x4 − 2x3 + x2 − ε = 0.

(3.36)

Let us build a graph, along the abscissa of which we plot the degrees of the terms of the equation, and along the ordinate - the degrees of a small parameter at these terms (Fig. 3.4). It is easy to see the possible limit systems from the figure. 1. The first two terms of equation (3.36) are equal, hence x1 = −ε −3 + ... 2. The second, third, and fourth terms of equation (3.36) are equal, hence x2,3 = 1 + ... 3. The fourth and fifth terms of equation (3.36) are equal, hence x4,5 = ±ε 1/2 + ... Let us now consider a generalization of the Newton’s diagram to the twodimensional case. The first example is the deformation of a membrane on an elastic foundation of high rigidity, described by the equation ε(wxx + wyy ) − w = Q(x, y),

ε 1 + 2α + 2β ,

1 + 4α > 1 + 4β ,

1 + 4α > 2α,

1 + 4α > 2β .

Conditions for the existence of a limit system εwxxxx − wyy = 0 have the form 1 + 4β = 2α, 2α > 2β , 2α > 1 + 2α + 2β , etc. Thus, instead of analyzing the terms of the equation themselves, we come to the analysis of the exponents of the orders of these terms. This technique was called “power geometry” [76]. In this case, one can use the symmetry of the system with respect to the replacement of x by y, which greatly simplifies the calculations. So, we write down the possible limit systems corresponding to equation (3.48): 1) εwxxxx = 0, 2) εwyyyy = 0, 3) wxx = 0, 4) wyy = 0, 5) ε∇4 w = 0, 6) ∇2 w = 0, 7) εwxxxx − wxx = 0, 8) εwyyyy − wyy = 0, 9) ε∇4 w − ∇2 w = 0. In this case, the plane αβ is divided into equivalence classes (Fig. 3.6), among which there is one point (−1/2, −1/2), four rays and four quarter-planes. There can be no other limiting systems. The found limit systems can be arranged in a hierarchical sequence in accordance with the degree of simplification: ε∇4 w − ∇2 w = 0, ∇2 w = 0,

ε∇4 w = 0,

69

SINGULAR ASYMPTOTIC EXPANSIONS

Figure 3.6 Plane of parameters α, β for problem (3.48).

∂ 4w ∂ 2w − 2 = 0, ∂ x4 ∂x ∂ 4w ∂ 2w ε 4 − 2 = 0, ∂y ∂y ε

∂ 2w = 0, ∂ x2 ∂ 2w = 0, ∂ y2

∂ 4w = 0, ∂ x4 ∂ 4w ε 4 = 0. ∂y ε

Let us describe the scheme of application of the found approximate relations. Let the variability of the right side of the inhomogeneous equation (3.47) be small: ∂Q ∼ Q, ∂x

∂Q ∼ Q. ∂y

Then, in the first approximation, one should use the limit system ∇2 w0 = Q.

(3.50)

As before, we need to split the boundary conditions. It is easy to show that for equation (3.50), the boundary conditions must be set at

x = 0, a w0 = 0,

at

y = 0, b w0 = 0.

Discrepancies in other boundary conditions are removed by solutions of the boundary layer type. At the edges x = 0 and x = a the boundary layer is described by the equation ∂ 2 wb1 ε − wb1 = 0 ∂ x2

70

Asymptotic Methods for Engineers

with boundary conditions wb1x = −w0x

at x = 0, a.

At the edges y = 0, b the boundary layer is described by the equation ε

∂ 2 wb2 − wb2 = 0, ∂ y2

and the boundary conditions are as follows: wb2y = −w0y

at y = 0, b.

The constructed boundary layers remove the residuals everywhere, except for narrow (extent of the order of ε) zones in the corners of the region. Here the so-called corner boundary layers appear, which are described in section 3.6. The examples given dealt with polynomials or series with positive integer exponents. However, the described method is easily generalized. Recall that Puiseux series are a generalization of power series that allow to use negative and fractional exponents. The generalized Newton algorithm is called Newton–Puiseux polygon.

3.3

AN EXAMPLE OF AN ASYMPTOTIC SPLITTING OF A PARTIAL DIFFERENTIAL EQUATION

In the variables ξ = x/R , η = y/R , the solving equation for an isotropic circular cylindrical shell has the form [129, part III]: ∂ 4Φ ∂ 6Φ 2 + (8 − 2ν ) + ∂ξ4 ∂ ξ 4∂ η 2 ∂ 6Φ ∂ 6Φ ∂ 4Φ ∂ 4Φ 8 2 4 +2 6 +4 2 2 + = 0, ∂ξ ∂η ∂η ∂ξ ∂η ∂ η4

∇4 ∇4 Φ + a−2

(3.51)


4 4 4 where Φ is the potential function, ∇4 = ∂∂ξ 4 +2 ∂ ξ ∂2 ∂ η 2 + ∂∂η 4 , a2 = h2 / 12(1 − ν 2 )R2 , R, h are the shell radius and thickness. We introduce the variation index α and β as follows: ∂Φ ∼ aα Φ, ∂ξ

∂Φ ∼ aβ Φ. ∂η

Let us describe all possible asymptotics of equation (3.51): a) limit equations containing one term each: 1)

∂ 8Φ = 0, ∂ η8

(3.52)

2)

∂ 8Φ = 0, ∂ξ8

(3.53)

71

SINGULAR ASYMPTOTIC EXPANSIONS

3)

∂ 4Φ = 0, ∂ξ4

(3.54)

4)

∂ 4Φ = 0, ∂ η4

(3.55)

b) limit equations containing two terms each: 5)

4 ∂ 8Φ −2 ∂ Φ + a = 0, ∂ξ8 ∂ξ4

from here

∂ 4Φ + a−2 Φ = 0, ∂ξ4 ∂ 8Φ ∂ 4Φ + a−2 4 = 0, 8 ∂η ∂ξ

6)

(3.56)

(3.57) (3.58)

c) limit equations containing three or more terms: 7) ∇8 Φ = 0,

(3.59)

this equation can be reduced to the following two ∇4 Φ = 0,

(3.60)

∇4 Φ = 0,

(3.61)

8) ∇8 Φ + a−2 9)

∂4 ∂ η4 10)

∂ 4Φ ∂ξ4

= 0,

2 ∂2 ∂ 4Φ + 1 Φ + a−2 4 = 0, 2 ∂η ∂ξ   2 ∂4 ∂2 + 1 Φ = 0. ∂ η4 ∂ η2

(3.62)



(3.63) (3.64)

All obtained limit equations have a physical meaning: equation (3.55) describes the membrane theory, equations (3.58), (3.63) describe the semi-membrane theory (with large and small variability in η, respectively). According to the terminology of Gol’denveizer [129], these are the equations of the fundamental state of stress. Equation (3.57) corresponds to the edge effect, equation (3.62) corresponds to the theory of shallow shells (often called the Donnell-Mushtari equation). Equation (3.59) describes the bending deformation of the ring; equations (3.52) and (3.53) - bending and longitudinal deformations of the rod in the direction of the η and ξ axes, respectively; equation (3.54), (3.55) - the longitudinal deformation of the rod in the direction of the η and ξ axis. Equations (3.60) and (3.61) describe the plane stress state and plate bending, respectively.

72

Asymptotic Methods for Engineers

It is also easy to determine the values of the parameters α and β corresponding to each of the limiting cases: 1 1) α < − , β > α; 2

1 1 1 2) β < − , β > α, and − < β < 0, α < + 2β ; 2 2 2

1 α 1 1 α 1 − , α >− ; 4) 0 < β < − , α > ; 2 4 2 2 4 2 1 1 1 5) β > α, α = − ; 6) β > − , α = + 2β ; 2 2 2 1 1 7) β < − , α = β ; 8) β = α = − ; 2 2 1 1 9) β = 0, α = ; 10) β = 0, α > . 2 2 Thus, the plane of parameters α and β is divided into a number of regions (Fig. 3.7). Among them there are two points, three rays, four “pieces of the plane” and one area consisting of a line segment without edge points and a ray. 3) β >

Figure 3.7 Asymptotic portrait of the equation of equilibrium of a circular cylindrical shell.

The most general of the obtained equations is (3.62), it includes almost all limit equations, with the exception of (3.63) and (3.64). The equation of the membrane theory is a degenerate equation lying on the spectrum. The semi-membrane theory equations are (3.58) and (3.63). The edge effect (3.56) makes it possible to satisfy the boundary conditions.

3.4

SETTING OF BOUNDARY CONDITIONS

Along with the asymptotic simplification of the original equations, it is necessary to formulate the boundary conditions for the simplified equations [11, 129]. This question has already been considered for ordinary differential equations in section 3.1. To illustrate the approach used in the case of partial differential equations, we consider the semi-membrane equation (3.58) ∂ 4 Φ1 ∂ 8 Φ1 + a2 = 0, 4 ∂ξ ∂ η8

(3.65)

73

SINGULAR ASYMPTOTIC EXPANSIONS

and edge effect (3.57) Φ2 + a2

∂ 4 Φ2 = 0. ∂ξ4

(3.66)

Equation (3.65) corresponds to the variation indexes α = 0, β = −0.25; equation (3.66) – to the α = −0.5, β = −0.25. Consider a semi-infinite shell with clamped edge at ξ = 0 :

T1 = T0m sin mη,

T12 = 0,

w = 0,

wξ = 0.

(3.67)

Here T1 , T12 are the longitudinal and shear forces; T0m ≡ const. The expressions of the functions the boundary conditions (3.67) in terms of the potential functions Φ1 and Φ2 are as follows [129]: (1)

=−

Eh Φ , (1 − ν 2 )R 1ξ ξ ηη

T12 =

(2)

=−

Ehν Φ , (1 − ν 2 )R 2ξ ξ ηη

T12 =

w(1) = Φ1ηηηη ,

T1

w(2) = Φ2ξ ξ ξ ξ ,

T1

(1)

Eh Φ , (3.68) (1 − ν 2 )R 1ξ ξ ξ η

(2)

Eh Φ . (3.69) (1 − ν 2 )R 2ξ ξ ξ η

Using formulas (3.68), (3.69), we estimate the orders of the functions in the boundary conditions: (1)

T1

∼ a−1/2 Φ1 ,

w(1) ∼ a−1 Φ1 , (1)

T12 ∼ a−1/4 Φ1 , (1)

wξ ∼ a−1 Φ1 ,

(2)

T1

∼ a−3/2 Φ2 ,

w(2) ∼ a−2 Φ2 ,

(3.70)

(2)

T12 ∼ a−7/4 Φ2 , (2)

wξ ∼ a−5/2 Φ2 .

The boundary conditions are arranged in ascending order of the difference between the exponents of the small parameter at the functions Φ1 and Φ2 . For the first two conditions it is 1, for the last two it is 3/2. Such arrangement of boundary conditions is called canonical [319]. Let us now introduce the parameter γ characterizing the relative order of magnitude of Φ1 and Φ2 : Φ1 ∼ a−γ Φ2 . The value γ is selected from the condition of the correctness of the formulation of boundary value problems, i.e., the number of boundary conditions must correspond to the order of the equation. In this case, we get γ = 1, and the boundary conditions have the form: at ξ = 0 :

(1)

T1

= T0m sin mη,

w(1) = 0,

(1)

(2)

T12 + T12 = 0,

(1)

(2)

wξ + wξ = 0.

74

Asymptotic Methods for Engineers

3.5

PAPKOVICH-FADLE METHOD (METHOD OF HOMOGENEOUS SOLUTIONS)

Often, the ratio of sizes can be used as an asymptotic parameter. Such a simplification is widely used in hydrodynamics and the theory of composites and is called the “lubrication theory approximation” [88, 290]. As an example of using this asymptotics, let us consider the bending of a clamped plate, in which the length of one side (L1 ) is much less than the length of the other (L). The original equation and boundary conditions are as follows: D∇4 w = Q(x, y),

(3.71)

at x = 0, L w = wx = 0, at y = 0, L1 w = wy = 0.

(3.72)

Let us make a change of variables: ξ = x/L ,

η = y/L1 .

Then equation (3.71) is reduced to the form ε 4 wξ ξ ξ ξ + 2ε 2 wξ ξ ηη + wηηηη = q,

(3.73)

where q = QL14 /D, ε = L1 /L 0, y = 1 at x = 0. The singularity of the problem lies in the fact that at ε = 0, we obtain a smooth solution y = 0, which does not allow us to satisfy the given initial condition. But one can look for a solution in the form of a non-smooth function. Namely, setting z(x) = H(x)y(x), we obtain from the original Cauchy problem εz0 = −z + εδ (x).

(3.98)

The solution of equation (3.98) is sought in the form of a regular expansion ∞

z=

∑ zn ε n .

n=0

As a result, we get z0 = 0,

z1 = δ (x),

zn+1 = (−1)n δ (n) (x),

n = 1, 2, ...

(3.99)

Note that from expressions (3.99), we can pass to smooth functions. To do this, we can apply the Laplace transform, use the Pad´e approximation in the dual space (see chapter 7.2), and then use inverse Laplace transform.

79

SINGULAR ASYMPTOTIC EXPANSIONS

Let us show one more application of the asymptotic method using generalized functions. Consider the equation of a membrane reinforced with threads of small but finite width 2ε: [1 + 2εΦ0 (y)]uxx + uyy = 0, (3.100) ∞

where Φ0 (y) = ∑ [H(y − kb + ε) + H(y − kb − ε)]. k=−∞

Let us expand the function Φ0 (y) into a series in ε. Applying the two-sided Laplace transform ¯ Φ(p, ε) =

Z∞

e−p|y| Φ(y, ε)dy,

−∞

¯ expanding Φ(p, ε) into a series in ε and performing the inverse Laplace transform, we obtain Φ0 (y) = 2εΦ(y) + 2ε ∑ ε n Φ(n) (y), (3.101) n=1,3,5,... ∞

where Φ(y) = ∑ δ (y − kb). k=−∞

Now we represent the solution of the original equation (3.100) in the form u = u0 + εu1 + ε 2 u2 + . . .

(3.102)

Substituting expressions (3.101) and (3.102) into equation (3.100) and splitting the resulting equation with respect to ε, we arrive at a recurrent sequence of boundary value problems [1 + 2εΦ(y)]u0xx + u0yy = 0, (3.103) [1 + 2εΦ(y)]u1xx + u1yy = −εu0xx Φy (y),

(3.104)

... Thus, in the first approximation we obtain the problem with one-dimensional threads (3.103), the effect of the thread width is taken into account in the following approximations (3.104). Let us now consider the construction of the asymptotics of the Schroedinger equation (2.184) with boundary conditions (2.183) as N → ∞. In this case, from the eigenvalue problem (2.183),(2.184), we obtain ψxx + Eψ = 0,

ψ(±1) = 0.

The solution of this boundary value problem gives the following energy levels En = 0.25π 2 (n + 1)2 , n = 0, 1, 2, ... Comparison of the lower energy level E0 with known numerical values [61] (see Table 3.1) shows that an acceptable accuracy is achieved only for large values of N, so it is necessary to construct an improved solution.

80

Asymptotic Methods for Engineers

Table 3.1 Comparison of the numerical values of E0 (N) [61] with the predictions of various approximate theories. Numerical E0 1.0000 1.0604 1.2258 1.5605 1.1052 2.3379 2.4058 2.4431 2.4558

N 1 2 4 10 50 200 500 1500 3500

Equation (3.113) roots 0.9100 1.0422 1.2385 1.5831 2.1074 2.3382 2.4032 2.4428 2.4555

Error % 9.00 1.72 1.04 0.81 0.10 0.02 0.01 0.01 0.01

Consider the function ϕ = x2N at 0 ≤ x ≤ 1. The expansion of this function in powers of 1/N using delta functions is as follows: ∞

ϕ = ∑ (−1)i δ (i) (x − 1)(2N + 1)−1 (2N + 2)−1 ...(2N + 1 + i)−1 .

(3.105)

i=0

Expression (3.105) is obtained as follows: we apply the Laplace transform to the function ϕ(x) Z∞

e−pt ϕ(t)dt = p−2N−1 γ(2N + 1, p),

0

where γ(..., ...) is an incomplete gamma function [1]. Expand the resulting expression in a series in powers of 1/(2N + 1) and using inverse Laplace transform, we obtain (3.105). In the region 0 ≤ x ≤ 1, equation (2.183) can be represented as follows: ψ1xx − ϕψ1 + Eψ1 = 0.

(3.106)

We are looking for a solution to equation (3.106) in the form ∞

ψ1 =

∑ ψ1k (2N + 1)−1 ...(2N + 1 + i)−1 , k=0 ∞

E=

∑ E (k) (2N + 1)−1 ...(2N + 1 + i)−1 . k=0

After substituting these expressions into equation (3.106) and splitting in (2N + 1)−1 , we obtain the following recurrent system of equations ψ10xx + E (0) ψ10 = 0,

(3.107)

81

SINGULAR ASYMPTOTIC EXPANSIONS

ψ11xx + E (0) ψ11 + E (1) ψ10 − δ (x − 1)ψ10 = 0, ...

(3.108)

The solution of equation (3.107) for the case of symmetry about the line x = 0 has the form (the antisymmetric case is considered similarly): ψ1 = C cos λ x,

1/2

λ = (E (0) )

.

(3.109)

Let us pass to the study of the region x > 1. In this case, in the zeroth approximation, the term Eψ can be omitted (0)

(0)

ψ2xx − x2N ψ2 = 0.

(3.110)

Equation (3.110) must be supplemented by the condition (0)

ψ2 → 0

at x → ∞.

(3.111)

The solution of equation (3.110) with the boundary condition (3.111) is (0)

ψ2 = C1 x1/2 Kµ (µxN+1 ), where Kµ is the modified Bessel function of the second kind [1], µ = 0.5/(N + 1). For x = 1, the solutions ψ1 and ψ2 must be matched, therefore, at x = 1 (i)

(i)

(i)

(i)

ψ1 = ψ2 ,

(3.112)

ψ1x = ψ2x , i = 0, 1, 2, ... From conditions (3.112) for i = 0, we have a transcendental equation for determining λ 4λ Kµ (µ) . (3.113) −cotλ = 2Kµ (µ) − K1−µ (µ) − K1+µ (µ) The minimum real roots of equation (3.113) for various values of N are given in Table. 3.1. It can be seen that the accuracy of the proposed approximation is quite satisfactory.

3.8

MULTIPLE SCALES METHOD The method of multiple scales is so popular that it is being rediscovered just about every 6 months A.H. Nayfeh [228]

Another approach to the study of singularly perturbed boundary value problems is the method of two (or several) scales. His idea is simple and clear. Let us explain it using the example of a strongly damped oscillator (a system with 1/2 degree of freedom [237]): ε x¨ + x˙ + x = 0, (3.114)

82

Asymptotic Methods for Engineers

at t = 0 x = a,

x˙ = 0.

(3.115)

Instead of one independent variable t, we introduce two - “slow”, coinciding with the original one (t1 = t), and “fast”, τ = ε −1t. The first one is used to describe the solution in the inner region, the second - the boundary layer. The time derivative according to the rule of the total derivative will be written as follows ∂ ∂ d + ε −1 . = dt ∂t1 ∂τ

(3.116)

The dependent variable x is now a function of two arguments: t1 and τ. Substituting expression (3.116) into equation (3.114) and initial conditions (3.115), we have   2 ∂ 2x ∂ x ∂ ∂2 2∂ x + + ε + 1 + 2 x + ε = 0, (3.117) ∂ τ2 ∂ τ ∂t1 ∂ τ∂t1 ∂t12   ∂x ∂ x x(0, 0) = a, +ε = 0. (3.118) ∂τ ∂t1 τ=0, t1 =0

Instead of a singularly perturbed ordinary differential equation, we formally come to a differential equation in partial derivatives, but no longer containing a small parameter at higher derivatives. This method of regularization is sometimes called lifting into the space of a higher dimension [182]. Now the solution can be sought in the form of a series x = x0 (t1 , τ) + εx1 (t1 , τ) + . . . (3.119) Substituting expression (3.119) into equation (3.117) and initial conditions (3.118), we obtain ∂ 2 x0 ∂ x0 + = 0, (3.120) ∂ τ2 ∂τ ∂ 2 x1 ∂ x1 ∂ x0 ∂ 2 x0 + =− − x0 − 2 , 2 ∂τ ∂τ ∂t1 ∂ τ∂t1 ...

(3.121)

x0 = a,

(3.122)

at t1 = 0, τ = 0 ∂ x0 = 0, ∂τ x1 = 0, ∂ x0 ∂ x1 =− , ∂τ ∂t1 ...

(3.123) (3.124) (3.125)

The general solution of equation (3.120) has the form x0 = C(t1 ) +C1 (t1 )e−τ .

(3.126)

83

SINGULAR ASYMPTOTIC EXPANSIONS

It follows from the initial conditions (3.122), (3.123) that C1 (0) = 0,

C(0) = a.

(3.127)

The function C(t1 ) is still undefined. To find it, consider equation (3.121): ∂ 2 x1 ∂ x1 ∂C ∂ 2 x0 + =− −C. +2 2 ∂τ ∂τ ∂ τ∂t1 ∂t1

(3.128)

If the right side of equation (3.128) does not vanish, then the solution x1 contains a secular term. Setting the condition for its absence, we arrive at the equation ∂C +C = 0. ∂t1 Hence, taking into account condition (3.127), we have C = ae−t1 . The general solution of equation (3.121) can now be represented as x1 = C1 (t1 ) +C11 (t1 )e−τ . From the initial conditions (3.124), (3.125), we obtain C1 (0) = −C11 (0),

C11 (0) = a.

The function C1 (t1 ) must be determined from the condition that there are no secular terms in the equations of the second approximation. An essential advantage of the multiple-scales method is the possibility of its application in cases of irregular degeneracy. Consider, for example, the equation ε 2 y000 + y0 + y = 0

(3.129)

with initial conditions at x = 0 y = a,

y0 = 0,

y00 = 0,

a = const.

(3.130)

It is easy to verify that in this case the conditions for the regularity of degeneracy (see section 3.1) are not satisfied, since the roots of the characteristic equation for the equation ε 2 y00 + y = 0 purely imaginary. Consequently, the solution of equation (3.129) cannot be divided into the fundamental state and the boundary layer. At the same time, the multiple scale method works in this case as well. Indeed, by choosing slow and fast variables in the form x1 = x and ξ = ε −1 x and representing the original function as y = y0 (x1 , ξ ) + εy1 (x1 , ξ ) + . . . , we obtain the following recurrent sequences of equations and boundary conditions: ∂ 3 y0 ∂ y0 + = 0, ∂ξ3 ∂ξ

(3.131)

84

Asymptotic Methods for Engineers

∂ 3 y1 ∂ y1 ∂ 3 y0 ∂ y0 + − y − 3 , = − 0 ∂ξ3 ∂ξ ∂ x1 ∂ ξ 2 ∂ x1 ... at x1 = ξ = 0 y0 = a, y1 = 0,

∂ 2 y0 = 0, ∂ξ2

∂ y0 = 0, ∂ξ

∂ y1 ∂ y0 =− , ∂ξ ∂ x1 ...

∂ 2 y1 = 0, ∂ξ2

(3.132)

(3.133) (3.134)

From the first equation (3.131), taking into account the initial conditions (3.133), we have y0 = C(x1 ), C(0) = a, and from the conditions for the absence of secular terms in the solution of equation (3.132), we find C(x1 ) = ae−x1 . Finally, the fast part of the solution of equation (3.132), taking into account the initial conditions (3.134), takes the form y1 = −a sin ξ . Thus, if the boundary layer method works well in cases where some part of the solution is localized in the vicinity of points or lines, then the multiple scales method allows one to separate solutions with significantly different variability even in cases where there is no localization. Sometimes it is advantageous to take a “fast” variable in a more general form, namely, as a function of a slow variable and a small parameter (for example, τ = ϕ(t, ε)/ε). Examples of the choice of such functions (called regularizing functions) containing the entire irregular dependence on ε are given in [182] (see also section 6.4).

3.9

METHOD OF MATCHED ASYMPTOTIC EXPANSIONS

Let us demonstrate this method using the example of an strongly damped oscillator (3.114). We take the initial conditions in the form at t = 0

x = 1,

dx = 1. dt We seek a solution to equation (3.114) in the form of a series x = x0 (t) + εx1 (t) + ...

(3.135) (3.136)

(3.137)

85

SINGULAR ASYMPTOTIC EXPANSIONS

Substituting expression (3.137) into equation (3.114), after splitting with respect to ε, we obtain dx0 + x0 = 0, (3.138) dt dx1 d 2 x0 + x1 = − 2 , dt dt ...

(3.139)

General solutions of equations (3.138) and (3.139) have the form x0 (t) = C0 e−t ,

(3.140)

x1 (t) = −C0te−t +C1 e−t .

(3.141)

The solution of the degenerate equation (3.138) makes it possible to satisfy only one initial condition, and therefore, it is necessary to construct an additional asymptotics. Let us introduce a stretching variable τ = t/ε, then the function x(t) passes into the function X(τ), and instead of the original Cauchy problem (3.135)–(3.137), we obtain d 2 X dX + + εX = 0, (3.142) dτ 2 dτ dX at τ = 0 X = 1, = ε. (3.143) dτ The solution of the Cauchy problem (3.142), (3.143) is sought in the form X(τ) = X0 (τ) + εX1 (τ) + ...

(3.144)

As a result, we arrive at the following recurrent system of equations: d 2 X0 dX0 + = 0, dτ 2 dτ d 2 Xi dXi + = −Xi−1 , dτ 2 dτ and initial conditions at τ = 0

i = 1, 2, . . . ,

(3.145) (3.146)

X0 = 1,

dX0 = 0, dτ

(3.147)

X1 = 0,

dX1 = 1, dτ

(3.148)

dXi = 0, i = 2, 3, ... dτ The solution of the Cauchy problem (3.145), (3.147) has the form Xi = 0,

X0 (τ) = 1,

(3.149)

(3.150)

86

Asymptotic Methods for Engineers

and for the initial problem (3.146), (3.148), we obtain X1 (τ) = −τ + 2(1 − e−τ ).

(3.151)

So, the solution of the original equation (3.114) is represented in the form of an inner (local) expansion (3.144), suitable at the initial time, and an outer (global) expansion (3.137), which describes the behavior of the system after some time. The idea of matching is that at t → 0 the outer expansion behaves the same way as the inner one at τ → ∞ [143, 310]. We note that the terms inner and outer expansions used most frequently, although, as noted by Van Dyke [312], the terms global and local expansions are more logical. At t → 0, from solutions (3.140), (3.141), we obtain   t2 (3.152) x0 (t) = C0 1 − t + − ... , 2!     t2 t2 t3 x1 (t) = −C0t 1 − t + − ... +C1 1 − t + − ... . (3.153) 2! 2! 3! Passing to the variable τ, from relations (3.152), (3.153), we obtain x0 (ετ) + εx1 (ετ) = C0 − εC0 τ + εC1 + O(ε 2 τ 2 ) at τ → 0.

(3.154)

For the inner solution, we have X0 (τ) + εX1 (τ) = 1 + ε(2 − τ) + ... at τ → ∞.

(3.155)

It is important that there is a region where both asymptotic expansions (3.154) and (3.155) are suitable. It is easy to check that in the region t ∼ ε α (τ ∼ ε α−1 ) at 0.5 < α < 1, the terms discarded when writing expressions (3.154) and (3.155) are smaller than those left. In the region of matching, we have x0 (t) + x1 (t)ε − [X0 (τ) + εX1 (τ)] = O(ε 2α ) at ε → 0 if we suppose C0 = 1, C1 = 2. Now the outer asymptotic expansion has the form x0 (t) + x1 (t)ε − e−t + ε(2 − t)e−t ,

(3.156)

X0 (τ) + εX1 (τ) ∼ 1 + ε[τ + 2(1 − e−τ )].

(3.157)

and the inner one is

Expressions (3.156) and (3.157) have common terms, namely, those that have been matched. Adding expressions (3.156) and (3.157) and subtracting this common part, we obtain a uniformly applicable asymptotic representation of the solution: x ∼ e−t + ε(2 − τ)e−t − 2εe−τ .

87

SINGULAR ASYMPTOTIC EXPANSIONS

3.10

ON BOUNDARY VALUE PROBLEMS OF NON-CLASSICAL THEORIES OF BEAMS, PLATES, AND SHELLS

Non-classical theories of beams, plates, and shells (Reissner, Timoshenko-Ehrenfest, Mindlin-Uflyand, etc.) have a large scope [136]. Non-classical theories differ from ordinary classical ones by the presence of additional terms in differential equations. Such theories often make it possible to describe effects, “lost” in classical theories. Usually such theories are based on physical considerations. Later, for many such theories it was shown that they can be treated as first approximations of some asymptotic processes. However, it should be remembered that, for example, taking into account three-dimensional boundary layers that appear near the ends of the shell (plate) can radically change the applicability of these theories [130, 152]. Let us consider simple examples, the equation of oscillations of a stretched beam ρFwtt − T wxx + EIwxxxx = 0.

(3.158)

We take the boundary conditions in the form w = wxx = 0

at x = 0, L.

(3.159)

Weptransform the system (3.158), (3.159) by setting ξ = x/L ; ε 2 = EI/(T L2 ) ; τ = t T /(ρF): wττ − wξ ξ + ε 2 wξ ξ ξ ξ = 0, (3.160) w = wξ ξ = 0

at ξ = 0, 1.

(3.161)

The string model is obtained from (3.158), (3.159) at ε = 0 wττ − wξ ξ = 0, w=0

at ξ = 0, 1.

(3.162) (3.163)

Equation (3.162) is of the second order in the longitudinal coordinate. Let us show that it is possible to improve the approximation accuracy with retaining the second order of the approximate equation. For the differential operator −∂ 2 /∂ ξ 2 + ε 2 ∂ 4 /∂ ξ 4 , we apply the Pad´e approximation, replacing it with the following (see section 7.2 for details): −∂ 2 /∂ ξ 2 . 1 + ε∂ 2 /∂ ξ 2 Then equation (3.160) can be rewritten up to ε 2 as follows:   ∂2 1 + ε 2 2 wττ − wξ ξ = 0. ∂ξ

(3.164)

The boundary conditions for equation (3.164) have the form (3.163). Equation (3.164) is called Love’s equation, although Love himself noted [183] that it was previously obtained by Rayleigh [287]. Model (3.162), (3.163) approximates the eigenvalues of the original problem up to ε 2 , model (3.163) - up to ε 4 , while the order of the equation in the spatial coordinate does not change.

88

Asymptotic Methods for Engineers

Equation (3.164) can be obtained in another way. To do this, act on equation (3.160) by the operator 1 + ε 2 ∂ 2 /∂ ξ 2 , and then discard terms of the order of smallness higher than the second. Let us now consider the question of the boundary conditions for the equation of the first approximation (3.164). In the case of simply support the solution of the degenerate equation satisfies the boundary conditions (3.161). Let now the beam is clamped at the ends w = wξ = 0 at ξ = 0, 1. Then the solution of the equation (3.164) does not satisfy the boundary conditions for the angle, and it is necessary to add the boundary layer: wb = wb0 + εwb1 + ε 2 wb2 + ... Given that wb0ξ ∼ ε −1 wb0 , we obtain in the first approximation from equation (3.160) ε 2 wb0ξ ξ − wb0 = 0, εwb0ξ = −w0ξ

at ξ = 0, 1.

The solution of equation (3.165) has the form   wb0 = C1 exp(−ε −1 ξ ) +C2 exp(ε −1 (ξ − 1)) exp(iωτ)

(3.165) (3.166)

(3.167)

and introduces a discrepancy of order ε into the boundary conditions with respect to w w0 + εwb0 6= 0 at ξ = 0, 1. Therefore, equation (3.164) with boundary conditions (3.163) cannot be used. Let us apply method described in section 3.1. Substitute the solution (3.167) to the boundary conditions (3.166), we find C1 = w0ξ (0), C2 = w0ξ (1). Now we can formulate the boundary conditions for equation (3.164) as follows: w0 − εw0ξ = 0

at ξ = 0, 1.

(3.168)

The boundary value problem (3.164), (3.168) gives a solution up to ε 2 .

3.11

KIRCHHOFF’S AND BOLOTIN’S APPROXIMATIONS IN THE THEORY OF NON-LINEAR BEAMS OSCILLATIONS

We have included this section in our book for the following reasons. First, Kirchhoff’s and Bolotin’s approximations are often used in non-linear beam dynamics. Secondly, their derivation allows us to once again demonstrate the asymptotic procedure for obtaining approximate models.

89

SINGULAR ASYMPTOTIC EXPANSIONS

Kirchhoff [158] proposed simple approximate equations of non-linear beam vibration, which became very popular [11, 154]. Let us briefly discuss this approximation. Consider the governing equations of non-linear beam vibration in the following form   ∂ 2W ∂ 2 M ∂ ∂W ρF 2 + − T = 0, (3.169) ∂t ∂ x2 ∂x ∂x ρF

∂ 2U ∂ T − = 0, ∂t 2 ∂x

(3.170)

 2 2 ∂W + 0.5 ; E is the Young’s moduwhere: M = EIκ , T = EFε, κ = ∂∂ xW2 , ε = ∂U ∂x ∂x lus; F, I are the area and the static moment of transversal beam cross section, respectively; κ is the curvature; U,W are the longitudinal and normal beam displacements; ρ is the density of beam material; t is the time, and x is the spatial coordinate. Below, we consider two cases of boundary conditions in the axial direction: prescribed end shortening or dead (forced) loading: U = U (0)

at x = 0,

U = U (L)

at x = L,

(3.171)

or T = T (0)

at x = 0,

x = L.

(3.172)

The Kirchhoff hypothesis is that the axial inertia in equation (3.170) can be neglected. Then one obtains   ∂T ∂U ∂W 2 = 0, i.e. ε = + 0.5 = N(t). (3.173) ∂x ∂x ∂x Upon integration of relation (3.170) with taking into account boundary conditions (3.171) we have 1 Ub + N= L 2L

ZL  0

∂W ∂x

2 dx,

Ub = U (L) −U (0) .

Using equations (3.169) and (3.174), one obtains    ZL  ∂ 2W ∂ 4W EF  1 ∂W 2  ∂ 2W ρF 2 + EI 4 + Ub − dx = 0. ∂t ∂x L 2 ∂x ∂ x2

(3.174)

(3.175)

0

Equation (3.175) describes the approximate Kirchhoff model [154, 158]. It is worth noting that in Kirchhoff’s book [158] equation (3.175) is not presented. Kirchhoff in spite of neglecting the longitudinal inertia in equation (3.170) has also omitted the second term in equation (3.169), and the original ’Kirchhoff equation’ has the form  ZL  ∂W 2 ∂ 2W ∂ 2W EF ρF 2 − dx 2 = 0. ∂t 2L ∂x ∂x 0

90

Asymptotic Methods for Engineers

May be that is why for equation (3.175) sometimes used term “Mettler equation”, referring to the work [209]. For axial boundary conditions (3.171) equation (3.169) is linearized, and takes the following form ρF

∂ 4W ∂ 2W ∂ 2W + EI 4 − T (0) 2 = 0. 2 ∂t ∂x ∂x

(3.176)

Bolotin [63] mentioned that in some cases (e.g., if compressed load is near the buckling value) the so-called “non-linear inertia” must be taken into account. Bolotin used some physical assumption and did not compare order of linear and non-linear terms in original equations. Below we propose asymptotic way for obtaining these equations, which we name “Bolotin’s equations”. We show that Bolotin’s approach leads to system of two equations; the first of them is Kirchhoff-type equation and the second one takes into account non-linear inertia of the system. These equations stand for the first and the second approximation of asymptotics, by using the quantity p δ = h/L > 1. This is due to the fact that when expanding in normal forms, summation over k from 1 to n is used for a discrete system, and integration over x from 0 to l is used

118

Asymptotic Methods for Engineers

for a continuous system. The situation can be corrected using the Euler-Maclaurin formula [106, 108] n+1 Z

n+1

1 f (x)dx + [ f (0) + f (n + 1)]+ 2 0   j ∞ (−1) j+1 d f (n + 1) d j f (0) − . B j ∑ dx j dx j j=1 j + 1

∑ f (k) = k=0

(5.49)

Here Bi are Bernoulli numbers, the partial values of which are as follows: B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0 [1]. The following recursive formula is also true [1]: Bn = −

1 n k+1 ∑ Cn+1 Bn−k . n + 1 k=1

When expanding solution of a discrete system in normal forms, it is necessary to calculate some sums, the values of which are determined by formulas 4.4.2.6, 4.4.1.5, and 4.4.1.7 from the handbook [254] n+1

kπ j

∑ sin2 n + 1 =

k=0 n+1 

∑ k=0

1−

n+1 , 2

 j kπ j 1 jπ sin = cot . n+1 n + 1 n + 1 2(n + 1)

(5.50)

(5.51)

The corresponding integrals are n+1 Z

n+1 π jx dx = , n+1 2

(5.52)

π jx 2 x sin dx = . l n+1 πj

(5.53)

sin2

0 n+1 Z 

1−

0

The values of the sum (5.50) and the integral (5.52) coincide. Using the EulerMaclaurin formula, one can obtain the value of the continuous projection, which is closer to the value of the sum (5.51) n+1  Z  j kπ j x π jx sin = 1− sin dx+ n+1 n+1 l n+1 0 " # 1 jπ 2 π 2 j2 . sin( jπ) − + ... = 1− 2 6(n + 1) πj 12(n + 1)2

n+1 

∑ k=0

1−

(5.54)

119

CONTINUALIZATION

Based on formula (5.54), one can construct a simple expression that approximates sum (5.51) well for any values of j from j = 1 to j = n. For this purpose, we change the second term on the right side of equality (5.54) as follows: " #  n+1  2 kπ j j2 j ∑ 1 − n + 1 sin n + 1 ≈ π j 1 − (n + 1)2 . k=0

5.8

LOGISTIC EQUATIONS: DISCRETIZATION AND CONTINUALIZATION

Verhulst is credited with formulating the ODE   N dN = rN 1 − , dt K

(5.55)

and coining the name logistic [314]. Later investigators proposed variations on Verhulst equation (e.g., difference logistic equation), sometimes continuing to refer to these as logistic models. The first application of ODE (5.55) was connected with population problems, and more generally, problems in ecology. If the Verhulst model is used for describing change in population size N over time t, then in equation (5.55) r is the Malthusian parameter (rate of maximum population growth) and K is the carrying capacity (i.e., the maximum sustainable population). Equation (5.55) is widely used in problems of ecology, economics, chemistry, medicine, pharmacology, epidemiology, etc. [201, 221]. As a rule, this model is oversimplified for quantitative estimations, but reflected the key qualitative features of processes under consideration. Equation (5.55) can be reduced to the form dx = rx(1 − x), dt

(5.56)

x(0) = a.

(5.57)

where x = N/K. Initial condition is The Cauchy problem (5.56), (5.57) has the exact solution x=

a . a + (1 − a)e− rt

(5.58)

The discrete logistic equation can be written as follows: xn+1 = Rxn (1 − xn ),

(5.59)

where parameter R characterizing the rate of reproduction (growth) of the population; R = rh, parameter h defines the time between consecutive measurements. The nonlinear difference equation (5.59) exhibits period doubling to chaos [79, 201, 221].

120

Asymptotic Methods for Engineers

We analyze a slightly different discrete logistic equation xn+1 − xn = Rxn (1 − xn ).

(5.60)

Difference equation (5.60) is obtained from ODE (5.56) using a forward difference scheme for the derivative with the step of discretization h. O∆E (5.60) close enough in behavior of solutions to the equation [201, 221] xn+1 = xn exp[R(1 − xn )].

(5.61)

Initial condition for O∆Es (5.59), or (5.60), or (5.61) is x0 = a.

(5.62)

The discrete Cauchy problems (5.59), (5.62), or (5.60),(5.62), or (5.61), (5.62) for sufficiently large values of the parameter R describe the complex, chaotic behavior of the system [201, 221]. For O∆E (5.60), for R = 2.3 the solution starts to oscillate periodically around the value x = 1. This solution is stable as long as R < √ 6 ≈ 2.449. For R = 2.500, the process comes to steady periodic oscillations with a period 4. It can be mentioned that chaotic threshold for O∆E (5.61) is 2.6824 [201, 221]. Both continuous and discrete logistic equations has been extensively investigated. Our study focuses on the discretization and continualization of nonlinear ODE and O∆E, while the logistic equations serve only as convenient and simple examples. First problem: is it possible to discretize ODE (5.56) in such a way that the resulting O∆E has only regular solutions? On the other hand, many researchers point that discrete logistic models to be more adequate to the essence of the physical, economic or biological processes precisely because they have chaotic regimes. In this regard, second problem: can be proposed such a continualization of the original O∆E, that the resulting O∆E has chaotic solution? It is difficult to expect that standard continualization, based on the use of the Taylor series, will provide the desired result. It remains to be hoped for the techniques based on the use of Pad´e approximants. As it is mentioned in [79], non-invertible maps, such as the logistic map, may display chaos. Of interest is the transformation of the original discrete logistic equation into a form leading to deterministic solutions. We rewrite O∆E (5.60) in the following form xn+1 − xn = Rxn − Rxn+1 xn .

(5.63)

This presentation makes it possible to express xn+1 not a polynomial, but a fractional rational function xn . Equation (5.63) with initial condition (5.62) has the exact solution of the form a . (5.64) xn = a + (1 − a)(1 + R)−n Thus, representation (5.63) allows one to obtain a difference scheme without chaotic behavior.

121

CONTINUALIZATION

Let us try to construct a continuous model, i.e., ODE, describing the chaotic behavior like original O∆E. As it is mentioned in [79], for generating chaotic behavior nonlinear ODE must have dimension D ≥ 3. For continualization of O∆E (5.60), let us introduce the continuous coordinate x scaled in such a way that xn = x(nh). Suppose x(t) slightly changing function we use Maclaurin expansion xn+1 − xn = hxt +

h2 h3 xtt + xttt + ... 2 6

(5.65)

third order ODE, obtained using usual continualization h3 xttt + 3h2 xtt + 6hxt − 6Rx + 6Rx2 = 0,

(5.66)

describes completely deterministic trajectories. Let us consider the second equilibrium position, xn = 1. Using changing of variables |yn | 0, δ = 0.

132

Asymptotic Methods for Engineers

The corresponding limit system is     1 2 2 2 2 1 + ε θ Fθ1θ − + θ − ε θ θ w 10 20 η η ξ 12 (1 − ν 2 ) θ θ θ θ ξ 0 (Fξ0ξ θη2 + 2Fξ0η θξ θη + Fηη θξ2 )w1θ θ + w1ττ = 0,  2   Fθ1θ θ θ θξ2 + θη2 − ε20 θξ2 + ε10 θη2 w1θ θ = 0,

1 2 (w1θ ) θξ θη , 2 2(1 − ν ) 1 2 ε −1 Fθ1θ θξ θη + Fξ0ξ = (w1 ) θ θ , 2(1 − ν 2 ) θ ξ η 1 2 (w1 ) θ θ . ε −1 Fθ1θ θξ θη + Fξ0η = − 2 (1 + ν) θ ξ η 0 ε −1 Fθ1θ θξ θη + Fηη =

(6.38)

(6.39) (6.40) (6.41) (6.42)

The derivatives of the function F 0 entering equation (6.38) are determined from equations (6.40)-(6.42) after averaging the latter over θ . Moreover, due to the periodicity of the function Fθ1 , the equality ZT

Fθ1θ dθ = 0

0

takes place. As a result, we get ZT h 1 2 2 = (w1θ ) (θξ2 + θη2 ) − 2 2 (1 − ν ) (6.43) 0 h ii 1 2 2 2w ε10 (θξ + νθη ) + ε20 (θη + θξ ) dθ .

− (Fξ0ξ θη2 + 2Fξ0η θξ θη

0 + Fηη θξ2 )w1θ θ

Taking into account relation (6.43), we obtain the following simplified equations in the original variables   1 ∂2 1 ∂2 D 4 ∇ w− + F− h R2 ∂ x2 R1 ∂ y2   Za Zb  Zb Zb    ν E 1 2 2 2 0.5∇ w wx + wy dxdy − wxx + wdy dx− ab(1 − ν 2 )  R1 R2 a a

Za Zb 

0 0



 

1 ν + wdy dx + ρwtt = 0, (6.44)  R1 R2 0 0   D 4 1 ∂2 1 ∂2 ∇ F− + w = 0. (6.45) h R2 ∂ x2 R1 ∂ y2 At R1 → ∞, R2 → ∞ equation (6.44) goes over into the Berger equation [52], and in the one-dimensional case, into the Kirchhoff equation (see section 3.12). wyy

133

AVERAGING AND HOMOGENIZATION METHODS

6.5

DIFFERENTIAL EQUATIONS WITH RAPIDLY VARYING COEFFICIENTS

Let us demonstrate the homogenization method on a simple one-dimensional problem [35, 48, 191, 236]     d x du a = q(x), (6.46) dx ε dx u=0

at

x = 0, L.

(6.47)

Here a(x/ε) is a function that is periodic in x with period ε. The variability of the right side of equation (6.46) is small, the variability of the coefficient a(x/ε) is large. Therefore, one can use the method of multiple scale (section 3.8) and introduce “fast” η = x/ε and “slow” y = x variables. Then the derivative can be rewritten like this: ∂ ∂ d = + ε −1 . (6.48) dx ∂ y ∂η We will seek solution in the form of expansion u = u0 (η, y) + εu1 (η, y) + . . . ,

(6.49)

where u0 , u1 , ... are functions periodic in η with period one. Substituting expressions (6.48), (6.49) into the original equation (6.46) and boundary conditions (6.47) and equating the terms at the same powers of ε, we arrive at the following recurrent system of equations   ∂ ∂ u0 a(η) = 0, (6.50) ∂η ∂η     ∂ u0 ∂ 2 u0 ∂ ∂ u1 ∂ a(η) + a(η) + a(η) = 0, (6.51) ∂η ∂y ∂ y∂ η ∂ η ∂η     ∂ u2 ∂ 2 u0 ∂ ∂ u1 ∂ 2 u1 ∂ a(η) + a(η) 2 + a(η) + a(η) = q(y), (6.52) ∂η ∂η ∂y ∂η ∂y ∂ y∂ η ... uj = 0

at y = 0,

η = 0, L/ε,

j = 0, 1, 2, 3, ...,

(6.53)

where [...] means ingeger part of a number (...). From equation (6.50), due to the periodicity of the function u0 in η, we have u0 = u0 (y), i.e., u0 is some average, independent of the fast variable, part of the function u. In a number of physical problems, the existence in the solution of the slowly changing part is clear from their formulations. In this case in the expansion (6.49), one can consider the first term to be independent of the fast variable. Equation (6.51) now takes the form   ∂ ∂ u1 ∂ a(η) du0 a(η) =− . (6.54) ∂η ∂η ∂ η dy

134

Asymptotic Methods for Engineers

This equation is considered on the period (0 ≤ η ≤ 1); it is called the “problem on the cell” or “local problem”. The solution of the problem on one cell is much simpler than for the entire region. In our case, we get ∂ u1 ∂ u0 C(y) =− + . ∂η ∂y a

(6.55)

From the periodicity condition u1 |10 = 0 we define the constant C(y): −1

 C = aˆ

du0 , dy

Z1

aˆ = 

a−1 dη  .

0

Eliminating the function ∂ u1 /∂ η from equation (6.52), we obtain     ∂ ∂ u2 ∂ ∂ u1 d 2 u0 a + a + aˆ 2 = q(y). ∂η ∂η ∂η ∂y dy

(6.56)

Now, to extract slow components from equation (6.56), it is natural to use the R1

homogenization operator (...)dη. The first two terms vanish as a result of averaging 0

due to periodicity, and finally from equation (6.56), we have aˆ

d 2 u0 = q(y). dy2

(6.57)

For equation (6.57), the boundary condition must be set u0 = 0

at

y = 0, L.

(6.58)

Let us now define the function u1 from relation (6.55)   Z1 du0  u1 = aˆ a−1 dη − η  , 0 ≤ η ≤ 1. dy 0

Further, the function u1 periodically continues along the coordinate η with a period one. The found value of u1 in general case does not satisfy the boundary conditions (6.47), while the corresponding residuals are of order ε. To compensate for them, we arrive at a problem that in the original variables has the form     d x du a = 0, dx ε dx

135

AVERAGING AND HOMOGENIZATION METHODS

u|x=0 = A = u1 |y=η=0 ,

u|x=L = B = u1 |y=L,η=|L/ε−[L/ε]| .

Applying the homogenization method to this problem again, we obtain in the first approximation aˆ

d 2 u01 = 0, dy2

u01 |y=0 = A,

u01 |y=L = B.

This naturally leads to the idea to seek the solution of problem under consideration in the form of expansion u = u0 (y) + ε[u01 (y) + εu02 (y) + ε 2 u03 (y) + . . .]+ ε[u1 (η, y) + εu2 (η, y) + ε 2 u3 (η, y) + . . .].

(6.59)

Here ui (η, y) are functions with zero mean over the period. Let us consider one more model example related already to the non-linear equation     x d x du a +b u3 = q(x), (6.60) dx ε dx ε u=0

at

x = 0, L.

(6.61)

Introducing, as before, the fast and slow variables η and y and representing the function u in the form (6.49), we obtain the following recurrence relations:   ∂ ∂ u1 da(η) du0 a(η) + = 0, (6.62) ∂η ∂η dη dy     ∂ ∂ u2 ∂ ∂ u1 a(η) + a(η) + (6.63) ∂η ∂η ∂η ∂y a(η)

∂ 2 u1 d 2 u0 + a(η) 2 + b(η)u30 = q(y), ∂ y∂ η dy ... u0 = 0

u1 = 0

at

at

y = 0, L,

y = 0, L,

η = 0, [L/ε],

(6.64)

... Equation (6.62) coincides with equation (6.51) - the local problem does not change when new terms are added to the equation without changing the higher derivatives. Using solution (6.55), we find the homogenized equation aˆ

d 2 u0 ˆ 3 + bu0 = q(y), dy2

bˆ =

Z1

b(η)dη. 0

The boundary conditions for equation (6.65) have the form (6.58).

(6.65)

136

Asymptotic Methods for Engineers

Let us pay attention to an important fact: u = u0 + O(ε),

but

du du0 ∂ u1 = + + O(ε). dx dy ∂η

In other words, although the solution of the homogenized equation u0 approximates the function u up to terms of the order of ε, the terms with u1 must be retained in the expression for the derivative, since the latter increase strongly upon differentiation. It is their presence that complicates the process of direct numerical solution. Let us pass to the physical interpretation of the coefficients of the homogenized equation (6.65). It can be seen that the stiffness b and compliance 1/a are averaged. Averaging of rigidity is often called Voigt averaging [88, 322], averaging of compliance - Reuss averaging [88, 259] (see also section 8.5). Voigt and Reuss estimations are the arithmetic mean and harmonic mean of the matrix and inclusion characteristics of the composites. For a wide class of problems, it is known that the true values of the coefficients of the averaged equations (6.65) a˜i j are between the values of the coefficients averaged according to Voigt (a¯i j ) and Reuss (aˆi j ): aˆi j ≤ a˜i j ≤ a¯i j .

(6.66)

Estimate (6.66) is often called the “Voigt-Reuss fork” or “Hill’s fork” [88], although it was apparently first obtained by Wiener [328]. Unfortunately, the interval between these estimates is often too large. For more precise bounds, see section 8.5.

Figure 6.2 Comparison of the results of homogenization of the Laplace equation in a periodically inhomogeneous medium (solid curve) with the estimates of Voigt (curve of points) and Reuss (dotted curve) [69].

For an example in Fig. 6.2 presents the results of calculating the averaged conductivity of the composite material, consisting of a matrix with square inclusions.

137

AVERAGING AND HOMOGENIZATION METHODS

The original problem is described by the Laplace equation in a periodically inhomogeneous medium. The periodicity cell is a square with a side of 1, the inclusions are located symmetrically with respect to the center of this square and have a side equal to 1/3, the ratio of the conductivities of the matrix and the inclusion is denoted by the letter d0 . The dotted curve shows the Reuss estimate, and the points show the Voigt estimate. The solid line shows the results of homogenization with the numerical solution of the problem on a cell [69]. Fig. 6.2 gives understanding of the applicability of estimates (6.66) in practice. Consider now the eigenvalue problem     x du d a + λ u = 0, dx ε dx (6.67) u = 0 at x = 0, L. We represent the desired eigenform in the form (6.59), and the eigenvalue λ in this way: λ = λ0 + ελ1 + ε 2 λ2 + . . . (6.68) Substituting the expansions (6.59), (6.68) into the original boundary value problem (6.67) and taking into account the expressions for the derivative (6.48), we obtain the following recurrent system of equations:   ∂ ∂ u1 ∂ a du0 + a = 0, (6.69) ∂ η dy ∂η ∂η     ∂ ∂ u2 ∂ ∂ u1 ∂ 2 u1 ∂ a du01 d 2 u0 a + a +a + + a 2 + λ0 u0 = 0, (6.70) ∂η ∂η ∂η ∂y ∂ y∂ η ∂ η dy dy     ∂ ∂ u3 ∂ ∂ u2 ∂ a du02 ∂ 2 u2 a + a + +a + ∂η ∂η ∂η ∂η ∂ η dy ∂ y∂ η (6.71) d 2 u01 + λ u + λ (u + u ) = 0, a 1 0 0 01 1 dy2 ...

u1 + u01 = 0

u0 = 0

at

y = 0, L

at

y = 0, L;

η = 0, L/ε − [L/ε],

(6.72) (6.73)

... Determining the value ∂ u1 /∂ η from equation (6.69), substituting it into equation (6.70) and boundary conditions (6.72) and averaging, we arrive at the boundary value problem for u0 , λ0 : aˆ

d 2 u0 + λ0 u0 = 0, dy2

u0 = 0

at y = 0, L.

138

Asymptotic Methods for Engineers

Now from equation (6.70), we find: ∂ u2 ∂ u1 du01 C1 (y) =− − + . ∂η ∂y dy a From the condition of the periodicity of the functions u2 on the variable η, we determine Z1 du01 ∂ uˆ1 C1 = aˆ + aˆ , where uˆ1 = u1 dη. dy ∂y 0

Substituting u1 , u2 into equation (6.71) and averaging, we have aˆ

d 2 u01 ∂ 2 uˆ1 + λ0 u01 + aˆ 2 + λ0 uˆ1 + λ1 u0 = 0. 2 dy ∂y

(6.74)

The boundary conditions for equation (6.74) are obtained from conditions (6.73) and have the form u01 = −b u1 at y = 0, L. (6.75) The correction to the oscillation frequency λ1 is determined as usual in the perturbation method (see section 2.2), after which the slow correction to the averaged solution u01 is found from the solution of the boundary value problem (6.74), (6.75). The approach outlined above makes it possible to determine the solution in any approximation with respect to ε. Another attractive feature is its generality. Indeed, if a solution to a local problem is found, then the solution of both the original problem and the eigenvalue problem can be easily determined. If we add non-linear terms to the equations (so that the higher derivatives do not change), then the construction of averaged relations does not become more complicated again. The local problem remains exactly the same as in the linear case; higher approximations will also be linear. All non-linearity is contained in averaged boundary value problems with smooth coefficients, which are conveniently solved numerically or by variational methods.

6.6

DIFFERENTIAL EQUATIONS WITH PERIODICALLY DISCONTINUOUS COEFFICIENTS

Consider the application of the homogenization method for solving differential equations with periodically discontinuous coefficients. They could also be characterized as problems with periodic barriers [318]. As a model example, let us study the deformation of a membrane reinforced with threads. We suppose 0 ≤ x1 ≤ H, −∞ < y1 < ∞. The equilibrium equations in the intervals kl < y1 < (k + 1)l can be written as follows: ∂ 2 u1 ∂ 2 u1 + = Q(x1 , y1 ). (6.76) ∂ x12 ∂ y21

139

AVERAGING AND HOMOGENIZATION METHODS

The contact conditions for neighboring sections, also called conjugation or jump conditions [318], have the form lim u ≡ u+ =

y1 →kl+0



∂u ∂ y1

lim u ≡ u− ,

y1 →kl−0

+

 −

∂u ∂ y1

− =p

k = 0, ±1, ±2, ..., (6.77)

∂ 2 u1 , ∂ x12

where p is the parameter characterizing the relative stiffness of the thread. The boundary conditions are as follows: u=0

at x1 = 0, H.

(6.78)

Let the external load be periodic in y1 , and let its period L be much greater than the distance between the threads. Then it is natural to use the averaged description, taking the value ε = l/L as a small parameter. Instead of the variable y1 , we introduce “fast” (η = y1 /l) and “slow” (y = y1 /L) variables, then   ∂ 1 ∂ −1 ∂ = +ε . (6.79) ∂ y1 L ∂y ∂η Function u can be represented by expansion: u = u0 (x, y) + ε α [u01 (x, y) + u1 (x, y, η)]+ ε α1 [u02 (x, y) + u2 (x, y, η)] + . . . ,

(6.80)

where 0 < α < α1 , x = x1 /L. Substituting (6.80) into equation (6.76) and matching conditions (6.77) and taking into account the expression for the derivative (6.79), we have ∂ 2 u1 ∂ 2 u1 ∂ 2 u2 + 2ε α−1 + + ε α1 −2 2 ∂η ∂ y∂ η ∂ η2 ∂ 2 u2 2ε α1 −1 + O(ε α ) = q(x, y), ∂ y∂ µ

∇2 u0 + ε α−2

(6.81)

[u0 + ε α (u01 + u1 ) + . . .]+ = [u0 + ε α (u01 + u1 ) + . . .]− , "    #  2  ∂ u1 − ∂ u0 ∂ u1 + α α α−1 − + O(ε ) = p1 + O(ε ) , ε ∂η ∂η ∂ x2

(6.82)

where q = L2 Q,

p1 = p/L,

∇2 u0 =

∂ 2 u0 ∂ 2 u0 + , ∂ x2 ∂ y2

(...)± = lim u. η→k±0

We note that the vast majority of works on the homogenization of periodic systems, especially purely mathematical ones, were carried out under the implicit assumption that the parameters included in the system are of the order of unity. However, the nature of the construction of the asymptotics essentially depends on the

140

Asymptotic Methods for Engineers

order of the relative rigidity of the thread p1 in comparison with the parameter ε. Let us introduce a parameter β characterizing this order (p1 ∼ ε β ), and analyze the possible form of limit systems depending on α, β . Judging by equation (6.81), different limit systems take place at 0 < α < 2, α = 2 and α > 2: ∂ 2 u1 = 0, (6.83) at 0 < α < 2, ∂ η2 at α = 2,

∇2 u0 +

at α > 2,

∂ 2 u1 = q, ∂ η2

∇2 u0 = q.

(6.84) (6.85)

The limit relations obtained from relation (6.82) at ε → 0 have the form ∂ 2 u0 = 0, ∂ x2     ∂ u1 + ∂ u1 − ∂ 2 u0 at β = α − 1, − = p1 ε 1−α , ∂η ∂η ∂ x2     ∂ u1 − ∂ u1 + = . at β > α − 1, ∂η ∂η at β < α − 1,

(6.86) (6.87) (6.88)

The “quarter-plane” of the parameters β > 0, α > 0 is divided into nine regions (Fig. 6.3).

Figure 6.3 Partitioning the quarterplane of parameters α, β which determines the zones of different asymptotics (asymptotic portrait of the problem under consideration).

Let us move on to their detailed study. Let β < α − 1. Physically, this means that the threads are rigid. From equation (6.86), we then have u0 = 0; hence, the homogenized description does not work here. The corresponding limit equation for zones 1 - 3 has the form ∂ 2 u1 = q. (6.89) ∂ η2

AVERAGING AND HOMOGENIZATION METHODS

141

The case β > α − 1 corresponds to zones 4 - 6. Physically, this is the case of weak threads, when their influence is small, and the limit equation has the form (6.85). For zones 7 and 8, systems are obtained that do not have a physical meaning. Of particular interest is the case α = 2, β = 1 (zone 9) - “average” threads in terms of stiffness. Then the limit system contains equations (6.84), (6.87), and the transition conditions take the form − u+ (6.90) 1 = u1 , +  −  ∂ u1 ∂ u1 ∂ 2 u0 − = p2 2 , (6.91) ∂η ∂η ∂x where p2 = p/l. From equation (6.84) it is easy to determine u1 : u1 = 0.5(q − ∇2 u0 )η 2 +C(x, y)η +C1 (x, y). The constant C1 (x, y) should be attributed to the component u01 , which is determined from the averaged equations of the following approximations. From conditions (6.90), we determine C(x, y): C(x, y) = −0.5(q − ∇2 u0 )L.

(6.92)

It is also necessary to satisfy condition (6.91), but there are no more arbitrary constants. However, it is easy to see that condition (6.91) gives the required homogenized equation. Indeed, substituting in (6.91) the found value u1 , we obtain ∇2 u0 + p2

∂ 2 u0 = q. ∂ x2

(6.93)

Equation (6.93) plays the role of solvability condition. It should be integrated under the boundary conditions u0 = 0

at

x = 0, H/L.

Physically, the transition to equation (6.93) corresponds to the “smearing” of the stiffness of the threads (transition to the structurally-orthotropic theory). The function u1 can be finally represented as u1 = 0.5p2

∂ 2 u0 η(η − 1). ∂ x2

The boundary conditions at the ends of the strip are generally not satisfied in this case. The edge discrepancy is rapidly variable in η and leads to the appearance of a boundary layer ub . To construct it, we introduce the fast variable ξ = x1 /l and the following ansatz ub = ε γ1 ub1 (x, y, ξ , η) + ε γ2 ub2 (x, y, ξ , η) + . . . , where 0 < γ1 < γ2 < . . .

142

Asymptotic Methods for Engineers

The equations for the function ub1 take the form ∂ 2 ub1 ∂ 2 ub1 + = 0, ∂ξ2 ∂ η2 ub1 |η=k = 0

k = 0, ±1, ...

The boundary conditions (we consider only one edge, for the second one everything turns out similarly) for x = ξ = 0 are as follows: ub1 = −u1 . To actually construct the boundary layer, one can apply the Kantorovich method [151], representing ub1 in a form that satisfies the boundary conditions at η = 0, l: ub1 = Φ(ξ )η(η − l). Next, the standard procedure of the Kantorovich method [151] is applied. Let us now clarify the concepts of “rapidly” and “slowly” changing loads [318]. The function f (ε, θ ) is called oscillating with the rate ε −1 on the period 2π if Zα Z2π f (ε, θ )dθ ≤ Cε, 0 < C1 ≤ | f (ε, θ )|2 dθ ≤ C2 < ∞, 0 ≤ α ≤ 2π, 0 0 where C,C1 ,C2 - some constants.

6.7

PERIODICALLY PERFORATED MEDIA

The Poisson equation is considered ∇2 u = f (x, y)

(6.94)

in the multiply connected region Ω (Fig. 6.4) [135]. The small parameter ε characterizes the ratio of the characteristic size of the repeating section to the characteristic size of the membrane. Neumann boundary conditions are set on the boundaries of the holes ∂u =0 ∂ ni

at

∂ Ωi ,

(6.95)

where ni is the outer normal to the contour of the i-th hole. The edges of the membrane are rigidly fixed u=0

at

∂ Ω.

(6.96)

We introduce “fast” variables ξ = x/ε, η = y/ε. The solution is sought in the form of expansion u = u0 (x, y) + εu1 (x, y, ξ , η) + ε 2 u2 (x, y, ξ , η) + . . . , where u j , j = 1, 2, ... are the functions periodic in ξ , η with period 1.

(6.97)

AVERAGING AND HOMOGENIZATION METHODS

143

Figure 6.4 Perforated medium.

The partial derivatives are now: ∂ ∂ ∂ = + ε −1 , ∂x ∂x ∂ξ

∂ ∂ ∂ = + ε −1 . ∂y ∂y ∂η

(6.98)

The view of a periodically repeating cell with a hole in fast variables is shown in Fig. 6.5.

Figure 6.5 Periodically repeated cell.

Substitute ansatz (6.97) into the boundary value problem (6.94)–(6.96) and taking into account expressions (6.98), after splitting with respect to ε one obtained a

144

Asymptotic Methods for Engineers

recurrent sequence of boundary value problems ∂ 2 u1 ∂ 2 u1 + =0 ∂ξ2 ∂ η2

in Ωi ,

∂ u1 ∂ u0 + = 0 at ∂ Ωi , ∂k ∂n  2  ∂ 2 u0 ∂ 2 u0 ∂ u1 ∂ 2 u1 ∂ 2 u2 ∂ 2 u2 + + 2 + =f + + 2 2 ∂x ∂y ∂ x∂ ξ ∂ y∂ η ∂ξ2 ∂ η2 ∂ u2 ∂ u1 + =0 ∂k ∂n ... ui = 0,

at ∂ Ωi ,

i = 0, 1, 2, ... at ∂ Ω,

(6.99) (6.100) in Ωi ,

(6.101) (6.102)

(6.103)

where k the outer normal to the hole contour in fast variables. We define the averaging operator as follows: ˜ y) = Φ(x,

ZZ

Φ(x, y, ξ , η)dξ dη.

(6.104)

Ωi

From equation (6.101), we obtain after applying the averaging operator (6.104)   2
∂ u0 ∂ 2 u0 + 1 − πa2 + 2 2 ∂x ∂y  ZZ  2 (6.105)
∂ u1 ∂ 2 u1 + dξ dη = 1 − πa2 f . ∂ x∂ ξ ∂ y∂ η Ωi

The averaged boundary condition has the form u0 = 0

at ∂ Ω.

(6.106)

Now we need to solve the problem on the cell (6.99), (6.100) with periodic continuation conditions, i.e., conditions for the equality of the function u1 and its first-order partial derivatives with respect to the corresponding variables on opposite sides of the cell. The reduction of periodic problems to boundary value problems is described, for example, in [35]. For the case under consideration, the periodic problem can be divided into two. In both cases, the displacements on two opposite sides of the outer boundary of the cell and the normal derivatives on the other two sides are equal to zero. Let the hole diameter 2a be small compared to the cell size. Then, in the first (1) approximation, for the function u1 (u1 ≈ u1 ), we can pass to the problem of a hole in an infinite plane (1) (1) ∂ 2 u1 ∂ 2 u1 + = 0, (6.107) ∂ξ2 ∂ η2

145

AVERAGING AND HOMOGENIZATION METHODS (1)

∂ u1 ∂ u0 + = 0 at ∂k ∂n

∂ Ωi ,

(6.108)

u1 → 0 at ξ 2 + η 2 → ∞.

(6.109)

(1)

In polar coordinates, the boundary value problem (6.107)–(6.109) is written as (1)

(1)

(1)

∂ 2 u1 1 ∂ u1 1 ∂ 2 u1 + + = 0, ∂ r2 r ∂r r2 ∂ θ 2 (1) ∂ u1 ∂ u0 ∂ u0 =− cos θ − sin θ , ∂r ∂x ∂y

(6.110)

(6.111)

r=a

(1)

u1 → 0 at

r → ∞.

(6.112)

Solution of the boundary value problem (6.111)–(6.112) are   a2 ∂ u0 ∂ u0 (1) u1 = cos θ + sin θ . r ∂x ∂y

(6.113)

(1)

The function u1 does not satisfy the periodicity conditions. To eliminate the (1) discrepancy, we obtain in the second approximation for the function u1 (u1 ≈ u1 + (2) u1 ) boundary value problem (2)

∆ u1 = 0 in Ω∗i , (2)

(2)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

(1)

(1)

u1 (0.5, η) − u1 (−0.5, η) = u1 (−0.5, η) − u1 (0.5, η) , u1 (ξ , 0.5) − u1 (ξ , −0.5) = u1 (ξ , −0.5) − u1 (ξ , 0.5) , u1ξ (0.5, η) − u1ξ (−0.5, η) = u1ξ (−0.5, η) − u1ξ (0.5, η) , u1η (ξ , 0.5) − u1η (ξ , −0.5) = u1η (ξ , −0.5) − u1η (ξ , 0.5) . (2)

We now represent u1 as (2)

(12)

u1 = u1

(22)

+ u1 ,

(6.114)

(12)

where the function u1 satisfies homogeneous boundary conditions in ξ and in(22) (12) homogeneous in η; and the function u1 is obtained from the function u1 by a change of variables (ξ ↔ η; x ↔ y). (12) Then we obtain the boundary value problem for determining u1 (12)

∆u1 (12)

(12)

u1 (0.5, η) = u1 (−0.5, η),

= 0 in Ω∗i , (12)

(6.115) (12)

u1ξ (0.5, η) = u1ξ (−0.5, η),

(6.116)

146

Asymptotic Methods for Engineers (12)

u1

(12)

(ξ , 0.5) − u1

(12)

(1)

(1)

(1)

(1)

(ξ , −0.5) = u1 (ξ , −0.5) − u1 (ξ , 0.5) ,

(12)

(6.117)

u1η (ξ , 0.5) − u1η (ξ , −0.5) = u1η (ξ , −0.5) − u1η (ξ , 0.5) . The general solution of equation (6.115) has the form: (12)

u1

∞

= A0 + B0 η + ∑ [(An cosh(2πnη) + Bn sinh(2πnη)) cos(2πnξ )+ n=1

(6.118)

Cn cosh(2πnη) + Dn sinh(2πnη)) sin(2πnξ )], where An , Bn , Cn , Dn are arbitrary constants. Let us now represent the boundary conditions (6.117) in the form (12)

u1

(12)

(ξ , 0.5) − u1

(ξ , −0.5) = −

∂ u0 2 2 −1 a (ξ + 0.25) , ∂y

(6.119)

∂ u0 2 −2 a ξ (ξ 2 + 0.25) . (6.120) ∂x Expanding the right-hand sides of expressions (6.119), (6.120) into Fourier series and substituting solution (6.118) into (6.119), (6.120), we find (12)

(12)

u1η (ξ , 0.5) − u1η (ξ , −0.5) = 2

An = Dn = 0, Bn = −

n = 0, 1, ...,

B0 = −

∂ u0 2 ∂ u0 ∗ πa = B , ∂y ∂y 0

 ∂ u0 ∗ ∂ u0 2a2  −πn e Im E1 (πn(i − 1)) − eπn Im E1 (πn(i + 1)) = B , ∂ y sinhπn ∂y n

∂ u0 ∂ u0 ⇒ , n = 0, 1, 2, ..., ∂y ∂x √ where E1 (...) is an exponential integral [1], i = −1. So   ∞ ∂ u0 ∂ u0 ∗ ∂ u0 (2) u¯1 = B0 η + ∑ B∗n sinh(2πnη) cos(2πnξ ) + cosh(2πnη) sin(2πnξ ) . ∂y ∂y ∂x n=1 Cn = Bn

subject to replacement

(22)

The function u1 is found similarly. (1) (2) Substituting the expression u1 = u1 + u1 into equation (6.105) gives the homogenized equation  2  ∂ u0 ∂ 2 u0 q + = f, (6.121) ∂ x2 ∂ y2 where q = 1 − πa2 +


8π 2 a4 ∞ n e−πn Im E1 (πn(i − 1)) − eπn Im E1 (πn(i + 1)) . ∑ 2 1 − πa n=1 sinhπn (6.122)

147

AVERAGING AND HOMOGENIZATION METHODS

The series in expression (6.122) is absolutely convergent with rapidly decreasing terms: | an+1 /an | → exp(−π). The averaged boundary condition for equation (6.121) has the form (6.106). Note that in the papers [215, 216], the complex flux and effective conductivity are given by series, the coefficients of which are written out exactly. The result is also based on an asymptotic expansion in terms of the contrast parameter and in terms of concentration. Let us dwell on a “paradox” discovered by Bakhvalov and Eglit [36]. They considered the following two cases. In the first one, averaging was performed for a medium with holes. In the second, the medium with some inclusions was averaged, and then, in the averaged relations, the characteristics of the inclusions were assumed to be equal to zero. The corresponding limit systems did not coincide. The point here is that the right-hand sides of the original Poisson equations for inclusions are averaged over the entire area of the cell, and for a medium with holes, only over the area of the cell without a hole. This once again reminds us of the caution when interchanging different passages to the limit (see section 1.2).

6.8

WAVES IN A PERIODICALLY INHOMOGENEOUS MEDIUM

The homogenization method for problems of wave propagation in a periodically inhomogeneous medium is often based on the representation of the desired solution as a product of a periodic function and some modulating one. Mathematicians in this case speak of the Floquet method [41], physicists - of the Bloch method [41, 91]. Recall how the solution of equations with periodic coefficients is constructed. Consider the Hill equation d 2 y(x) − ϕ(x)y(x) = 0, dx2 where ϕ(x) is a periodic function of period a. A particular solution of the Hill equation, according to the Floquet theorem [41], is represented as y1 (x) = Φ(x) exp(iµx), where Φ(x) is a periodic function of the period a, µ is a characteristic exponent, generally speaking, complex. Let us demonstrate the homogenization method for wave problems using the example of a spatially one-dimensional composite material (Fig. 6.6) [86]. The equations of motion of neighboring sections of the composite have the form Ek ukxx − ρk uktt = 0,

k = 1, 2.

(6.123)

At the boundaries of the contact, natural conjugation conditions are set: u1 = u2 ,

E1 u1x = E2 u2x .

(6.124)

148

Asymptotic Methods for Engineers

Figure 6.6 Scheme of a one-dimensional composite material.

In addition, the quasi-periodicity conditions must be specified uk (x + d,t) = uk (x,t) exp(ird),

(6.125)

where k = 1, 2, r = 2π/L, L is the wavelength. We represent the solution on the sections in the form: uk (x,t) = Ak exp[i(pk x + ωt)] + Bk exp[i(−pk x + ωt)], (6.126) p where pk = ω/Ck , Ck = Ek /ρk , k = 1, 2. Substituting expressions (6.126) into boundary conditions (6.124), (6.125), we obtain a system of 4 linear homogeneous algebraic equations for unknown coefficients Ak ,Bk . Equating the determinant of this system to zero, we arrive at the dispersion equation [14, 41] cos(rd) = cos Ω cos(Ωa) −

b2 + 1 sin Ω sin(Ωa), 2b

where Ω=

ωL1 , C1

a=

L2C1 , L1C2

(6.127)

√ E1 ρ1 b= √ . E2 ρ2

In equation (6.127), the parameter b represents the ratio of the impedances of the components of the composite, and the parameter a is the ratio of the times required for the wave to pass through them. The homogenized solution is obtained from equation (6.127) for small Ω and rd, while the parameters a and b are assumed to be of the order of unity. Expanding the right and left sides of equation (6.127) in a Maclaurin series and leaving only the first terms, we obtain " #−1/2 (b − 1)2 a 2 Ω = rd (1 + a) + . (6.128) b

AVERAGING AND HOMOGENIZATION METHODS

149

To solve the transcendental equation (6.126), one can also use perturbation methods other than the homogenization method. Let, for example, b = 1 + ε, ε > 1, one can introduce a small parameter ε2 = 1/b and present equation (6.127) in the following form ε2 cos rd = ε2 cos Ω cos(Ωa) −


1 1 + ε22 sin Ω sin(Ωa). 2

(6.130)

Possible simplifications of this equation depend on the order of a. If a ∼ 1, i.e., L1 /L2 ∼ ε2 (the length of one section of the composite is significantly less than the length of the other), then the solution of equation (6.130) can be sought in the form of expansion √ Ω = ε2 Ω0 + ε2 Ω1 + ... (6.131) Substituting expansion (6.131) into equation (6.130), we obtain in the first approximation

√ √ 1 sin ε2 Ω0 sin ε2 Ω0 a . (6.132) cos rd = 1 − 2ε2 Expanding the right-hand side of expression (6.132) into a series in Ω0 , preserving the terms of the second and fourth orders, it is possible to approximate the original system with a sufficiently high accuracy by a chain of two periodically alternating masses connected by identical springs [222].

150

6.9

Asymptotic Methods for Engineers

HIGHER ORDER ASYMPTOTIC HOMOGENIZATION FOR DYNAMICAL PROBLEMS

Let us consider the eigenvalue Neumann problem     d x du a + pu = 0, dx ε dx

(6.133)

du = 0 at x = 0, 1. (6.134) dx


Here a εx = a εx + 1 ; 0 < a0 ≤ a εx ≤ a1 , u(x) is the eigenform, p is the eigenvalue. Going over to “fast” η and “slow” x variables and taking into account relation d ∂ ∂ = + ε −1 , dx ∂ x ∂η

(6.135)

instead of the original ODE one obtains PDE. We seek its solution as expansions of the original eigenform and eigenvalue u = u0 (η, x) + εu1 (η, x) + ε 2 u2 (η, x) + . . . ,

(6.136)

p = p0 + ε p1 + ε 2 p2 + . . .

(6.137)

Functions ui , i = 1, 2, 3, ... subject to conditions Z1

ui dη = 0, ui (0, x) = ui (1, x).

(6.138)

0

Substituting the expansions (6.136), (6.137) into the original eigenvalue problem (6.133), (6.134) and splitting with respect to ε, one obtains in the zero and first approximations: u0 ≡ u0 (x), (6.139)     ∂ ∂ u1 ∂ du0 a + a = 0, (6.140) ∂η ∂η ∂η dx ∂ u1 du0 ah du0 =− + , ∂η dx a dx where

−1



Z1

ah = 

(6.141)

a−1 dη  .

(6.142)

0

From (6.141), one obtains du0 du0 u1 = − η + ah dx dx

Zη 0

a−1 dη +C1 (x).

(6.143)

151

AVERAGING AND HOMOGENIZATION METHODS

Using condition (6.138), one obtains 1 du0 du0 C1 (x) = − ah 2 dx dx

Z1





Zη

a−1 dη  dη.

 0

(6.144)

0

Finally, one gets u1 = u f 1 (η)

du0 , dx 

Zη

1 u f 1 = − η + ah  2

−1

a dη −

0

Z1





(6.145)

−1

a dη  dη  .

 0



Zη 0

Equation of the second approximation can be written as follows:     ∂ ∂ u2 ∂ ∂ u1 ∂ 2 u1 d 2 u0 a + a +a + a 2 + p0 u0 = 0. ∂η ∂η ∂η ∂x ∂ x∂ η dx

(6.146)

Boundary conditions are du0 ∂ u1 =− dx ∂η

at

x = η = 0;

x = 1,

η = |ε −1 − [ε −1 ]|,

(6.147)

where [...] means integer part of a number. Substitute expression (6.141) to PDE (6.146) and BCs (6.147), after using averR1

aging operator (...)dη one obtains: 0

ah

d 2 u0 + p0 u0 = 0, dx2

(6.148)

du0 =0 at x = 0, 1. dx Solution of eigenvalue problem (6.148), (6.149) is p0 = π 2 n2 ah ,

u0 = A cos(πnx),

n = 1, 2, 3, ...

(6.149)

(6.150)

The area of applicability of the homogenized solution are determined by the inequality n m2 + n2 k i 2n2 i 0 (−1)  + , 2 , ∑ o γim αi coth β coth(−1)i (β /2)  ψ n i=1,3,5,... 2 i < m2 + n2 k i

i=2,4,6.....

i

where m2 ψ =n + 2 , k 2

r α=

m2 2 2 + n2 , k

r αi =

i2 + m2 + n 2 , βi = π k2

r

 (−1)m   sin(2π µm), for i = m, 2(0.5 − µ) +    πm)  "(    i 4 1 sin(π µi) cos(π µm)− γim = π (m2 − i2 ) m    ( ) #    m   sin(π µm) cos(π µi) for i 6= m,   i the symbol ∑ 0 means the sum without the term i = m.

m2 − i2 + n2 , k2

APPLICATIONS OF PADE APPROXIMATIONS

181

Figure 8.4 Oscillation frequencies of a square plate with partially clamping contour. The upper curve corresponds to the symmetric case, the lower one to the asymmetric one.

AP has the form (8.35). Comparison of the results of calculations by formula (8.35) for a square plate with experimental data [234] (points and triangles in Fig. 8.4) shows its high accuracy. For the symmetric case, a comparison was also made with the solution by the dual series method [299]. On the scale of the figure, the corresponding curves almost coincide.

8.7

REDUCTION OF THE GIBBS-WILBRAHAM PHENOMENON

Here we show acceleration of the convergence of the Fourier series and reduction of the Gibbs-Wilbraham phenomenon with the help of AP. Fourier expansions possess polynomial rate of convergence; meanwhile, the Fourier-Pad´e approach provides the exponential rate of convergence. This makes it possible to reduce the number of terms of expansion and, consequently, to increase the minimum period of the retained harmonics. The AP of a function g(z) is introduced as follows. Let g(z) = c0 + c1 z + c2 z2 + ... + cN zN + O(zN+1 ).

(8.50)

182

Asymptotic Methods for Engineers

Then, the diagonal AP is g(z) =

a0 + a1 z + a2 z2 + ... + aN/2 zN/2 1 + b1 z + b2 z2 + ... + bN/2 zN/2

+ O(zN+1 ),

(8.51)

where the coefficients ai , bi are obtained from the rule: Taylor series expansion of the AP (8.51) must coincide with expansion (8.50) up to the O(zN ). There exist different algorithms for the evaluation of AP. one of the simplest procedures is as follows. The denominator coefficients are determined from the system of algebraic equations      c1 c2 c3 ... cN/2 bN/2 cN/2+1  c2     c3 c4 ... cN/2+1    bN/2−1  cN/2+2   c3     c4 c5 ... cN/2+2  bN/2−2  = cN/2+3  (8.52)  ,  ... ... ... ... ...   ...   ...  cN/2 cN/2+1 cN/2+2 ... cN−1 b1 cN afterward, the numerator coefficients are calculated by the formulas i

a0 = c0 ,

ai = ci + ∑ bk ci−k ,

i = 1, 2, 3, ..., N/2.

(8.53)

k=1

Oryginally, the AP was developed for power series expansions. In recent years, many authors have attempted to extend this procedure to general series of functions [147]. Let us represent Fourier expansion in the exponential form !   N N 2πx in 2πx = ℜ ∑ ck,n e L f (x) = ∑ ck,n cos n L n=0 n=0 and make a substitution ei f (x) = ℜ

2πx L

→ z. Then, the Fourier-Pad´e approximant (FPA) is ! a0 + a1 z + a2 z2 + ... + aN/2 zN/2 , 1 + b1 z + b2 z2 + ... + bN/2 zN/2 + O(zN+1 )

where the coefficients ai , bi are determined by expressions (8.52), (8.53). As an model example, let us consider the piecewise-continuous function  0 for −50 < x < −10, f (x) = 1 for −10 ≤ x ≤ 10,  0 for 10 < x < 50.

(8.54)

In Fig. 8.5, the partial sum of the Fourier expansion is compared with the corresponding FPA at N = 32. We observe that the FPA almost coincides with the original function (8.54). The absolute error of the approximation η is presented in Fig. 8.6 in a logarithmic scale for N = 32 and N = 64. The FPA accelerates the convergence of the Fourier

183

APPLICATIONS OF PADE APPROXIMATIONS

Figure 8.5 Approximation of a piecewise-continuous function; solid - the Fourier expansion, dashed - the FPA.

Figure 8.6 Absolute error of the approximation of a piecewise-continuous function; FE - the Fourier expansion, FPA - the Fourier-Pad´e approximant: a) N = 32, b) N = 64.

series. Moreover, increasing the number of terms N leads to a more effective acceleration. At the points of discontinuity, x = ±10, the Fourier series exhibit local overshoots due to the Gibbs-Wilbraham phenomenon. Application of the FPA allows us to reduce this effect, which is illustrated in Fig. 8.7 (N = 256).

8.8

AND THAT’S NOT ALL!

AP are widely used in perturbation methods [23, 11, 29]. In addition to the above mentioned examples, we show a few possible application of AP. Let us start with problems that have localized solutions. As a simple model, consider the nonlinear boundary value problem y00 − y + 2y3 = 0, y(0) = 1,

y(∞) = 0,

(8.55) (8.56)

having an exact solution y = ch−1 (x).

(8.57)

184

Asymptotic Methods for Engineers

Figure 8.7 Approximation at the points of discontinuity; solid - the Fourier expansion, dashed - the Fourier-Pad´e approximant.

The quasi-linear asymptotics gives the solution in the following form: y = Ce−x (1 − 0.25C2 e−2x + 0.0625C4 e−4x + ...),

C = const.

(8.58)

It is easy to check that, transforme truncated series (8.58) in AP and determining the constant C from the boundary conditions (8.56), we arrive at the exact solution (8.57). In the same way, solitons and other localized solutions of nonlinear problems can be constructed, in connection with which the term “padeon” appeared [172, 173]. It is interesting to apply AP to problems with the “blow-up” phenomenon, when the solution goes to infinity at a finite value of the argument. Such is, for example, the Cauchy problem dx = αx + εx2 , x(0) = 1, (8.59) dt where 0 < ε