Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications [1 ed.] 9781613247433, 9781611228588

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

MATHEMATICS RESEARCH DEVELOPMENTS

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PARTIAL DIFFERENTIAL EQUATIONS: THEORY, ANALYSIS AND APPLICATIONS

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MATHEMATICS RESEARCH DEVELOPMENTS

PARTIAL DIFFERENTIAL EQUATIONS: THEORY, ANALYSIS AND APPLICATIONS

CHRISTOPHER L. JANG Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, FRQVHTXHQWLDO RU H[HPSODU\GDPDJHV UHVXOWLQJ LQZKROHRU LQSDUW IURP WKH UHDGHUV¶ XVH RIRU reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Partial differential equations : theory, analysis, and applications / editor: Christopher L. Jang. p. cm. Includes index. ISBN H%RRN 1. Differential equations, Partial. 2. Mathematical physics. I. Jang, Christopher L. QC20.7.D5P38 2010 515'.353--dc22 2010043947

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CONTENTS Preface Chapter 1

Time-Spectral Solution of Initial-Value Problems Jan Scheffel

Chapter 2

A Stochastic Agent-Based Approach to the Fokker-Planck Equation in Human Population Dynamics Minoru Tabata and Nobuoki Eshima

51

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films Aomar Hadjadj

71

The Role of the Method of Characteristics in the Solution of Estimation and Control Problems for Hyperbolic PDE Systems Efrén Aguilar-Garnica and Juan P. García-Sandoval

97

Chapter 3

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vii 1

Chapter 5

Soliton Solutions of One KdV Equation Zhao-ling Tao and Yang Yang

141

Chapter 6

Numerical Solution of Fractional Partial Differential Equations Jiunn-Lin Wu

151

Chapter 7

Boundary Control of Systems Described by Partial Differential Equations by Input-Output Linearization Ahmed Maidi, Moussa Diaf and Jean-Pierre Corriou

173

Robust No Parametric Identifier for a Class of Complex Partial Differential Equations R. Fuentes, I. Chairez, A. Poznyak and T. Poznyak

201

The Generalized Weierstrass System Inducing Surfaces in Euclidean Three Space and Higher Dimensional Spaces Paul Bracken

223

Natural Convection and Its Effect on Diffusion Measured with Nuclear Magnetic Resonance Aleã Mohoriþ

265

Chapter 8

Chapter 9

Chapter 10

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vi Chapter 11

Chapter 12

Contents Partial Differential Equations as a Tool for Evaluation of the Continuous Wavelet Transform Eugene B. Postnikov

277

The Blowup Mechanism in Nonlinear Partial Differential Equations: Scaling and Variation Takashi Suzuki

313

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Index

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339

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PREFACE Partial differential equations are used to formulate and thus aid the solution of problems involving functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. This book presents current research in the study of partial differential equations, including a generalized fully spectral weighted residual method (GWRM) for solution of initial value partial differential equations; the Fokker-Planck equation in human population dynamics; solition solutions to one KdV equation; boundary control of systems described by partial differential equations by inputoutput linearization; the Weierstrass system and partial differential equations as a tool for evaluation of the continuous wavelet transform. Chapter 1 ± A generalized fully spectral weighted residual method (GWRM) for solution of initial value partial differential equations is presented. For all temporal, spatial and physical parameter domains, the solution is represented by Chebyshev series, enabling global semi-analytical solutions. The method avoids time step limitations. The spectral coefficients are determined by iterative solution of a system of algebraic equations, for which a globally convergent root solver has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by the use of temporal and spatial subdomains. Example applications include the diffusion, Burger and forced wave equations as well as a system of ideal magnetohydrodynamic (MHD) equations. A stiff ordinary differential equation introduces the method. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Potential applications of the GWRM are, for example, advanced initial value problems in fluid mechanics and MHD. Chapter 2 ± Various kinds of mathematical models are constructed in order to quantitatively describe migration. For example, a partial differential equation of parabolic type (an integro-partial differential equation, respectively) is employed in order to describe the phenomenon. The partial differential equation (the integro-partial differential equation, respectively) is called the Fokker-Planck equation (the master equation, respectively). We need to apply microscopic foundations for the descriptions given by these functional equations. As described in their chapter, the authors take a stochastic agent-based approach to the master equation, i.e., they construct a stochastic agent-based model to apply a microscopic foundation for the description given by the master equation. Hence, in the present paper, they take a stochastic agent-based approach to the Fokker-Planck equation, i.e., they construct a stochastic agent-based model and apply a microscopic foundation for the description given by the Fokker-Planck equation.

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Christopher L. Jang

Chapter 3 ± Hydrogen plays a crucial and antagonist role in hydrogenated amorphous silicon (a-Si: H). Its ability to move into, out of and within a-Si:H leads to both beneficial and undesirable properties. On the one hand it bonds to weak and broken silicon bonds and thus reduces the defect density to a level compatible with doping and useful electronic devices. On the other hand it is linked to metastable defect formation that limits the applications of a-Si:H. In addition, all models describing the growth of hydrogenated microcrystalline silicon (Pc-Si :H) by plasma-enhanced chemical vapour deposition involve either short-range effects of hydrogen or long-range effects, or both of them. A considerable number of studies have been devoted to hydrogen transport in a-Si:H and related materials. Both effusion and diffusion measurements were used for that purpose. The diffusion is performed either from a solid source (a deuterated a-Si:H layer) or from a gas-phase source (a hydrogen or deuterium plasma or heated filament). Secondary-ion mass spectroscopy (SIMS) is the most used technique to determine the hydrogen (or deuterium) diffusivity from the concentration profile. Other techniques such as infrared spectroscopy, electron spin resonance, elastic-recoil detection analysis, small-angle X-ray scattering, electrical measurements and ellipsometry were used to characterize the hydrogen diffusion. Chapter 4 ± In this chapter the authors summarize the use of the Method of Characteristics (MC) in the solution of estimation and control problems for processes whose model is given by First-Order Partial Differential Equations or Hyperbolic systems. First, a dynamical robust observer is built with the MC for a non-isothermal plug flow reactor whose state variables travel along one characteristic line. Then, this observer in its integral equation version is designed for a solid-waste anaerobic digestion process which is described by a set of variables that follow different characteristic lines. In both cases the aim of the observers is to estimate the evolution of certain specific variables which is necessary to monitor but whose measurement becomes difficult either because do not exists the suitable analytical techniques or because such measurement is relatively expensive. Finally, the chapter is closed with an analysis of a robust control law derived with the MC for a heat exchanger in which a timevariant characteristic line is involved. The performance of the observers and the control law is analyzed with simulation experiments obtaining satisfactory results. Chapter 5 ± The author know KdV equations are important, so the authors join to study the soliton solutions of one KdV equation. Three simple and effective methods are employed, the exp-function method, the tanh method and solitary wave ansatz method for the soliton solutions in varied form. To their knowledge, not only same solutions as those in the open literature are obtained, but also some new are arrived at by the help of the symbolic computation software. Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body. Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this chapter, the numerical methods for fractional partial differential equations will be reviewed, especially the approach based on the operational matrices of the orthogonal functions. It transforms the problem to a simple Lyapunov matrix equation solving. Advantages of the operational method include (1) the computation is simple and computer oriented, (2) it can solve the partial differential equations numerically, even the

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Preface

ix

ones with fractional order, (3) the scope of application is wide and (4) the step size used could be large and the result obtained is still satisfactory. The numerically unstable problem does not occur in the operational method. Chapter 6 - Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body. Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this chapter, the numerical methods for fractional partial differential equations will be reviewed, especially the approach based on the operational matrices of the orthogonal functions. It transforms the problem to a simple Lyapunov matrix equation solving. Advantages of the operational method include (1) the computation is simple and computer oriented, (2) it can solve the partial differential equations numerically, even the ones with fractional order, (3) the scope of application is wide and (4) the step size used could be large and the result obtained is still satisfactory. The numerically unstable problem does not occur in the operational method. Chapter 7 ± The characteristic variables and parameters of many systems depend on space, so that their dynamic behavior leads to models by partial differential equations (PDE). PDE are adequate models for problems arising in many modern applications, for instance chemical and biological processes, thermochemical flow phenomena, fluid dynamics and vibrations. This kind of systems are termed distributed parameter systems or infinite dimensional systems and include the transport-reaction processes, particulate processes, processes involving fluid flows and wave equation problems. Chapter 8 ± In this chapter a strategy based on differential neural networks (DNN) for the identification of a class of models described by partial differential equation with a complexvalued state is proposed. The identification problem is reduced to finding an exact expression for the weights dynamics using the DNNs properties. In this case, the DNN can be viewed as two coupled networks where one of them reproduces the real part of the complex valued equation and the other provides the identification of the imaginary part, where each stimated state is a complex valued state. The adaptive laws for complex weights ensure the convergence of the DNN trajectories to the PDE complex-valued states. To investigate the qualitative behavior of the suggested methodology, here the non parametric modeling problem for two distributed parameter plants is analyzed: the Sch¨odinger and GinzburgLandau equations. Chapter 9 ± An inducing mechanism for describing minimal surfaces imbedded in threedimensional Euclidean space was first formulated by Enneper andWeierstrass one and a half centuries ago. More recently, this idea has been substantially generalized by B. Konopelchenko, who established the connection between certain classes of constant mean curvature surfaces and the trajectories of an infinite dimensional Hamiltonian system. Here the authors begin by reviewing his formulation of the generalized Weierstrass system which consists of nonlinear Dirac-like partial differential equations. It is the solutions of this system which can be used to generate surfaces in Euclidean three space which have constant mean curvature. The correspondence between this system and the two-dimensional nonlinear sigma model is presented. A linear spectral problem is established. The integrability of the system is discussed and symmetry reduction is systematically applied to derive several classes of

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invariant solutions for both the two-dimensional nonlinear sigma model and the generalized Weierstrass system. Extensions of this system to higher dimensional spaces are considered. Finally, a physical application in the area of string theory will be presented, including an application in Minkowski space, and the path integral quantization in three space using the Weierstrass representation. An introduction to the case of inducing surfaces of nonconstant mean curvature is briefly treated. Chapter 10 – Natural convection is an effect occurring in fluids where temperature gradient causes flow on macroscopic scale if the fluids density changes significantly. If the gradient is strong enough, the convection turns chaotic. Equations governing the flow dynamics are nonlinear Navier-Stokes, continuity, and heat conduction equation which in the first approximation reduce to the Lorenz system of equations. The equations describe the Lorenz oscillator and the long-term behavior of the oscillator corresponds to a fractal structure known as Lorenz attractor. The system was introduced to describe convection roles in the atmosphere. Chaotic velocity fluctuations can be observed in magnetic resonance images of self-diffusion constant distribution. To describe specific measurements, a model of natural convection in a horizontally oriented cylinder, cooled from above, can be derived. The Lorenz model of natural convection is derived for a free boundary condition, so its validity is of a limited value for the natural no-slip boundary condition. It can be shown, that even a slight temperature gradient can cause measurable enhancement of the apparent self-diffusion constant of the liquid. Self-diffusion is a process of particle random motion causing the spread in statistical distribution of these particles in space. Nuclear magnetic resonance is one of the best suited methods for measuring self-diffusion since it non-invasively measures particle motion in magnetic field gradient through attenuation of spin echo. Chapter 11 – The presented review is dedicated to the consideration of the Continuous Wavelet tarsnsform from the diffusion signal and image processing point of view. Such an approach is based on the consideration of diffusion smoothing via the solution of proper partial diferential equations. Within this group of methods the real and complex wavelet transform with the wavelets of Gauss and Morlet families are considered. Especial attention is concentrated on the variety of numerical examples considering the processing of regular and irregular (random samples, chaotic ODE solutions etc.) signals. All of them are graphically illustrated. Chapter 12 – The authors study the blowup mechanism of the solution to nonlinear partial differential equations derived from their scaling and variational properties. Then this method is applied to several mean field equations provided with the free energy decreasing and conservation laws. Among them is a Smoluchowski-Poisson equation describing the kinetic motion of self-gravitating fluids which exposes two different blowup patterns.

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In: Partial Differential Equations Editor: Christopher L. Jang

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Chapter 1

TIME-SPECTRAL SOLUTION OF INITIAL-VALUE PROBLEMS Jan Scheffel Division of Fusion Plasma Physics (Association EURATOM/VR), Alfvén Laboratory, Royal Institute of Technology, Stockholm, Sweden

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ABSTRACT A generalized fully spectral weighted residual method (GWRM) for solution of initial value partial differential equations is presented. For all temporal, spatial and physical parameter domains, the solution is represented by Chebyshev series, enabling global semi-analytical solutions. The method avoids time step limitations. The spectral coefficients are determined by iterative solution of a system of algebraic equations, for which a globally convergent root solver has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by the use of temporal and spatial subdomains. Example applications include the diffusion, Burger and forced wave equations as well as a system of ideal magnetohydrodynamic (MHD) equations. A stiff ordinary differential equation introduces the method. Comparisons with the explicit LaxWendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Potential applications of the GWRM are, for example, advanced initial value problems in fluid mechanics and MHD.

Keywords: initial-value problems, PDE, WRM, semi-analytical, Chebyshev polynomials, spectral method, fluid mechanics, MHD

1. INTRODUCTION A novel spectral method for solution of time-dependent partial differential equations is presented. The method differs from earlier spectral methods for initial-value problems in that also the time domain is treated spectrally. In this introduction we will provide a brief

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background, give an outline of the method, compare with earlier work, discuss the method from a philosophical perspective and present the organisation of the sections to follow. Solutions to initial-value problems, formulated as partial differential equations, are usually purely numeric and obtained by advancing the solution using a large number of time steps. Since the solutions are numeric rather than analytic, sampled series of well chosen runs are required in order to determine detailed parametrical dependences. The time steps of explicit time advance methods are restricted to small values by constraints such as the Courant-Friedrich-Lewy (CFL) condition [1], and implicit schemes require time-consuming matrix operations at each time step. Semi-implicit methods [2,3] allow large time steps and more efficient matrix inversions than those of implicit methods, but may feature limited accuracy. In the present approach, a fully spectral method is used to obtain semi-analytical solutions of initial-value partial differential equations [4]. The fundamental novelty of the present work is that also the temporal evolution is modelled spectrally, using series expansion. By semi-analytical is meant that finite, approximate analytical solutions in time, space and physical parameters are obtained as analytical spectral Chebyshev expansions with numerical coefficients. This approach is, for example, of great interest for carrying out scaling studies, in which the detailed parametrical dependence preferably is computed in analytical form rather than purely numerically. It is not unusual that the envelope of the characteristic dynamics is of more interest than, for example, fine scale turbulent phenomena. The possibility to filter out, or average, fine structures in the interest of higher computational efficiency has been the main inspiration in this project. It has subsequently turned out that the method may also provide high accuracy when chosing to compute more detailed phenomena, using more spectral terms. In all cases, the resulting functional solutions are immediately available for differentiation, integration or other analytic subsequent usage. We are not aware of any counterpart to the present method, here called the generalized weighted residual method (GWRM), in the literature. Some ideas relating to the spectral expansion of the time domain appears to have been put forth early [5], but were not developed further. The idea to employ orthogonal sets of basis functions to globally minimize spatial spectral expansions (weighted residual methods, WRM), is however far from new [6,7]. The GWRM is a Galerkin WRM, using the weak formulation of the differential equation, just like finite element methods (FEM). An important difference to FEM is the use of more advanced trial functions, valid in much larger domains. This is of particular use in “smearing out” small scale fluctuations. Interestingly, we will also show that our implementation of the GWRM is equivalent to least-square weighted residual methods. Another method, called proper orthogonal decomposition (POD), analyzes multidimensional data (such as found in fluid flow patterns) in an optimized least square sense in order to determine lower dimensional representations of the data (see for example Ref. [8]). Although being data dependent, rather than dependent on the data generating process or algorithm, POD can also be combined with a Galerkin projection procedure to generate lower dimensional models of dynamical systems that have very large phase spaces. Applications may be found in various disciplines of physics. A main difference between the GWRM and POD methods is, however, that the GWRM is designed to solve partial differential equations, including dependence on physical parameters, whereas POD methods are mainly tools for reducing already obtained numerical or experimental data.

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Time-Spectral Solution of Initial-Value Problems

3

The GWRM is based on Chebyshev polynomial expansions, having several desirable qualities. They converge rapidly to the approximated function, they are real and can easily be converted to ordinary polynomials and vice versa, their minimax property guarantees that they are the most economical polynomial representation, they can be used for non-periodic boundary conditions (being problematic for Fourier representations) and they are particularly apt for representing boundary layers where their extrema are locally dense [9,10]. The method is spectral; essentially all calculations are carried out in spectral space. The powerful minimax property of the Chebyshev polynomials [10] implies that the most accurate n-m order approximation to an nth order approximation of a function is a simple truncation of the terms of order > m-n. Thus nonlinear products are easily and efficiently computed in spectral space. An alternative is to employ a pseudospectral method, which handles nonlinearities efficiently by computing them in physical space. The transformation back and forth between spectral and physical space can be efficiently done using FFT routines adapted to Chebyshev polynomials. The pseudo-spectral methods can, however, cause aliasing errors, which are removed by using only two thirds of available spectral space [7]. Since the GWRM efficiently uses rapidly convergent Chebyshev polynomial representation for all time, space and parametrical dimensions, pseudospectral implementation has so far not been pursued. The GWRM eliminates time stepping and associated, limiting grid causality conditions such as the CFL condition. The method is acausal, since the time dependence is calculated by a global minimization procedure (the weighted residual formalism [6,7]) acting on the time integrated problem. Recall that in standard WRM methods, initial value problems are tranformed into a set of of coupled ordinary, linear or non-linear, differential equations for the time-dependent expansion coefficients. These are solved using finite differencing techniques. From a more philosophical point of view, the GWRM offers a somewhat different methodology to solving initial-value problems. Having observed regularity in Nature, we as scientists formulate laws of Nature that are believed to be causal. The time-stepping approach to solve these differential equations in a way mimics the cause and effect that we have encountered. The laws that we formulate are usually, however, extremely powerful in that they, in their mathematical form, already encompass the entire chain of cause and effect from the initial to the end state. There is thus no need to solve these equations deterministically using time-stepping. We should not confuse the phenomena that we are observing with the methodology used to solve the representing equations. In the GWRM, the initial-value partial differential equations are regarded as implicit formulations of the world lines of the system and are solved globally in time, physical space and parameter space simultaneously. The number of representing modes for each space can be chosen at will; approximative and simple as well as very accurate solutions may be calculated. In explicit time stepping methods, the time step restriction requires the time steps to be small enough to protect information to travel with characteristic speeds to grid points where this information is not taken into account. If violated, the condition causes numerical instability. A reason for chosing local, time stepping methods to solve initial-value problems is that they may efficiently use computer resources in the sense that required memory storage is limited when only a few time levels are involved in each calculation. The GWRM is also well suited for solving linear or nonlinear ordinary differential equations, that may be formulated as initial-value problems. As an introductory example, we solve a stiff, time-dependent ordinary differential equation representing flame propagation caused by lighting a match.

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In this paper, the GWRM is first outlined. We show, in sections 2-4, how integration, differentiation, nonlinearities, as well as initial and boundary conditions are handled in multivariate Chebyshev spectral space for arbitrary solution intervals. Having transformed the initial-value problem, a set of algebraic equations result. These represent the coefficients of the Chebyshev expansions, and should be solved iteratively. A new, global nonlinear equation solver has been developed for this purpose and is briefly described in section 5. The introduction of temporal and spatial subdomains, for increasing efficiency, is discussed in section 6. Detailed comparisons with the time differencing Lax-Wendroff (explicit) and Crank-Nicolson methods (semi-implicit) are next presented in section 7. For studying accuracy, the nonlinear Burger equation and its exact solution is used. A forced wave equation is subsequently used for studying the methods’ ability to handle strongly separated time scales. Application of the GWRM to a stability problem formulated within the linearised magnetohydrodynamic equations is then described in some detail. The paper ends with a summary and conclusions.

2. THE METHOD Consider a system of parabolic or hyperbolic initial-value partial differential equations, symbolically written as

u  Du  f t

(1)

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where u  u(t, x;p) is the solution vector, D is a linear or nonlinear matrix operator and

f  f (t, x;p) is a source (or forcing) term. Note that D may depend on both physical variables (t, x and u) and physical parameters (denoted p) and that f is assumed arbitrary but non-dependent on u. Initial u(t0,x;p) as well as (Dirichlet, Neumann or Robin) boundary u(t,xB;p) conditions are assumed known.

2.1. Multivariate Chebyshev Spectral Expansion Our aim is to determine a spectral solution of Eq.(1), using Chebyshev polynomials [10] in all dimensions. To avoid inverting a matrix solution vector, associated with the time derivative, Eq.(1) is now integrated in time. This formulation of the problem is conveniently coupled to the fixed point algebraic equation solver SIR described in Section 5. Thus we obtain t

u(t, x;p)  u(t 0 , x;p)   {Du(t , x;p)  f (t´, x;p)}dt  t0

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(2)

Time-Spectral Solution of Initial-Value Problems

5

The solution u  u(t, x;p) is approximated by finite, multivariate first kind Chebyshev polynomial series. Recall that Chebyshev polynomials of the first kind (henceforth simply referred to as Chebyshev polynomials) are defined by Tn (x)  cos(n arccos x). These functions can be written as real, simple polynomials of degree n and are orthogonal in the interval [-1,1] over a weight (1  x 2 )1/2 . Thus T0 (x) = 1, T1 (x) = x, T2 (x) = 2x2 - 1 etc. For simplicity, we will now restrict the discussion to a single equation with one spatial dimension x and one physical parameter p. Schemes for several coupled equations and higher dimensions may subsequently be straightforwardly obtained. Thus, K

L

M

u(t, x; p)   ´  ´  ´ aklmTk ( )Tl ( )Tm (P)

(3)

k 0 l 0 m0

With



p  Ap t  At x  Ax ,  , P Bt Bx Bp

At  (t1  t 0 ) / 2, Ax  (x1  x0 ) / 2, Ap  ( p1  p0 ) / 2

(4)

Bt  (t1  t 0 ) / 2, Bx  (x1  x0 ) / 2, Bp  ( p1  p0 ) / 2

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where indices ’0’ and ’1’ denote lower and upper computational domain boundaries, respectively. The basic Chebyshev polynomials have the limited range of variation [-1,1] and they require arguments within the same range. The variables  ,  and P used here allow for arbitrary, finite computational domains. We note that, at the spatial boundaries, and

 (x0 )  1

 (x1 )  1. Primes on summation signs in Eq. (3) indicate that each occurence of a zero

coefficient index should render a multiplicative factor of 1/2. Next, we use a weighted residual formulation to determine the unknown coefficients

aklm

of the solution ansatz (3).

2.2. Weighted Residual Formulation The weighted residual method (WRM) is based on the idea that a residual, when using the ansatz (3) in Eq.(2), is to be minimized globally. The residual is integrated over the entire solution domain in order to produce a set of equations for the coefficients of Eq. (3). In the Galerkin WRM approach, the simplifying continuous orthogonality properties of the basis functions are employed through first multiplying the partial differential equation by basis functions of the same kind as those of the ansatz. Weight functions may also be included. Thus, similarly as for FEM, the weak formulation of the pde is used. The Galerkin WRM of the present method, providing the equations for the coefficients aqrs , is

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Jan Scheffel t1 x1 p1

   R T ( )T ( )T (P)w w w dtdxdp  0 q

r

s

t

x

(5)

p

t 0 x0 p0

where the residual R is defined as t   R  u(t, x; p)  u(t 0 , x; p)   {Du  f }dt    t0 

with

wt  (1   2 )1/2 , wx  (1   2 )1/2 , w p  (1  P 2 )1/2 . The solution ansatz (3) is now inserted into Eq. (5). The separation of variables inherent in the Galerkin WRM approach, enables separately performed integrations. Consequently t1 K

 a

K

´

t1

T ( )Tq ( )wt dt   aklm  Tk ( )Tq ( )(1   2 )1/2 dt  ´

klm k

t0 k  0

k 0

t0



      ´ aklm  Bt cos k cos q d   ´ aklm Bt  kq   k 0 q0   Bt aqlm . 2 2  2 k0 k0 0

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K

 ik

K

(6)

We have used the variable transformation   cos  and introduced the Kronecker delta (being 1 if i = k and 0 otherwise). The integrals over the spatial and parameter domains

may be computed likewise, and the first term of Eq. (5) becomes t1 x1 p1



   u(t, x; p)T ( )T ( )T (P)w w w dtdxdp  B B B ( 2 ) a q

r

s

t

x

p

t

x

p

3

qrs

(7)

t 0 x0 p0

where the indices obey 0  q  K,

0  r  L, 0  s  M . For the second term of Eq.

(5) the initial condition is expanded as L

u(t 0 , x; p)  

´

M

b ´

T ( )Tm (P)

lm l

(8)

l 0 m0

with K

K

blm   aklmTk ( 0 )   ´ aklm (1)k . k0

´

k0

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(9)

Time-Spectral Solution of Initial-Value Problems where

7

 0   (t 0 ) has been used. We find

t1 x1 p1



   u(t , x; p)T ( )T ( )T (P)w w w dtdxdp  B B B ( 2 )  0

q

r

s

t

x

p

t

x

p

2

b .

q0 rs

(10)

t 0 x0 p0

Next, we apply the expansions t

K 1

t0

k0 l0 m0

L

M

´ ´ ´  Du(t , x; p)dt      AklmTk ( )Tl ( )Tm (P) t

K 1

L

M

 f (t , x; p)dt     

t0

k0

´

´

´

(11)

FklmTk ( )Tl ( )Tm (P)

l0 m0

which yield

 t   3 Du( t , x; p)d t     Tq ( )Tr ( )Ts (P)wt wx w p dtdxdp  Bt Bx Bp ( ) Aqrs t x p t 2 0 0 0  0  t1 x1 p1

(12)

and similarly for the coefficients Fqrs . The final expression for the GWRM coefficients of Eq.(3) becomes simply, using Eqs. (5), (7), (10) and (12),

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aqrs  2 q0 brs  Aqrs  Fqrs , defined for all

(13)

0  q  K  1, 0  r  LBC , 0  s  M . Note that the number of modes

in the temporal domain is extended to K + 1 due to the time integration, that the initial conditions are included at this stage and that the high end spatial modes with r  LBC are saved for implementation of boundary conditions. Since the solution ansatz (3) extends to K temporal modes only, the K + 1 mode needs special attention as discussed below. It may also be noted that Eq. (13), the equation for the Chebyshev coefficients, is the same as that which would obtain if the residual R, using Eqs. (3), (8) and (11), is set identically to zero. How can the global WRM solution of Eq. (13) correspond to a “local” Chebyshev approximation? The explanation is that Chebyshev approximations are not local in contrast to, for example, Taylor series expansions of ordinary polynomials. They are always defined on an interval (see Eqs. (3-4)). Due to the uniqueness given by their minimax property, “local” or “global” Chebyshev approximations are identical once a domain is defined. Whether solving a single differential equation or a system of coupled linear/nonlinear differential equations, Eq. (13) will constitute a set of coupled linear/nonlinear algebraic equations. Here the coefficients Aqrs are themselves functions of the coefficients aklm , whereas the coefficients Fqrs are uniquely determined from the forcing term f. For example,

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8 if

Jan Scheffel

f equals a constant C, then Fqrs  4Bt r 0 s0 (2 po   p1 )C . This is shown using Eq.

(11). Thus Eq. (13) specifies a complete, implicit relation for the coefficients aqrs of the solution together with the boundary conditions, which will be discussed in the next section. Methods for efficient iterative solution of Eq. (13) will be introduced in section 5. Convergence is controlled by requiring that the absolute values sum of the first few coefficients of the solution (3) deviates less than some value  from one iteration to the next. Usually we assume

  1·10 6 .

2.3. Integration and Differentiation By also letting K

L

M

Du   ´  ´  ´ cklmTk ( )Tl ( )Tm (P)

(14)

k 0 l 0 m0

we may employ the Chebyshev representation of time integration:

Aklm 

Bt (ck 1,l,m  ck 1,l, m ), cK 1,l, m  cK  2,l, m  0, 0  k  K  1 2k K 1

A0lm  2  Aklm (1)

(15)

k

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k 1

We have generalized expressions in Ref. [10] for the coefficients of an integrated Chebyshev series to arbitrary intervals [t 0 ,t1 ] . We have also chosen A0lm so that the time integral over a zero time interval vanishes. For nonlinear partial differential equations, the coefficients cklm are nonlinear functions of aqrs , and for more complex forms of operators D, they are obtained from repeated Chebyshev approximations (see Eqs (21-22)). As a simple example, cklm  aklm for D  1. If the operator D contains spatial derivatives, the following formulas are used [10]: L 1 1 d L ´ GklmTl ( )   ´ gklmTl ( ), gklm   Bx dx l  0 l0

L



 l 1  l odd

L2 1 d2 L ´ ´ H T (  )  hklmTl ( ), hklm  2  klm l 2  dx l  0 Bx l0

2 Gk  m

L



 l  2  l even

 ( 2  l 2 )H k  m

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(16)

(17)

Time-Spectral Solution of Initial-Value Problems Note that the coefficients

l  L  2, respectively.

9

gklm and hklm are valid for the intervals l  L  1 and

2.4. Minimax Considerations The finite number of spectral terms introduces some subtleties. Although Eq. (13) may be solved to order K + 1, the solution ansatz (3) is limited to order K. Assuming that K is the highest temporal mode number used in the computation, the term AK 1,lm in the sum of Eq. (15) must still be retained so that

A0lm is correctly calculated for a true K order minimax

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approximation. This is also the standard procedure used in this work. Since the integrals in Eq. (11) are then truncated to order K, the initial condition u(t 0 , x; p) will not be exactly reproduced when setting t = t0 in the solution Eq. (3) due to that the integrals will become non-zero (to order K + 1 they are indeed zero). Thus, in a K order computation, one has to consider a trade-off between either exactly reproducing initial conditions for t = t0 with somewhat reduced global accuracy, or maintaining optimal minimax accuracy with slightly inexact reproduction of initial conditions in the solution Eq. (3) for t = t0. The former case is obtained by simply summing to order K in Eq. (15). In most calculations this subtlety does not present a problem. For example, one may wish to chose a low value of L, which would nevertheless be insufficient for an exact representation of the initial state. Alternatively, if one choses to represent the initial condition exactly, the sub-optimal approximate solution (3) will only weakly deviate from the minimax solution. There is, however, a remedy to save both minimax accuracy and true representation of the initial condition. By letting aK 1,lm  0 everywhere except for on the left hand side of Eq. (13), a solution is obtained which exactly reproduces the initial condition for t = t0. We do not give a formal proof here; rather we conjecture (after having studied a few cases) that this solution exactly coincides with the WRM solution produced directly from the original differential form Eq. (1). This is not overly surprising, since the information used by both the differential and the integral formulations becomes the same in this case. Note that this procedure amounts to a cancellation of the residual error |AK+1TK+1(t)|, being discussed in Section 2.3, that would otherwise arise. Recall also that we prefer the integral form (2) since the resulting equations (13) are then conveniently solved with a fixed point root solver. In practice the computation is carried out within the present formulation simply by solving Eq. (13) to order K + 1 and by subsequently extending Eq (3) to order K + 1, including the additional term with coefficient aK 1,lm . This brings us to an additional subtlety. Naively, one may draw the conclusion that a GWRM solution to an initial-value problem would be identical to a minimax, same order Chebyshev approximation of the true solution function. As one can easily convince oneself, by solving simple problems with known solutions, this is not true. The reason is that the residual being minimized represents not only the target function but also the problem formulation. Thus what is minimized in the GWRM formulation is an integral or differential form, implicitly representing the solution. As a consequence, the computed solution may differ somewhat from the corresponding same order Chebyshev approximation of the true

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10

Jan Scheffel

solution. This is usually of minor importance, since solution accuracy is improved simply by increasing the number of temporal, spatial or parameter modes.

2.5. Comparison with Least Square WRM Formulation The accuracy of WRM solutions to initial-value problems are dependent on the choice of spectral functions and the number of spectral terms included as well as on the formulation of the residual and the corresponding weight factors. Thus, the question comes to mind whether a least-square based GWRM would provide more accurate results than does the same order Galerkin GWRM. We will now show that the two approaches will generally result in identical solutions (3). This is gratifying, since the Galerkin GWRM is computationally a more straightforward method than a least-square GWRM. In a least-square GWRM formulation, the coefficients aklm would be determined from

 aklm

t1 x1 p1

R

2

wt wx w p dtdxdp  0

t 0 x 0 p0

or t1 x1 p1

R

   R a

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t 0 x0 p0

wt wx w p dtdxdp  0

(18)

klm

It may appear that a difference between the two methods is that the integrand of Eq. (5) is allowed to become locally large due to the possibility of sign changes over the temporal domain, whereas the residual may appear to be minimized in a stricter, absolute sense in Eq. (18). This is, however, not the whole story. Note that by expanding each term of R spectrally, R  0 over the entire domain to order K independently of whether the corresponding approximate solution (3) is a close approximation to the true solution or not. Since R  0 everywhere to this order, Eqs. (5) and (18) do indeed result in the same solution. The residual becomes non-zero to order K + 1, however: L

M

R   ´  ´ AK 1,lmTK 1 ( )Tl ( )Tm (P)

(19)

l0 m0

Because of this, the least-square and Galerkin GWRM are expected different results from Eqs. (5) and (18). As noted earlier, the situation computing a Chebyshev expansion to order K + 1 through solving Eq. aK 1,lm  0 everywhere on the right hand side of the equation. The residual

to give slightly is remedied by (13) by letting term Eq. (19) is

then eliminated. In summary, least-square and Galerkin GWRM solutions to initial-value problems are identical to order K, and also to order K + 1 if the solution technique just described is applied. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Time-Spectral Solution of Initial-Value Problems

11

The computations of this work are performed using the computer mathematics programme Maple, since all editing, compilation, linking, execution, plotting and debugging are here conveniently performed within the same environment. For some computations, like when solving Eqs. (13), analytic differentiation and analytic simplification of expressions, being easily carried out in Maple, is desirable. The GWRM is easily coded in numerically efficient languages like Matlab or Fortran. The computational speed per se is not important for the benchmarking and comparisons with other methods given in Section 7; rather it is here more important that all comparisons are carried out within the same environment.

3. BOUNDARY AND INITIAL CONDITIONS We now turn to a discussion of implementation of initial and boundary conditions in the GWRM. Their number depends on the number of equations in (1) and by the order of the spatial derivatives. It is already shown that initial conditions enter directly into Eq. (13). Boundary conditions should be applied at coefficients aklm at the upper end of the spatial mode spectrum. This can be seen in several ways. From Eqs. (16-17) it is clear that the Chebyshev representation of functions differentiated l times is only valid up to order L - l. Thus the coefficient equations (13) do not apply for higher spatial mode numbers.

K5

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k

4

D1

I 3

2

D2 1

0

l 0

2

4

6

8

L-1

10 L

Figure 1. Diagram, illustrating the flow of information in Chebyshev space to a modal point (k,l), associated with the coefficient aklm (the modal point is marked with a cross) from nearby modes when performing integration (I) in time as well as single differentiation (D1) or second differentiation (D2) in space. Modes that are used for implementing initial conditions (empty squares) and boundary conditions (filled squares) are also indicated (two boundary conditions are shown).

Furthermore, it is instructive to consider the flow of information in Chebyshev space, associated with temporal integration and spatial differentiation during iteration of Eq. (13); Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

12

Jan Scheffel

see Fig. 1. Note that for differentiation, only higher order modes contribute to the value of the Chebyshev coefficient at a specific modal point whereas for integration, the Chebyshev coefficient at modal point k samples information from modal points both at k - 1 and k + 1. Modes that contribute to the values of the integral or derivatives are marked. Modes outside the computational domain (dashed region and beyond) are defined to give zero contribution. The spatial domain behaviour is consistent with that the solution (13) is defined only to spatial orders less than LBC. Thus L - LBC is the number of boundary conditions that should be imposed for all k and m. In the diagram, modal points used for two boundary conditions are shown (filled squares). It is seen that any error occuring at high spatial mode numbers is amplified trough the multiplicative terms in Eqs. (16)-(17), and numerical instability could result. Since Chebyshev coefficients usually converge rapidly with mode numbers and since the boundary conditions are considered known, numerical stability is in practice not compromised by this effect. The initial condition is imposed at k = 0 for all modes with 0 ≤ l < LBC ≤ L and 0 ≤ m ≤ M and are marked by empty squares. The initial condition may be chosen arbitrarily. If the initial condition requires many, or all, temporal modes for sufficient resolution, care must be taken not to conflict with the boundary conditions applied at high l values. Preferably, initial conditions are chosen so that they satisfy the boundary conditions.

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3.1. Implementation of Boundary and Initial Conditions – Chebyshev Approximation The boundary conditions are implemented into the GWRM in the following way. We choose here to describe the case of Dirichlet boundary conditions; one at each end of a 1D spatial interval. Other types of boundary conditions may straightforwardly be implemented once this case is understood. For systems of equations with many boundary equations, subroutines for handling this are preferably programmed. The boundary conditions are Chebyshev expanded as K

M

u(t, x0 ; p)   (t; p)   ´  ´ kmTk ( )Tm (P) k0 m0 K

M

(20)

u(t, x1; p)   (t; p)   ´  ´  kmTk ( )Tm (P) k 0 m0

We choose to apply discrete Chebyshev interpolation both for initial and boundary conditions since this procedure has precisely the same effect as taking the partial sum of a Chebyshev series expansion and is easily computed [10]. We have generalized the well known one-dimensional Chebyshev polynomial interpolation of a function to three variables in time, physical space and parameter space, being shifted so that t [t 0 ,t1 ] , x [x0 , x1 ] and

p [ p0 , p1 ] . This formula can then be reduced in an obvious way to two variables for

Chebyshev expansion of the boundary and initial conditions discussed here, or further generalized to include more variables. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Time-Spectral Solution of Initial-Value Problems We thus approximate a function K

L

13

 (t, x; p) with the finite Chebyshev series

M

 (t, x; p)   ´  ´  ´ cklmTk ( )Tl ( )Tm (P)

(21)

k0 l0 m0

with coefficients

cklm 

2 2 2 K 1 L 1 M 1 * * *     (tq , xr ; ps )Tk (tq )Tl (xr )Tm ( ps ) K  1 L  1 M  1 q 1 r 1 s 1

(22)

where

t q*  Bt t q  At , xr*  Bx xr  Ax , t q  cos(

ps*  Bp ps  Ap



 1 1 (q  )), xr  cos( (r  )), 2 2 K 1 L 1

ps  cos(



1 (s  )) 2 M 1

The Chebyshev approximation given by Eqs. (21-22) can be shown to be, under rather mild conditions, an accurate polynomial approximation of  (t, x; p) [10]. The boundary condition Chebyshev expansion coefficients

 km

and

 km

are obtained by using the two-

dimensional version of Eqs. (21-22) with the known functions if

 (t; p)   (t; p)  0 then all coefficients  km

and

 km

 (t; p) and  (t; p) . Clearly,

must be zero. From Eqs. (3) and

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(20) we obtain the relations L

 km   ´ aklmTl ( (x0 )) l0

(23)

L

 km   ´ aklmTl ( (x1 )). l0

Since

 (x0 )  1 and  (x1 )  1, we use Tl (1)  (1)l

and

Tl (1)  1 to implement

the two boundary conditions at the highest modal numbers of the spatial Chebyshev coefficients;

(24)

for L being even (upper sign) or odd (lower sign), respectively, with

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14

Jan Scheffel L2

L2

l0

l0

S   ´ aklm (1)l , S   ´ aklm

(25)

blm in Eqs. (8) and (13), for the initial condition expansion, are computed by using the analytical form for u(t 0 , x; p) in a two-dimensional formulation The Chebyshev coefficients

of Eqs. (21-22) in physical and parameter space. A simplification occurs for periodic boundary conditions, that is when u(t, x0 ; p)  u(t, x1 ; p). This relation is only satisfied for even Chebyshev polynomials. Here, considerable computation time is saved by only computing coefficients aklm with even values of l. In summary, initial and boundary conditions are initially transformed into Chebyshev space by use of Eqs. (21-22) in suitable dimensional forms. All subsequent computations are performed in Chebyshev space, using Eqs. (13) and equations for the boundary conditions of the form (25). When sufficient accuracy in the coefficients aklm is obtained, the solution Eq. (3) is obtained in physical variables. For periodic boundary conditions, coefficients l odd can be neglected.

aklm with

3.2. Example – 1D Diffusion Equation

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To illustrate the GWRM including application of boundary and initial conditions, we solve the linear diffusion equation

u 2u  2 t x with

boundary

(26) conditions

u(t,0; )  u(t,1; )  0

and

initial

condition

u(0, x; )  x(1 x). Using the solution ansatz (3), with P  (  A ) / B , and the relations given above it is found that, for 0 ≤ r ≤ L - 2 and 0 ≤ s ≤ M K 1

a0rs  2(brs   Aqrs (1)q ) for q = 0

(27)

q 1

aqrs  Aqrs for 1 ≤ q ≤ K or 1 ≤ q ≤ K+1 with

Aqrs

2B  (2 s 0 A   s1 B ) t q

L



k(k 2  r 2 )[aq1, k, s  aq 1, k,s ]

kr2 k  r even

aK 1,r,s  aK  2,r, s  0. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(28)

Time-Spectral Solution of Initial-Value Problems

15

The boundary conditions enter as, using upper/lower signs for L being even/odd, respectively;

(29)

with L2

L2

S   aqls (1) , S   ´ aqls l0

´

l

(30)

l0

The initial condition is given by (31) b0s  1 / 4, b2s  1 / 8 with all other brs  0 . The linear system of equations (27-31) is solved either using a

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linear solver or in one iteration step, using a nonlinear solver.

Figure 2a. GWRM solution u(t, x) of the diffusion Eq. (26) with initial condition u(0, x)  x(1  x) and boundary condition u(t, 0)  u(t,1)  0 . Difference between exact and GWRM solution is shown versus t and x. Here K = 5, L = 5, t0 = 0 , t1 = 20 and  is set to a constant value 0.01. K modes have been used in the sum of Eq. (27), and K modes are used in Eq. (13).

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16

Jan Scheffel

To illustrate the difference between using K or K + 1 temporal modes in Eq. (27), we have produced the graphs in Figure 2. A GWRM solution u(t, x) of Eq. (26), without expansion in the parameter  , is compared to an analytical solution found by using separation of variables: 

u(t, x; )   al e l  t sin(lx) 2

2

(32)

l1

with coefficients 1

al  2 u(0, x; )sin(lx )dx .

(33)

0

The first 50 terms of the sum in Eq. (32) provide sufficient accuracy. In Figure 2a K modes have been used in the sum of Eq. (27), and also K modes are used for the coefficients aqrs to obtain the solution (13). As discussed in Section 2.2, this corresponds to a correct

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representation of the initial condition, but incorrect minimax Chebyshev expansion. In Figure 2b K + 1 modes are used in Eq. (27), leading to incorrect initial condition representation, but correct minimax expansion. Unless otherwise stated, this is the approach used in the remainder of the work presented here. The coefficients used in Figure 2c include also a K + 1 term in (13), being in line with the “remedy procedure” discussed in Section 2.2 which leads to correct initial value and minimax representations. It is seen for this example that the accuracy of all cases are comparable, with cases a) and b) slightly more accurate than case c).

Figure 2b. As Fig. 2a, but here K + 1 modes are used in the sum of Eq. (27). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Time-Spectral Solution of Initial-Value Problems

17

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Figure 2c. As Fig. 2a, but here K + 1 modes are used in the sum of Eq. (27), and K + 1 modes are used in Eq. (13).

In practice, the manual work in solving a specific partial differential equation is strongly reduced by having designed a set of procedures, that can easily be called and configured to generate a representation of the problem at hand. The mathematics of the GWRM has, in this work, been coded as a set of Maple procedures that generate Chebyshev coefficients for differentiation, integration, products as well as for initial and boundary conditions.

4. NONLINEARITY Nonlinear terms of the operator D are treated fully spectrally in this method, in contrast to in pseudo-spectral schemes [9], where the nonlinearities are transformed to physical space, multiplied there and then transformed back to spectral space. This procedure causes the problem of aliasing, which is avoided in the present scheme. In the GWRM, as nonlinear products are produced in spectral space, Chebyshev modes that lie outside the modal representation (K,L,M) will be truncated with associated loss of accuracy. As mentioned earlier it can be shown that truncated Chebyshev polynomials, because of their minimax properties, are the most accurate polynomial representation to this order [10]. For the sake of simplicity, we now discuss the handling of a second order nonlinearity in one-dimensional Chebyshev spectral space. Higher dimension cases are easily obtained from

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18

Jan Scheffel

that of one dimension. We also omit the arguments of the Chebyshev functions, which are assumed identical. Thus we wish to determine the coefficients M



m0

´

N

amTm   bnTn  ´

n 0

M N



´

ck in

ck Tk

(34)

k0

A basic and useful relationship is the identity

TmTn  (Tm  n  T|m  n| ) / 2 , which

’linearizes’ expressions containing simple products of Chebyshev polynomials. Since all variable expansions have the same number of modes within the same space (temporal, physical or parameter space), we may assume that N  M in Eq. (34). After some algebra, the following exact expression is determined: M

ck  ( fk1 / 2)  ´ am [(1   km )b|k  m|  (1   k 0 )bk  m ]

(35)

m0

being valid for 0 ≤ k ≤ 2M. Here

fk  1 / 2 for k  0 , and fk  1 for k  0 . The prime on

the summation sign denotes that all occurences of a zero index of a and of b should render a multiplying factor of 1/2. Note that only the coefficients for the employed spectral space are computed (we thus compute ck for 0 ≤ k ≤ M); other terms are truncated. The computation is

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best facilitated by creating a procedure that can be repeatedly called also when computing coefficients for higher order nonlinearities.

5. ROOT SOLVER The GWRM solution to an initial-value problem is found when the Chebyshev coefficients of Eq. (13) are determined to sufficient accuracy. For a linear problem, the coefficients can be obtained by a simple Gaussian elimination procedure. Nonlinear differential equations, however, lead to nonlinear algebraic equations and these may be difficult to solve numerically [11]. We thus need a robust nonlinear solver that converges both globally and rapidly. Although various such methods already exist [11], we have found it rewarding to develop a new semi-implicit root solver, SIR [12]. The GWRM is well adapted for solution using iterative methods for two reasons. First, Eq. (13) can be immediately cast in the fixed-point iterative form

x   (x)

(36)

where the solution vector x here contains the Chebyshev coefficients aqrs to be determined, and the vector function



reflects the functional forms of Aqrs and Fqrs . Second, all iterative

methods require an initial estimate of the solution vector, and if this deviates too much from Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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the solution to be determined, numerical instability results. For the GWRM, the coefficients that correspond to the solution for the entire time domain (the roots of the equations) may deviate strongly from the coefficients of the initial state. One of the simplest and most frequently used solvers, Newton’s method, features a fairly limited domain of convergence [11,13,14], however. Because the initial guess in the case of the GWRM is precisely the initial condition, there always remains the possibility to reduce the solution time interval [t 0 ,t1 ] , for example by using subdomains as described below, so that the solution Chebyshev coefficients become sufficiently close to the initial Chebyshev coefficients. This, incidentally, shows that a GWRM formulation of a well posed initial-value problems in principle always will lead to a solution, although we do not prove this rigorously at present. Newton’s method is usually globally improved by the addition of line-search methods, in which the iteration step size is decided from the minima of the function, the roots of which are to be determined. Unfortunately, these methods may land on spurious solutions, corresponding to local minima rather than to true zeroes of the function. We have thus developed the semi-implicit root solver (SIR), being an iterative method for globally convergent solution of nonlinear equations and systems of nonlinear equations. By ”global” is here meant that correct global solutions are usually (but not always) found, having the the new feature that they are never local, non-zero minima. It is shown in [12] using a set of test problems, that global convergence is at least as good as for Newton-like line-search methods. Convergence is quasi-monotonous and approaches second order in the proximity of the real roots. The algorithm is related to semi-implicit methods, earlier being applied to partial differential equations. We have shown that the Newton-Raphson and Newton methods are limiting cases of the method. This relationship enables efficient solution of the Jacobian matrix equations at each iteration. The degrees of freedom introduced by the semi-implicit parameters are used to control convergence. Let us take a short look at the philosophy behind semi-implicit root solving. The fundamental algorithm of SIR lends inspiration from developments in semi-implicit methods for solving partial differential equations in fluid dynamics [9] and in magnetohydrodynamics (MHD) [2,3]. In the latter approach, two new (essentially equivalent) terms are added to each time discretized equation, one being evaluated at the old time step and one at the advanced time step. By a proper choice of the added terms, unconditional numerical stability can be achieved for any choice of time step size. The error introduced by the extra terms tends to zero for near steady state solutions. For other cases, accuracy may sometimes be an issue. If semi-implicit terms are applied to algebraic equations, however, there is no deteriorating effect on accuracy whatsoever, once convergence has been reached. Details of SIR are given in Ref. [12]; we here only briefly describe the basic formulation. Instead of direct iteration, using Eq. (36), the semi-implicit method leads instead to the problem of finding the roots to the N equations N

xm    mn (xn   n (x))   m (x)   m (x;A) n 1

or, in matrix form

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(37)

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Jan Scheffel

x  A(x   (x))   (x)  (x;A)

(38)

The system (38) has the same solutions as the original system, but contains free parameters in the form of the components  mn of the matrix A. These parameters are determined by specifying the values of

 m / xn , the gradients of the hypersurfaces  m .

The latter gradients control global, quasi-monotonous and superlinear convergence. In SIR,  m / xn  0 for all m  n , whereas  m / xm is finite and is chosen to produce limited step lengths and quasi-monotonous convergence; it usually approaches zero after some initial iterations. Since Newton’s method is a limiting case of the present method, namely when all  m / xn  0 , rapid second order convergence is generally approached

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after some iteration steps. The relationship to Newton’s method fortunately leads to approximately similar numerical work, essentially that of solving a Jacobian matrix equation at each iteration step. There are two aspects of the GWRM that are of particular importance for the root solver. First, the algebraic equations to be solved are polynomials of the same order as the nonlinearity of the original differential equations. For example, second order nonlinear pde’s lead to the solution of a system of second order polynomial equations by SIR. Since a large class of problems in physics, formulated as pde’s, feature second (or third) order nonlinearities, there is a potential to device more efficient versions of SIR where this fact has been utilized. Second, most of the computational time in SIR, when applied to the GWRM, is not spent on matrix equation solution, but rather on function evaluation. If the functions  n are formulated and evaluated more economically, computational efficency may be improved. We conclude this section by stating that the SIR algorithm has turned out to be robust and well suited for all GWRM applications tried to date. Further development would focus on the possiblity to enhance SIR efficiency by economizing the handling and evaluation of the polynomial equations (13).

6. SUBDOMAINS The number of arithmetical operations involved in matrix inversion typically features a cubic dependence on the number of unknowns. The root solver, applied to Eq. (13), may consequently dominate computational time. Straightforward application of GWRM and SIR operations for each iteration would require approximately when solving Eq. (13). Using LU decomposition rather than matrix inversion, the number of [11]. As shown in the examples of the next operations for solving Eq. (13) is reduced to section, this may sometimes be an acceptable amount of work. As mentioned earlier, it is worthwhile to simplify the  m expressions analytically before starting SIR iterations. For more complex calculations, however, efficiency requires the temporal and spatial domains to be separated into subdomains. This enables a linear rather than a cubic dependence of efficiency on, for example, the number of spatial modes applied to the entire domain, given that the number of subdomains is proportional to L. Assume that the temporal and spatial domains are divided into Nt and Nx subdomains, respectively. This reduces

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operations to only operations when solving a particular problem, assuming that the same total number of modes are sufficient in both cases. As an example, for K = L = 11, M = 2 and Nt = Nx = 3 there would be a reduction from about 2.7·107 to 3.3·105 operations. Is really the required total number of modes the same, for a given accuracy, when using subdomains? The answer is: it depends. Clearly, a region including strong gradients is preferentially resolved into subdomains since one global Chebyshev approximation would require a high number of modes to resolve the critical region while at the same time faithfully represent the entire domain, whereas far less modes would suffice locally within this region. The overall number of modes required may thus be less when subdomains are used. If the gradients are spread over the entire domain perhaps a globally averaged solution, where accuracy is sacrificed, would be preferable. In this case, there is no need for subdomains. If high accuracy is required, also the subdomains must retain a reasonable number of modes. Hence, the total number of subdomain modes used will exceed those of the global solution in these cases. The conclusion becomes that if efficiency need be optimised with respect to the number of subdomains and total number of modes, this should be decided individually for each problem. As will be shown in the next Section, an elegant solution to this problem is to automatize this decision by using an adaptive scheme. Finally, as an additional comment on efficiency, it should be remembered that the functions  m will become substantially less complex when subdomains are used, with resulting reduced computational effort. Temporal and spatial subdomains are implemented quite differently. Intuitively, a temporal domain using K modes could simply be split up into Nt subdomains with K/Nt modes in each. As initial condition for each domain the end state of the previous one is used. This is indeed also the case. Some temporal subdomains may need more than K/Nt modes for sufficient accuracy, however. This may occur for stiff differential equations. Recall that a GWRM (as well as any WRM) solution is not per se a Chebyshev approximation of the true solution, but rather stems from a minimization of the residual, including information concerning the differential formulation of the problem, over the solution domain. Simple averaging (by using a few modes) over regions with strong gradients is likely to produce large errors, due to the poorly approximated differential character of the problem. As mentioned, a preferred solution is to use an adaptive scheme, which uses few modes by default in each subdomain, but increases this number whenever accuracy so requires. One may also note that the use of temporal subdomains is beneficial for SIR convergence, since the initial condition for each domain will be closer to the final solution than what would be the case using a single temporal domain instead. Spatial subdomains must be treated in another fashion. The reason is that boundary conditions are usually only known at the exterior, rather than at interior, boundaries. Thus a computation cannot progress successively through a sequence of spatial subdomains, as for temporal subdomains. Instead the boundary conditions should be imposed on the outermost spatial subdomains, and the subdomains are connected at interior boundaries through continuity conditions. For most problems, the functions and their first derivatives should be continuous across each subdomain (interior) boundary. All spatial subdomains are updated in parallel at each solution iteration. Computationally, the choice of procedure is a nontrivial task. Due to the large coefficients, appearing in higher order derivatives (see Eqs. (16-17)), derivative matching is sensitive to small errors and numerical instability may result. Instead

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Jan Scheffel

we have found that a ”handshaking procedure” where the functions are allowed to overlap into neighbouring domains, and are doubly connected, yields improved stability over derivative matching. An important question is whether the set of boundary conditions for all external and internal boundaries should be consistently solved for at each iteration or if only local matching of boundary conditions should be performed. Our experience is that global consistency does yield faster convergence, but that the corresponding matrix inversion is very costly due to the large number of terms involved. Independent solution of each spatial subdomain at each iteration is far less costly but may, however, require up to about twice as many iterations. The latter procedure is nevertheless the most efficient. Further details of optimised procedures for spatial subdomains, including the possibility to adjust the number of modes in each subdomain, are not covered here. Suffice it to say that we have successfully developed and applied the handshaking scheme to some of the examples illustrated in the next section. Similarly as for the spatial domain, subdomains may be introduced into the physical parameter domain. Should a global, semi-analytic solution be desired, a Chebyshev approximation (19) covering all subdomains may easily and efficiently be carried out at the end of the computation. Taken together, we have seen that the use of temporal, spatial and parameter subdomains substantially reduces GWRM computational time, because of the transition from cubic to near linear dependence on the number of modes.

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7. APPLICATION - ACCURACY AND EFFICIENCY We now turn to the important questions of accuracy and efficiency. In this section, the GWRM is compared by example to other methods for solving partial differential equations, that use time discretization in the form of finite differencing. Clearly, besides from producing semi-analytic solutions, the GWRM must be comparable to these standard methods with regards to accuracy and efficiency to be of practical use. We have already discussed GWRM solution of the linear 1D diffusion equation. To introduce application of the GWRM to nonlinear problems and study performance, a stiff ordinary differential equations is first solved. Adaptive, temporal subdomains are here showed to provide high accuracy and efficiency. Next we turn to the nonlinear, 1D viscous Burger equation, which features a shock-like structure near the boundary. It is shown that GWRM accuracy is comparable to that of the (explicit) Lax-Wendroff and (implicit) Crank-Nicolson schemes, for a similar number of floating operations. The GWRM has, of course, the additional feature of providing a finite semi-analytical solution. For the subsequent cases of a wave equation without and with a forcing term, characterised by two strongly separated time scales, the GWRM turns out to be considerably more efficient than both the Lax-Wendroff and the Crank-Nicolson solution methods when tracing the dynamics of the slower time scale. Finally we demonstrate successful application of the GWRM to the demanding problem of solving the linearized system of ideal magnetohydrodynamic (MHD) equations. This

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23

problem is of great interest when studying the stability of magnetically confined plasmas for purposes of controlled thermonuclear fusion. We conclude this section with a brief discussion on GWRM solution of other systems of nonlinear partial differential equations.

7.1. Introductory Example; the Match Equation When a match is lighted, the flame grows rapidly until the oxygen it consumes is balanced by the oxygen that comes through the surface of the ball of flame. A simple model for the flame propagation in terms of the ball radius u(t) is

du / dt  u 2  u 3

(39)

with

u(0)   , 0  t  2 / 

(40)

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For small values of  this problem becomes very stiff through the presence of a ramp at t  1/  , representing the explosive growth of the ball towards its steady state size [16]. We have solved this problem by using Eq. (35), transforming it to the form of Eq. (2), yielding a set of equations corresponding to Eq. (13) in which spatial and parameter modes are omitted. A solution with   0.0001 is presented in Figure 3. We have imposed an accuracy of  = 1.0·10-4. The GWRM solution is compared with the exact solution to Eqs. (39-40) which is

u(t )  1/[W (ae a t )  1] where

(41)

a  1/   1 and W is the Lambert W function.

Figure 3a. Solution to the match equation (39), with   0.0001 , K = 6,   1.0  10 , using an initial subdomain length of (2 /  ) / 10 . 4

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Jan Scheffel

Clearly, for this small value of  the ramp is very distinct and hard to resolve. Consequently, explicit finite difference methods will need extremely small time steps to resolve this problem. An optimised Matlab solution to the problem uses implicit methods that may reduce the computational effort to about 100 time steps, taking a few seconds on a tabletop computer.

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Figure 3b. As Fig. 3a, but with the ramp region near t  1 /  enlarged.

Figure 3c. Absolute error for the computation of Fig. 3a.

The GWRM solution in Fig. 3 uses 69 time domains and takes just about the same amount of computational time, but has the additional feature to provide analytical approximations to the solution in each domain. These may be of particular interest in the ramp region. For efficiency, the temporal domain length has been automatically adapted as follows. Throughout we have used the relation | Tn (t ) | 1, formulating the accuracy criterion as

(| aK 1 |  | aK |) /(| a0 |  | a1 |)   .

In performing the computation, a default of 10 time subdomains is assumed and K = 6 is used throughout. If the accuracy criterion is satisfied, the subdomain length is doubled at the next domain, and if not it is halved. In the latter case, the calculation is repeated for the same subdomain until the accuracy criterion is satisfied. This goes on as the calculation proceeds in time until near the endpoint, where the subdomain length is adjusted to land exactly on the predefined upper time limit. Due to the stiffness of the problem in Figure 3, the subdomains are concentrated near t = 1.0·104 where the subdomain length may be as small as about 2 time

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Time-Spectral Solution of Initial-Value Problems

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units. The automatic extension of the subdomain length in smoother regions saves considerable computational time; at the end of the calculation the subdomain length is several thousand time units. The essential information in the computations of Figure 3 is the transformation from the plateau u = 0 to the plateau u = 1 at t = 1.0·104. Perhaps we are willing to sacrify accurate details of the transition region, and would be satisfied with a global GWRM solution that only approximately models the transition, using only a few modes. As discussed earlier, GWRM solutions are not identical to Chebyshev approximations of the true solutions, but also mirrors the effect that results from the differential formulation of the problem. In other words – an implicit formulation of a function as a differential equation plus initial and boundary conditions will render approximate solutions that are, in some sense, imprints of the formulation. This imprint will, of course, diminish as the true solution is approached. In Eq. (39) there are also quadratic and cubic nonlinearities. As a result a global, low mode approximation of the solution is not trivially obtainable. The transition region needs a certain amount of resolution to “tie” with the solutions at lower and higher times t. This is an interesting topic for future studies. In summary, a stiff ordinary differential equation has been solved to high accuracy using the GWRM. Due to use of automatic domain length adaption, high efficiency is also obtained, comparing well with highly optimised Matlab routines for implicit finite difference methods.

7.2. Accuracy; Burger’s Equation

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Burger’s equation is of particular interest since it is nonlinear and contains two time scales as a result of the competition between convection and diffusion. The one-dimensional Burger partial differential equation

u u 2u  u   2 x t x

(42)

thus contains essential physics, such as convective nonlinearities and dissipation, expected to be encountered also in more complex problems of fluid mechanics and MHD. Here  denotes (kinematic) viscosity. Since this equation has an analytical solution, it provides excellent benchmarking.

7.2.1. Exact Solution The exact solution to Eq. (42) is found by performing a transformation by use of the Cole-Hopf transformation [7]

u  2

 / x

(43)

to produce a standard diffusion equation in (t, x) and then by using the Fourier method. The result, for the boundary conditions u(t,0)  u(t,1)  0 is Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

26

Jan Scheffel 

u(t, x) 

2  ´ mAm e m  t sin(m x) 2

m0  ´



Am e

2

(44)  m2  2 t

cos(m x)

m0

with coefficients 1

Am  2   (x)cos(m x)dx

,

(45)

0

where

 (x)  (0, x) . As an example, the initial condition

u(0, x)  x(1  x)

(46)

results in

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 (x)  e(3x

2

 2 x 3 )/(12  )

.

(47)

It should be noted that the sums of the exact solution Eq. (44) may need to be carried out over a large number of terms for sufficient accuracy, because of the poor convergence at low viscosity. As   0.005 at least 100 terms are required to compute a solution that gives a reasonably accurate solution near t = 0. Furthermore, in contrast to polynomials or Chebyshev polynomials, the exponential and trigonometrical functions of Eq. (44) are costly to evaluate numerically. This is one example of an exact solution that is of limited practical use. The most challenging aspect of the Burger equation, from the modelling point of view, is the shock-like structure that evolves for weak dissipation. The associated gradients are often difficult to resolve in spectral representations. We here study two cases. The first case develops a strong gradient near the boundary x = 1, and is representative of the gradients in, for example, edge pressure or temperature, encountered in magnetohydrodynamic computations in fusion plasma physics modelling. In the second case, we form an internal shock-like structure, with its spatial position moving in time. This structure is, for example, typical when modelling localized resistive instabilities in the tokamak and reversed-field pinch magnetic fusion configurations. It is desired that the GWRM should be able to resolve these structures for limited values of mode numbers. The two cases are compared to solutions obtained using the standard explicit LaxWendroff and implicit Crank-Nicolson finite difference schemes for partial differential equations [11].

7.2.2. GWRM Solution In Figure 4a, the GWRM solution of Burger’s equation for boundary conditions u(t,0)  u(t,1)  0 and initial condition (x) = x(1-x) is shown. Nine temporal (K = 8), eleven spatial (L = 10) and three parameter (viscosity) modes (M = 2) were required to obtain

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an error better than 1% after 7 iterations. The solution is valid within the domain [t 0 ,t1 ]  [0,5] , [x0 , x1 ]  [0,1] and [ 0 , 1 ]  [0.01, 0.05], and is displayed for fixed t =

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2.5. Note that the solution was attained for all viscosities in the given range in a single computation. It is seen that a sharp gradient builds up near the edge, being most profound for small values of viscosity. If the number of temporal or spatial modes are reduced somewhat, the same accuracy is retained everywhere except for near the edge x = 1. Since the ”exact” Cole-Hopf solution converges slowly at these low values of  , the obtained GWRM semianalytical solution is actually computationally more economical to use in applications.

Figure 4a. GWRM solution of Burger’s Eq. (42) with initial condition (x) = x(1 - x) and boundary condition u(t, 0)  u(t,1)  0 , shown versus x and  at time t = 2.5. Here K = 8, L = 10, and M = 2.

To enable comparisons with explicit and implicit finite difference partial differential equation solvers, we will now fix viscosity to   0.01 and compute the solutions as functions of t and x. The Burger equation, defined as above but now using t1 = 10, is solved using all GWRM, explicit Lax-Wendroff [11], and implicit Crank-Nicolson [11] methods. The latter two schemes are accurate to second order in both time and space. For the GWRM solution, two spatial subdomains with internal boundary at x = 0.75 are used. A similar result would be obtained using only one spatial domain with slightly higher number of spatial modes. With mode numbers K = 9, L = 7 an absolute global accuracy of 0.001 is obtained after 10 iterations, with a tolerance of 1.0·10-6 for the coefficient values. Results are displayed in Figure 4b-d. The peak near t = 0 in Figure 4d is due to the poor convergence of the exact solution of which 50 terms are used.

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Jan Scheffel

  0.01 .

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Figure 4b. GWRM solution of Eq. (42) with (x) = x(1 - x) and u(t, 0)  u(t,1)  0 , for Two spatial subdomains are used, with internal boundary at x = 0.8, and K = 9, L = 7.

Figure 4c. Snap shots of the solution in Fig. 4b, showing the gradient that builds up due to convection near the boundary x  1 , for times t / t1 = 0, 0.1, 0.2, 0.3, 0.4 and 0.5, with t1 = 10. The solution continuously decreases in amplitude due to viscous dissipation. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Time-Spectral Solution of Initial-Value Problems

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Figure 4d. Difference between the exact solution of Burger’s equation (first 60 terms of Eq. (44)), and the GWRM solution.

7.2.3. Finite Difference Solutions We now turn to solution of the Burger equation using finite difference methods. Accurate solutions are not straightforwardly obtained beacuse of the strong edge gradient. Let us estimate the spatial step length required for a global error  = 0.001. A second order estimate of the mid-point error resulting from finite spatial differencing with spacing ∆ x is

[ f (x  12 ∆ x )  f (x  12 ∆ x)] / 2  f (x) 

1 8

f´´(x) (∆ x)2   ,

(48)

where a prime denotes spatial differentiation. From the exact solution it is found that max f ´´(x) = 20.3 at (t,x) = (2.05,0.94). A maximum global error of  = 0.001 thus requires ∆ x < 0.02. The Lax-Wendroff finite difference scheme is widely used because of its reliability and because it is accurate to second order in both time and space. Since it is explicit, the maximum time step becomes limited, however. A von Neumann analysis of the LaxWendroff method applied to the Burger equation (42) features the limiting cases of strong convection or strong diffusion. When convection dominates, the standard CFL condition ∆ t  ∆ x / uc , where uc is a characteristic fluid velocity, results. This condition characterises the required causality on the solution grid for hyperbolic problems. When the diffusion term dominates, the problem is parabolic and the time step turns out to be limited by

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Jan Scheffel

causality to t  (x)2 / (2 )  (t)crit . Computations show that the latter criterion is the more relevant one for the present Burger problem. Recall that accuracy requires ∆ x < 0.02 according to Eq. (48). This is in reasonable agreement with the value x ≤ 1/70, that was found numerically. For t  0.98(t)crit , the number of time steps becomes 1000 for the

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given accuracy. The absolute error of a Lax-Wendroff computation is shown in Figure 4e. High accuracy is obtained everywhere except near the maximum spatial gradient. Using Maple 12 on the same platform for both methods, the Lax-Wendroff method needs 50% less time than the GWRM. It is thus somewhat more accurate for the same number of computational operations in this case. Note, however, that the discussion in Section 3.1 shows that for the case of a single spatial domain, the boundary conditions would be periodical (or homogeneous) in which case odd spatial mode numbers can be omitted and an eight-fold gain in efficiency would be attainable. The GWRM solution has also the advantage of being in analytic form whereas the Lax-Wendroff solution is purely numeric.

Figure 4e. Difference between the exact and the Lax-Wendroff solutions of Burger’s equation, for   0.01 . Here 1000 time steps were used, and ∆ x = 1/70. Only each 20th time step and each 2nd spatial step are shown.

Next, we solve the Burger equation using the Crank-Nicolson method. This scheme allows for arbitrarily large time steps by using an implicit approach where the functional values are determined both at the previous and present time steps. On the spatial scale, the resolution x ≤ 1/70 is, however, still needed to obtain a global accuracy of 0.001. To avoid costly matrix inversion at each time step, due to the implicit finite difference formulation, a tridiagonal matrix solution procedure has been developed [11] that radically speeds up the

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calculations for linear equations. To be able to use this scheme for the nonlinear Burger equation, we advanced the linear diffusive term using the standard Crank-Nicolson method, but advanced the nonlinear convective term explicitly. As a result, a von Neumann analysis shows that the time step is no longer unrestricted, but must obey the relation ∆ t  2 / uc 2 .

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Note that this relation is independent of x . For a time step t = 1/50 and with x = 1/70, an accuracy of 0.001 was achieved for the Burger equation, as shown in Figure 4f. The computer time used was about half that of the Lax-Wendroff method. For general nonlinear problems, when a linear higher order term that can be advanced explicitly does not exist, this method may be less accurate however. The reason is that, for making use of efficient tridiagonal matrix solving, the differential equation must be time linearized, which inherently introduces errors.

Figure 4f. Difference between the exact and the Crank-Nicolson solutions of Burger’s equation, for   0.01 . Here 500 time steps were used, and ∆ x = 1/70.

7.2.4. Internal Gradients To test GWRM accuracy at somewhat more extreme conditions, we now investigate resolution of internal gradients. In Figure 4g-h, the Burger equation is solved for a case with  = 0.01, but with different initial condition (x) = sin(2x) +1 and boundary conditions u(t,0)  u(t,1)  1. We have used K = 8 and L = 20 to resolve the moving, internal shocklike gradient that evolves. Note that at t = 0.1 the spatial resolution may need to be improved somewhat if higher accuracy is required. This case also shows the potential of spatial subdomains; the number of spatial modes can be reduced substantially if (preferally adaptive) subdomains are used to resolve the steep gradient region. Similar structures may be

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encountered when, for example, modelling the dynamics of reversed-field pinch fusion plasmas [15].

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Figure 4g. GWRM solution of Burger’s equation with initial condition (x) = sin(2x) + 1 and u(t, 0)  u(t,1)  1, shown for   0.01 . The mode numbers K = 5 and L = 20 have been used.

Figure 4h. Snap shots of the solution in Fig. 4g, showing the internal gradient that builds up due to convection, for times t / t1 = 0, 0.5 and 1.0, with t1 = 0.1. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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In summary; although the original purpose for designing the GWRM was to develop a global method for approximate solution of multiple time scale initial value problems we have found, using the match and Burger equations as representative examples, that the GWRM compares well with the explicit Lax-Wendroff and the implicit Crank-Nicolson methods with regards to accuracy for stiff problems. This is hopeful for future studies within fluid mechanics and MHD. We now turn to model examples for studying GWRM efficiency when small time scale behaviour is of secondary interest to that of longer time scales.

7.3. Efficiency; the Forced Wave Equation Problems in physics often feature multiple time scales, whereas it may be of main interest to follow the dynamics of the slowest time scale. Efficient partial differential equation solvers therefore must be able to employ long time steps, retaining stability and sufficient accuracy. By omitting resolution of the finer time scales, improved efficiency and the possibility to study complex systems are expected. As a test problem, we choose a wave equation with a forcing (source) term, also called the inhomogeneous wave equation:

2u  2u   2  f (t, x) t 2 x

(49)

with boundary and initial conditions

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u(t,0)  u(t,1)  0 u(0, x)  sin(nx) u (0, x)  Asin( x) t Here A, n, and are free parameters, and f (t, x)  A( 2   2 )sin( t)sin( x) is the forcing function. The exact solution is

u(t, x)  cos(n 0.5t)sin(n x)  Asin( t)sin( x) , for

(50)

  m , with m an integer. This problem has the separate system and forcing function  ) and 2 / . Using the parameter values   1, A  10 ,    / 15 , n  3 , the ratio of these time scales becomes R   /(n  ) = 1/45. Thus

time scales 2 /(n

  3

and

the forcing term in (50) has here introduced a time scale much longer than that of the ”unperturbed” system. The exact solution is displayed in Figure 5a for the temporal domain [t 0 ,t1 ]  [0, 30]. The separate dynamics of the two time scales is clearly distinguishable.

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Figure 5a. Exact solution, Eq. (50) of the forced wave equation as defined in Eq. (49).

7.3.1. GWRM Solution We now wish to solve Eq. (49) using all GWRM, Lax-Wendroff and Crank-Nicolson methods. The problem is thus posed as a set of two first order partial differential equations:

U  2u   2  f (t, x) t x u U t

(51)

with boundary and initial conditions corresponding to those of Eq. (49). The GWRM solution, using one spatial domain with K = 6 and L = 8, is rapidly obtained within a single iteration with a tolerance of 1.0·10-6 for the coefficient values. It is displayed in Figure 5b. The solution behaves as desired; it averages over the faster (system) time scale and follows the slower (forced) time scale in an averaging sense. This is shown in more detail in Figure 5c, where the temporal evolutions of both the exact and the GWRM solutions are shown jointly for fixed x.

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Figure 5b. GWRM solution of the forced wave equation (49), using a single spatial domain, with K = 6 and L = 8.

Figure 5c. GWRM temporal evolution of the forced wave equation (49) for x = 0.2 (smooth curve) as compared to the exact solution (oscillatory curve). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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7.3.2. Finite Difference Solutions Next, we solve Eq. (49) using The Lax-Wendroff scheme. Being an explicit method, it must obey the CFL condition, which for this case becomes ∆ t  ∆ x . Reasonable accuracy is obtained for ∆ x ≤ 1/30. Thus the maximum allowed time step is 1/30, and the number of time steps becomes 900 for the domain [t 0 , t1 ]  [0, 30] . The calculation requires about ten times more computer time than the GWRM. It can be seen in Figure 5d that it initially traces the exact solution, but thereafter follows the slower time scale. The solution appears not to average as accurately as the GWRM over the fast time scale.

Figure 5d. Lax-Wendroff temporal evolution of the forced wave equation (49) for x = 0.2 (smooth curve) compared to exact solution (oscillatory curve). Here 900 time steps were used, and ∆ x = 1/30.

The Crank-Nicolson method, being implicit, has no time step restriction and no amplitude dissipation and would perhaps intuitively be well suited for the present problem. Additionally, to avoid time-consuming large matrix equations, the so-called Generalized Thomas algorithm [7] uses a block-tridiagonal matrix algorithm that substantially speeds up the calculations at each time step. If the associated sub-matrix equations are solved for, rather than computing inverse matrices, a gain from Gauss elimination O((MN)3/3) operations to O(5M3N/3) operations is possible, that is the speed gain factor becomes N2/5. Here the number of equations M = 2 and the number of spatial nodes N = 30. The handling of a number of sub-matrix equations, required at each time step, is still limiting performance however. With ∆ x = 1/30, temporal resolution requires at least 50 time steps. Using matrix inversion, the corresponding computation is about three times slower than Lax-Wendroff and thus about 30 times slower than that of the GWRM. A speed gain of a factor three is expected by solving the sub-matrix equations rather than determining inverse matrices, but the GWRM remains considerable faster. The solution is shown in Figure 5e and in Figure 5f the solution is Chebyshev interpolated at x = 0.2 to facilitate a comparison with the exact solution. One

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Time-Spectral Solution of Initial-Value Problems

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notes that the Crank-Nicolson solution appears to strive to follow the exact solution, and does not accurately average over the fast time scale.

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Figure 5e. Crank-Nicolson temporal evolution of the forced wave equation (49) for x = 0.2. In this case 50 time steps were used with ∆ x = 1/30.

Figure 5f. Chebyshev interpolated solution of Fig. 5e as compared to the exact solution (oscillatory curve). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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7.3.3. Conclusions on Forced Wave Equation In conclusion it was found that the GWRM is well suited for slow time scale solution of the present wave equation test problem, which features both a slow and a fast time scale. For suitable mode parameters, it traces the slower dynamics using substantially less computational time than the Lax-Wendroff and Crank-Nicolson schemes. An important factor, contributing to the efficiency, is that whereas the Lax-Wendroff and Crank-Nicolson schemes must solve two first order equations (51) that represent the second order wave equation, the GWRM integrates both these equations formally in spectral space into one equation before the coefficient solver is launched. If results are sought for longer times, temporal subdomains are preferably used for the GRWM, in order to guarantee constant computational effort per problem time unit. For problems with wider separation of the time scales, the GWRM will be an increasingly advantageous method as compared to the LaxWendroff scheme since the latter must follow the faster time scale. This forced wave equation example featured an imposed, periodic, time scale that was longer than the system time scale. How will the GWRM perform when the imposed time scale is shorter than that of the system? At present it appears difficult to adequatly handle such problems using the GWRM. A major complication is that, for efficiency, the number of modes used in the calculation would not adequately resolve the forcing function, so that the problem would not be well defined for the GWRM. This is a subject that should be investigated further. As we saw in the case of the Burger equation, and will discuss further below, multiple time scales may also be inherent in the system we are modelling.

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7.4. Large System of Initial-Value PDE’s - Magnetohydrodynamic Stability As a next step, the GWRM is applied to an advanced research problem featuring a large set of coupled pde’s. In fusion plasma physics research, the stability of magnetically confined plasmas to small perturbations is of considerable importance for plasma confinement. Stability can be studied using a combined set of nonlinear fluid and Maxwell equations, magnetohydrodynamics (MHD). An absolute requirement is that the configuration is arranged so that the plasma is stable on the fastest MHD time scale – the so-called Alfvén time, being of the order fractions of microseconds. If plasma resistivity is included in the MHD model, new instabilities (on the milliseconds time scale) are accessible for the plasma, and remedies should be sought also for these. Let us now apply the GWRM to study the stability of a so-called screw pinch plasma configuration within the traditional set of ideal MHD (zero resistivity) equations. For simplicity the plasma is assumed to be surrounded by a close fitting, ideally conducting wall. It can be shown that such a wall provides maximal stability. The plasma equations are the continuity and force equations, Ohm’s law, the energy equation supplemented with Faraday’s and Ampere’s laws, respectively:

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    ( u)  0 t du  j  B  p dt EuB 0



(52)

d ( p  )  0 dt B E t   B  0 j

Here E and B are the electric and magnetic fields respectively, u is the fluid velocity, j is the current density, p is the kinetic pressure,  is the mass density, = 5/3 is the ratio of specific heats and 0 is the permeability in vacuum. Furthermore, d / dt   / t  u· . To determine the temporal evolution of these equations in circular cylinder geometry, they are linearized and Fourier-decomposed in the periodic coordinates  and z. All dependent variables q of Eq. (52) are assumed to be superpositions of an equilibrium term q0 and a (small) perturbation term q1. Perturbations are assumed proportional to exp[i(kz  m )], where k and m denote axial and azimuthal perturbation mode numbers, respectively. Next, a non-dimensionalization is carried out using the characteristic values (denoted with index ”c”) of plasma radius a, Alfvén velocity

u A  Bc /  0  , with t c  a / u A , pc  Bc2 /  0, and

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Ec  uc Bc. The resulting non-dimensional equations are not given here but become identical

to those using Eq. (52) with 0 = 1. We wish to solve for the time evolution of the perturbed terms for a given equilibrium and a specified perturbation (m,k). If these feature an exponential growth, the plasma is unstable to small perturbations for the assumed equilibrium. The equilibrium is here, for illustration purposes, taken to be that of a simple screw pinch with constant mass density:

B0r  0 B0  r B0 z  0.2

(53)

p0  1 r 2

0  1 After eliminating E and j in Eq. (52) there result seven complex-valued coupled partial differential equations for u1, B1 and p1 respectively as functions of the independent variables time t and cylindrical variable r [16]. They are all written on the component form q1i / t  ... to conform with the GWRM formulation of Eq. (1). The seven dependent variables were all Chebyshev expanded in t, r and in resistivity  (which was here set equal to

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Jan Scheffel

a constant). Since the GWRM is (so far) developed for solution of real valued equations, u1, B1 and p1 are finally split up in real and imaginary parts, resulting in a system of 14 simultaneous equations to be solved by the GWRM. Let us now discuss boundary and initial conditions. It can be shown that, in circular cylinder geometry, the following conditions must hold for m = 1 perturbations near the internal boundary r = 0:

du1r du1 db1r db1    0 dr dr dr dr u1z  b1z  p1  0

(54)

We have chosen to study m = 1 perturbations because they are often the most critical ones with respect to stability. At the outer, ideally conducting, boundary r = 1 the normal components of the fluid velocity and magnetic fields as well as the tangential component of the electric field must be zero. Since p0 (1)  0 here, it also follows that p1 (1)  0. The relation between the initial conditions can be chosen somewhat arbitrary. The reason is found from studies of the corresponding system of eigenequations and is that, for unstable behaviour, a competition between modes with different number of radial nodes will take place until the fastest mode (with zero radial nodes) will dominate the behaviour. The memory of the initial perturbation is then lost. For consistency with respect to the boundary conditions we however choose the following set of initial conditions:

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u1  i(1 r 2 ) u1z  r(1 r )

(55)

b1r  b1  b1z  p1  0 7.4.1. Benchmarking Using Simplified Model The 14 coupled pde’s we are about to solve are linearized, but nevertheless contain nonlinear terms with products between equilibrium and perturbed variables. Further complications include terms divided by r. These arise due to the choice of coordinate system and must be carefully handled to avoid singularities near r = 0. In order to test the GWRM code with respect to handling of these features as well as to test procedures devised to perform integration, multiplication and implementation of initial and boundary conditions, the code was first used for a test problem with the basic features included. We have chosen to solve the problem

1 p1 u1r  r r t p1  3r 2 u1r t

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(56)

Time-Spectral Solution of Initial-Value Problems

41

for 0 ≤ r ≤ 1 and t ≥ 0 using the initial and boundary conditions

u1r (0,r )  1 r 2 p1 (0,r )  r 2 (1 r )

(57)

p1 (t,0)  0 . The exact solution to this problem is

u1r (t,r)  

6 sin( 6t )  cos( 6t )  r sin(3t )  r 2 cos(2 3t ) 3

6 3 4 p1 (t,r )  r ( sin( 6t )  cos( 6t ))  r 3 cos(3t )  r sin(2 3t ) 2 2

(58)

2

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GWRM solutions to the problem given by Eqs. (56-57) are displayed in Figure 6a-d. Parameters are K = 9, L = 5 and three equidistant temporal domains were used. A single SIR iteration is sufficient for each time domain due to linearity in u1r and p1. The comparisons with the exact solutions show that accurate solutions are indeed produced.

Figure 6a. Test problem Eqs. (56-57) solved using the GWRM with K = 9, L = 5 and three equidistant temporal domains. GWRM and exact (Eq. (58)) u1r solutions shown. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 6b. Difference between GWRM and exact solutions of Fig. 6a.

Figure 6c. Graphs of the GWRM and exact p1 solutions to the problem in Fig. 6a. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 6d. Difference between GWRM and exact solutions of Fig. 6c.

But how was the singularity of Eq. (56) near r = 0 avoided? From the exact solution it is seen that the singularity is only apparent. Anticipating this, the spectral representation of p1 / r is transformed to ordinary polynomial space, where the lowest order constant term in r is removed, whereafter a back transformation to Chebyshev spectral space is made. Thus, before solving the system (13), the coefficients are properly prepared to avoid representing equation singularities. Likewise, before solving the full set of 14 ideal MHD equations, a similar preparation has been performed. First, a separate study is carried out where all equations are expanded to low order as ordinary polynomials. Secondly, all internal boundary conditions are imposed. , is Thirdly, the condition u1r  imu1  0, corresponding to finite compressibility imposed at r = 0. Fourthly, the resulting expansions near r = 0 are studied to determine whether the singularities imposed by the cylindrical coordinate system have vanished. This is indeed the case for each of the 14 ideal MHD component equations. Thus all apparent singularities may be safely removed using the same procedure as for the test problem before the coefficent Eq. (13) is solved.

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7.4.2. GWRM Solution of MHD Stability Problem and Comparison with Eigenvalue Approach The screw-pinch stability problem defined by Eqs. (52-55) is now solved for the perturbation (m,k) = (1,10) using the GWRM. Parameters are K = 5, L = 5, M = 0 and five equidistant temporal domains were used. A single SIR iteration is again sufficient due to linearity in u1, b1 and p1. We will not specifically compare efficiency with other methods here, but merely note that for this case, SIR solved only 372 coupled equations for the coefficients (13). This is considerably less than the 14 · 6 · 6 = 504 equations that obtain before the boundary conditions are applied. Plots of u1r and p1 vs t and r are given in Figure 7a-b. For comparison, we have also solved the same problem using an eigenvalue approach where time dependence has been eliminated through the asymptotic assumption  / t  i [17]. In this approach, the linearized ideal MHD equations are reduced to two simultaneous equations that are solved by a shooting procedure where the growth rate |  | is guessed until the boundary conditions are

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satisfied. The resulting eigenfunctions for u1r and p1 are shown in Figs. 7c-d. Clearly, there is a good correspondence between the GWRM and eigenvalue solutions.

Figure 7a. GWRM solution obtained by solving Eqs. (52) through (55) for perturbation (m,k) = (1,10). Parameters are K = 5, L = 5, M = 0; five equidistant temporal domains were used and a single SIR iteration. The perturbed radial plasma flow u1r is shown vs t and r.

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Figure 7b. As Fig. 7a but here the perturbed pressure p1 is shown vs t and r.

Figure 7c. Eigenfunction u1r for the problem of Fig. 7a, using an eigenvalue method [17]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 7d. Eigenfunction p1 for the problem of Fig. 7a, using an eigenvalue method [17].

Of particular interest in MHD stability analysis is the value of the obtained growth rate. Assuming an exponential time behaviour for the GWRM solution during the last 10% of the temporal evolution, a normalized growth rate |  | = 0.83 is obtained in this highly unstable case (recall that a normalization to the Alfvén time, being less than a microsecond, is used). The computed value exactly coincides with that obtained from the eigenvalue code. Also for other perturbations (m,k) very good agreement is obtained. Considering that the two methods are radically different in idea and implementation, these results certainly confirm the applicability of the GWRM to complex systems of initial value partial differential equations. The equilibrium used in this example is easily changed within the existing computer code to more realistic cases without changing the basic GWRM performance demonstrated here. In fact, we have also solved the above equations including resistivity for realistic reversed-field pinch equilibria. This class of equilibria is stable on the MHD time scale but may (for strong current gradients or for high pressure) become resistively unstable, on a much slower time scale. The GWRM works equally well also including Chebyshev expansion of resistivity  in this more advanced case. Detailed results will, however, be presented elsewhere.

7.5. Other Systems of Time-Dependent PDE’s A natural next step is to apply the GWRM to studies of stability using the full nonlinear system of resistive MHD equtions (52). Interestingly, relatively high efficiency can be expected since the number of MHD equations to be solved then reduces from 14 to 7 due to that complex variables need not be handled.

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Time-Spectral Solution of Initial-Value Problems

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The GWRM has also been applied to the dynamics of the Rayleigh-Taylor instability in a 2D medium modelled by the compressible Navier-Stokes equations, enclosed in a rectangular box. Early results show that globally averaged dynamics (in time and space) can be readily computed even for a small number of Chebyshev modes. Efficiency requires, however, that spatial subdomains are employed. Thus this problem is valuable for developing efficient methods for handling internal boundaries. Results, including comparisons with standard methods, will be published elsewhere.

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7.6. Summary - Applications The examples of this Section shows application of the GWRM to basic linear and nonlinear initial value problems in the forms of ordinary or partial differential equations. Accuracy and efficiency was studied by comparing with exact solutions. Improved performance using temporal and spatial subdomains was discussed and shown by example. Comparisons with standard explicit and implicit finite difference methods showed positive results with regards to computational efficiency. Finally we successfully solved an advanced fusion plasma stability problem formulated within ideal magnetohydrodynamics as 14 simultaneous initial value partial differential equations. Excellent agreement was found with results found by employing established eigenvalue methods. The computations in the present paper have all been carried out in Maple 12. Although faster computational environments exist, exact comparisons of efficiency is not essential here. The examples we have given show that the efficiency of the GWRM is comparable to both that of explicit and implicit finite difference schemes in the given environment. Further optimisation of both GWRM and finite difference codes could increase efficiency, but the point has been to determine whether development of a time-spectral method for initial-value pde’s is of interest. It may be mentioned that the computational speed of Maple is now enhanced; as an example an inversion of a dense 1000 x 1000 matrix takes less than a second on a lap top computer.

CONCLUSION A fully spectral method for solution of initial value ordinary or partial differential equations has been outlined. The time- and parameter generalized weighted residual method, GWRM, represents all time, spatial and physical parameter domains by Chebyshev series. Thus semi-analytical solutions are obtained, explicitly showing the dependence on these variables. The essence of the GWRM is its ability to transform the implicit dependencies inherent in physical laws formulated as differential equations to solutions of explicit, semianalytical form. The method is fully global and avoids time step limitations due to acausality. The characteristic form of the problem (hyperbolic, elliptic or parabolic) is thus irrelevant. This fact makes the method potentially applicable to a large class of problems. The spectral coefficients are determined by iterative solution of a linear or nonlinear system of algebraic equations, for which a new and efficient semi-implicit root solver (SIR) has been developed.

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Accuracy is explicitly controlled by the number of modes used and the number of iterations. The use of subdomains further increases efficiency and accuracy. If desired, global solutions, valid for the entire computational domain, may be obtained by carrying out Chebyshev interpolation over the set of subdomains. The practical solution of single or systems of partial differential equations is conveniently handled in spectral space by the use of procedures for differentiation, integration, products as well as initial and boundary conditions. The GWRM is shown by example to be accurate and efficient and to have potential for applications in fluid mechanics and in MHD. A model example shows that the method averages over rapid time scale phenomena, and follows long time scale phenomena. Thus problems with several time scales may be modelled, using only a limited number of temporal modes. Besides further development work we foresee, as a next step, solution of more challenging nonlinear problems in MHD. The GWRM was partly developed with the aim of finding new methods for obtaining operational limits of reversed-field pinch and tokamak fusion plasmas in mind. These limits depend on nonlinear plasma instabilities at finite plasma pressure in stochastic magnetic field geometries. The method will also be applied to problems of kinetic plasma stability theory. It appears that there are many potential future applications for the GWRM, and it will be exciting to explore them.

ACKNOWLEDGEMENT

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Special thanks go to Mr Daniel Lundin and Mr David Jackson for several constructive comments.

REFERENCES [1]

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-value Problems, Krieger Publishing, 1994. [2] D. S. Harned and W. Kerner, J. Comp. Phys. 60, 62 (1985). [3] D. S. Harned and D. D. Schnack, J. Comp. Phys. 65, 57 (1986). [4] J. Scheffel, TRITA-ALF-2004-03, Royal Institute of Technology, Stockholm, Sweden, 2004. [5] R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow, Springer, New York 1983. [6] B. A. Finlayson and L. E. Scriven, Appl. Mech. Rev. 19(1966)735. [7] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics,Vols 1-2, Springer, 2000. [8] A. Chatterjee, Current Science 78(2000)808. [9] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1987. [10] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall, 2003. [11] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes, Cambridge University Press, 1992.

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[12] [J. Scheffel and C. Håkansson, Appl. Numer. Math. 59(2009)2430. [13] G. Dahlquist and Å. Björck, Numerical Methods, Prentice-Hall, 1974. [14] A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York and London, 1966. [15] J. Scheffel and D. D. Schnack, Nuclear Fusion 40(2000)1885. [16] C. Moler, Matlab News & Notes, The MathWorks, Inc., May 2003. [17] G. Bateman, MHD Instabilities, The MIT Press, 1978.

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 ©2011 Nova Science Publishers, Inc.

Chapter 2

A STOCHASTIC AGENT-BASED APPROACH TO THE FOKKER-PLANCK EQUATION IN HUMAN POPULATION DYNAMICS Minoru Tabata1* and Nobuoki Eshima2Á 1

Osaka Prefecture University, Osaka, Japan 2 Oita University, Oita,Japan

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1. INTRODUCTION Various kinds of mathematical models are constructed in order to quantitatively describe migration. For example, a partial differential equation of parabolic type (an integro-partial differential equation, respectively) is employed in order to describe the phenomenon. The partial differential equation (the integro-partial differential equation, respectively) is called the Fokker-Planck equation (the master equation, respectively). We need to apply microscopic foundations for the descriptions given by these functional equations. As described in [8-11], we take a stochastic agent-based approach to the master equation, i.e., we construct a stochastic agent-based model to apply a microscopic foundation for the description given by the master equation. Hence, in the present paper, we take a stochastic agent-based approach to the Fokker-Planck equation, i.e., we construct a stochastic agentbased model and apply a microscopic foundation for the description given by the FokkerPlanck equation. This paper is organized into four sections in addition to our Introduction. In Section 2, in order to describe the migration, we construct an agent-based model consisting of a large number of agents that relocate stochastically within a discrete bounded domain. We consider that each agent represents an individual. We assume that each agent relocates in order to obtain higher utility, where the utility denotes a quantity representing socioeconomic desirability. In a similiar way used by [10,11], we assume that the utility is equal to a * corresponding author: Department of Applied Mathematics, Faculty of Engineering, Osaka Prefecture University, Osaka 599-8531 Japan, [email protected] Á Department of Statistics, Faculty of Medicine , Oita University, Oita 879-5593 Japan, [email protected] Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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52

Minoru Tabata and Nobuoki Eshima

sufficiently smooth known function that can be represented by a density of agents. Moreover, in the same way as shown in [10,11], we assume that the utility changes stochastically, i.e., that the utility contains a random variable. We assume that each agent relocates discontinuously in time for a discrete bounded domain, i.e., that the agent-based model has a discrete time variable and a discrete space variable. Furthermore, we introduce a quantity that represents the influence of one agent on the whole model, and we call this quantity the size of agents. The product of the total number of agents and the size of agents can be regarded as the total amount of influence for all the agents within the whole model. These assumptions have been already made [10,11]. Making use of the assumptions, as in [10,11], we can construct stochastic agent-based models which can define a microscopic foundation for the description given by the master equation. However, the agent-based models constructed as shown in [10,11] can apply no microscopic foundation for the description given by the Fokker-Planck equation. In order to accomplish this sort of agent-based model, we need to assume that no agent can go far away as an addition to the required assumptions. Hence, the model constructed in the present paper is essentially different from those already done in [10,11]. In Section 3, we state the main result as follows: if the scaling limit is taken, then the agent-based model constructed in Section 2 is close (in probability) to a deterministic continuous model described by the Fokker-Planck equation, where if we follow the processes below (p1-p3), then we say that the scaling limit is taken. (p1): We make the total number of agents tend to infinity and the size of agents converge to 0+0 with the condition that the product of the total number of agents and the size of agents is identically equal to a positive constant. (p2): We make the least unit of discrete time variable converge to 0+0. (p3): We make the least unit of discrete space variable sufficiently small. We can say that if the number of agents is sufficiently large, and if the least unit of discrete time variable and the least unit of discrete space variable are sufficiently small, then the stochastic agent-based model is close (in probability) to the deterministic continuous model described by the Fokker-Planck equation. It follows from the main result that the stochastic agent-based model applies a microscopic foundation for the description of migration given by the Fokker-Planck equation. In Section 4, by making use of the method developed in [8-11], we prove that if the scaling limit is taken, then the stochastic agent-based model converges (in probability) to a deterministic continuous model described by the master equation. Furthermore, in Sections 4 and 5, we prove that the mixed problem for the master equation has a unique solution which is close to that of the mixed problem for the Fokker-Planck equation (see Remark 1.1, (ii)), where we impose the periodic boundary condition on the mixed problems. Making use of these results, we can prove the main result.

2. THE AGENT-BASED MODEL Let us define the discrete space variable, the discrete time variable, and the density of agents. Each agent relocates in a bounded domain D. For simplicity, we assume that D:= [±1,1)u[±1,1),

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(2.1)

A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

53

and we impose the periodic boundary condition on D. Hence we can regard the 2u2 square D as a 2-dimensional torus. Therefore we need to define the distance between x and y, x,yD, as the length of the geodesic curve between x and y, and we denote it by |x±y|; no confusion will arise. In order to make our model discrete in space, we divide D into small disjoint squares as follows: D = ‰i , j =1,...,2N[±1+(i±1)/N,±1+i/N)u[±1+( j±1)/N,±1+j/N),

(2.2)

where N is a sufficiently large natural number. We call these small squares sections. We number the sections from 1 to 4N2, and we denote them by dj, j  I, where we define I:= {1,...,4N2}.

(2.3)

We see that D = ‰jId j.

(2.4)

We regard 1/N as the least unit of discrete space variable. If we make N sufficiently large, then the process (p3) is followed (see Section 1 for (p3)). Considering that agents are distributed uniformly in each section, and noting that the area of each section is equal to 1/N 2, we define the density of agents f = f(t,x), (t,x)  >’ uD, as follows:

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f = f(t,x):= (J/R)Rj(t)/(1/N2) if xdj, j  I,

(2.5)

where Rj = Rj(t) denotes the number of agents located in a section dj, j  I, at time t •:H assume that Rj = Rj(t), j  I, are random variables depending on the time variable t • By R we denote the total number of agents contained in D, i.e., we define R := ™jI R j(t).

(2.6)

We assume that each agent has the same size J/R, where J is a positive constant. We easily see that J is equal to the positive constant defined in the process (p1) (see Section 1 for (p1)). We can regard J as the total amount of influence of all the agents on the whole model. From (2.5-6) we easily obtain the conservation law of total amount of agents, i.e., ||f(t,·)||L1(D) = J for each t •

(2.7)

where ||·||L1(D) denotes the norm of L1(D). We reasonably assume that there exists no error in counting the total number of agents. Hence, from Remark 1.1, (i), we assume that R is not a random variable but a positive integer. If we make R tend to infinity, then the process (p1) is followed. In order to make our model discrete in time, we make the following assumption in the same way as shown in [10,11]:

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54

Minoru Tabata and Nobuoki Eshima

Assumption 2.1. No agent relocates at the initial time t = 0. Each agent attempts to relocate at each time t of the following form: t = n¨W, n ,

(2.8)

where ¨t is a sufficiently small positive constant representing the least unit of discrete time variable. No agent can relocate for each time interval of the following form: (n¨W,(n+1)¨W), n ‰{0}. Let us discuss the utility. In a real world agents will encounter unpredictable various fluctuations, which have a large influence on the behavior of agents. We cannot neglect such fluctuations. Hence, our agent-based model needs to be stochastic. In order to make our model stochastic, in the same way as described by [10,11], we assume that the utility can be decomposed as follows (see [10, Eq, (3.1)]):

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U(t,j) = Ud(t,j) + Us(t,j), (t,j)[0,+’)uI,

(2.9)

where we denote the utility of a section dj at time t •by U = U(t,j), (t,j)[0,+’)uI, Ud = Ud(t,j) is a function of (t,j)  >’ uI which is called the deterministic utility, and Us = Us(t,j) is a random variable depending on (t,j)  >’ uI which is called the stochastic utility. Let us discuss the deterministic utility Ud = Ud(t,j). It is known in socioeconomics that the following socioeconomic and economic variables (1-5) have a large influence on the desirability in business: (1) unemployment rate, (2) export structure index (industrial minus agricultural export divided by the total export), (3) overnight stays per capita, (4) percentage of total employment in the tertiary sector, and (5) population density (see, e.g., [7, Chapter 4] for the details). However, it is almost impossible to introduce all the variables into our model as endogenous variables, because it makes the model extremely complicated to do so. Hence, as in [10,11], we treat the variable (5) (the variables (1-4), respectively) as an endogenous variable (exogenous variables, respectively). Furthermore, for simplicity we regard the exogenous variables (1-4) as constants independent of the time variable and the space variable. Hence, similar to [10,11] we assume that the deterministic utility depends only on the variable (5), i.e., that Ud = Ud(t,j) depends only on f = f(t,x), (t,x)[0,+’)udj, jI. Hence we make the following assumption: Assumption 2.2. The deterministic utility has the following form: Ud(t,j):= U(f(t,x)) for each (t,x)[0,+’)udj, j  I,

(2.10)

where U = U(z) is a sufficiently smooth nonnegative-valued function of z  . Let us discuss the stochastic utility. As indicated in [10,11], for simplicity we assume that the random variables Us = Us(t,j), t = Q¨W, n  ‰{0}, j  I, are independent of each other, and that the density functions of those random variables are the same. We denote the density function by U = U(Us). We will follow this assumption in the same manner as [10,11]: Assumption 2.3. U(Us):=exp(Us) for each Us ”DQGU(Us):=0 for each Us >0. We reasonably assume that if an agent moves from one section to another, then she/he needs to bear the cost of moving. Hence, we follow this with the next assumption:

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A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

55

Assumption 2.4. The cost incurred in moving from a section dj to a section di is equal to C = C(|Xi ± Xj|) for each i, j  I, where we denote the center of dj by Xj, j  I, and C = C(r) is a sufficiently smooth increasing function of r  such that C(0) = 0 and C(r) > 0 for each r > 0. If an agent attempts to relocate, then she/he needs to choose sections. In the present paper, for simplicity, we assume that she/he chooses only one section at each attempt. However, as already mentioned in Section 1, we assume that no agents can go far away. Hence, we use this assumption: Assumption 2.5. If an agent contained in dj attempts to relocate, then she/he chooses a section di with the following probability, i,j  I: Pr{the agent chooses a section di} =

œyd G (y±X )dy, H

j

(2.11)

i

where we denote the probability of an event E by Pr{E}, and GH = GH(y) is a function of yD defined as follows:

GH = GH(y):= (1/H2)G1(y/H), y  D.

(2.12)

Here, H is a parameter such that 0 < H < 1,

(2.13)

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and G1 = G1(y) is a nonnegative-valued given to the sufficiently smooth function which satisfies the following conditions:

G1 = G1(y) depends only on |y|, i.e., G1(y) = G1(x) if |y| = |x|,

(2.14)

G1(y) = 0 if |y| > 1,

(2.15)

œyDG (y)dy = 1.

(2.16)

1

Let us discuss Assumption 2.5. We can easily see that GH = GH(y) converges to Dirac's delta function in the sense of distribution as H o 0+0, and that

GH(y) = 0 if |y| > H,

(2.17)

œyDG (y)dy = 1 for each H  (0,1].

(2.18)

H

For example, if we define G1 = G1(r) as follows: G1(y):= c2.1exp{±1/(1±|y|2)} if |y| < 1; G1(y):= 0 if |y| > 1, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(2.19)

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Minoru Tabata and Nobuoki Eshima

then it satisfies Eqs, (2.14-2.16), where c2.1 is a positive constant defined by the following equality:

œ|y| 0, where D is the torus defined in Section 2. Let m be a nonnegative integer. By Cm(D) we denote the Banach space of all real-valued functions u = u(x) such that Diu(x ”_i_”m, are continuous in D and ||u||C m(D) ™0 0,

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(3.1)

(3.2)

A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

57

i is a multi index such that i = (i1,i2), |i| is the length defined as follows: |i|:= i1+i2,

(3.3)

and xi denotes the i-th component of x, i.e., x = (x1,x2).

(3.4)

By ||·||L’(D) we denote the norm of L’(D). Let n and m be nonnegative integers. By C n , m([0,T]uD) we denote the Banach space of all real-valued functions v = v(t,x) such that (˜/˜W) jv(t,x) and D i v(t,x), 0 < j ”n, 0 < |i_”m, are continuous in [0,T]uD and such that ||v||C n,m([0,T]uD) ™0” j ”n||(˜/˜W) jv||L’([0,T]uD) ™0 0, where c3.2  (0,1] is a constant independent of z > 0. Let us discuss Assumption 3.1. If the utility increases with the density of population, then we say that imitative processes work (see, e.g., [4]). In a real world we often observe that the socioeconomic desirability increases with the population density. Hence, it is plausible to assume that imitative processes work at a certain degree. However, in a real world, we can observe that if the density of population is extremely large, then the utility ceases with the increase in the population density. Moreover we often observe that over population makes the utility decrease. If such a phenomenon is observed, then we say that avoidance processes work (see, e.g., [4]). If avoidance processes work, then we reasonably regard the utility as a strictly concave function of the concerning population density that monotonically increases (decreases, respectively) when the population density is smaller than a positive constant (greater than the positive constant, respectively). For example, in [30, Assumption 1.4] we assume that the utility is equal to a concave quadratic function of the population density. However, it follows from Assumption 3.1 that U(z ”^1 ± c3.2)/2}logz + U(1) for each z  [1,+f),

(3.12)

U(z •^1 ± c3.2)/2}logz + U(1) for each z  (0,1].

(3.13)

Hence, from these inequalities we see that not only imitative processes can work at a certain degree but also avoidance processes can work. Hence, we reasonably accept Assumption 3.1. We consider the mixed problem for the Fokker-Planck equation with the following initial condition: gH(0,x) = g0(x),

(3.14)

where g0 = g0(x) is a given function which satisfies the following condition: g0 = g0(x)  C m(D), m > 0.

(3.15)

Proposition 3.2. If (3.15) holds for some m > 2, then for each H(0,1] the mixed problem for the Fokker-Planck equation has a unique classical solution gH = gH(t,x)  C 1,2([0,S1/D(H)]uD) such that ||gH||C1,2([0,S1/D(H)]uD) < c3.3||g0||C2(D),

(3.16)

where S1 and c3.3 are positive constants independent of H. Proof. We consider the following new mixed problem: ˜K(t,x)/˜W = (1/D(H))FH(h(t,·))(x),

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(3.17)

A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ« h(0,x) = g0(x),

59 (3.18)

where we denote an unknown function by h = h(t,x). From (3.7) we see that (1/D(H))FH(·) is independent of H. Hence we do not attach the subscription of H to the unknown function. Making use of Assumption 3.1 and [31, Chapter V, Sections 5-6], we deduce that the new mixed problem has a unique local classical solution h = h(t,x)  C 1,2([0,S1]uD) such that ||h||C1,2([0,S1]uD) < c3.4||g0||C2(D),

(3.19)

where S1 and c3.4 are positive constants independent of H(0,1]. We see that gH = gH(t,x):= h(D(H)t,x) satisfies the present proposition. For each x  D there exists a unique section that contains x. By \ = \(x;·) we denote the characteristic function of the section containing x  D, i.e., if y belongs to the section containing x  D, then \(x;y) = 1; if not, then \(x;y) = 0. We define an operator ZN = ZN(·) as follows: ZN = ZN(k(·))(x):=

œyD\(x;y)k(y)dy/(1/N ). 2

(3.20)

Recalling that the area of each section is equal to 1/N2, we easily see that if x  dj, jI, then ZN(k(·))(x) is equal to the mean-value of k = k(x) in dj. By IN we denote the set of all step

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functions of x  D that are constant in each section dj, j  I. For an variable r and an 1-valued function g = g(x)  IN, we define

2

4N

-valued random

P(r,g(·);T):= Pr{|r ± (g(X1),...,g(X4N2))| •T}, T > 0.

(3.21)

See Assumption 2.5 for Pr{·}. See Assumption 2.4 for Xj, jI. Let g = g(x) be an 1valued Lebesgue-integrable function of xD. If P(r,ZN(g(·))(·);T) o 0 for some T > 0, then we say that the random variable r is close to g = g(x) in probability. If P(r,ZN(g(·))(·);T) o 0 for each T > 0, then we say that the random variable r converges to g = g(x) in probability, and we simply write it as follows: r o g(x).

(3.22)

No confusion will arise. We see that the following discrete random variable can describe the stochastic agentbased model constructed in Section 2: r = r(t):= (r1(t),...,r4N2(t)),

(3.23)

where rj(t):= f(t,Xj) for each (t,j)  [0,+’)uI. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(3.24)

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Minoru Tabata and Nobuoki Eshima

See Eq.(2.5) for f = f(t,x). The following theorem is the main result of this paper, and will be proved in the next section. Theorem 3.3. If the initial function g0 = g0(x) satisfies the following conditions and Eq.(3.15) for some m > 4: g0(x) > 0 for each x  D,

(3.25)

||g0(·)||L1(D) = J,

(3.26)

then the random variable r = r(t) is close to the solution gH = gH(t,x) of the mixed problem for the Fokker-Planck equation in probability as follows: for each H  (0,1] there exists an integer NH > 0 such that if N > NH , then lim¨tp0limr(0)og0(x)limRo’sup0”t”S P(r(t),ZN(gH(t,·))(·);4) = 0,

(3.27)

4:= H4O(||g0||C 4(D),S),

(3.28)

where r(0) o g0(x) denotes the convergence in probability, S is a positive constant independent of H, and O = O(·,·) is a nonnegative-valued function independent of H and increases monotonically with each argument.

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4. THE MASTER EQUATION The master equation is an integro-partial differential equation as shown in the following form: ˜fH(t,x ˜t = MH(fH(t,·))(x),

(4.1)

where fH = fH(t,x) is an unknown function, and MH = MH(·) is a nonlinear integral operator defined as follows: MH(k(·))(x):= ±wH(k(·);x)k(x) +

œyDW (k(·);x|y)k(y)dy. H

(4.2)

Here the kernel WH = WH(k(·);x|y) and the coefficient wH = wH(k(·);x) are defined as follows: WH = WH(k(·);x|y):= c3.1GH(x±y)exp{U(k(x)) ± U(k(y)) ± C(|x±y|)}, wH = wH(k(·);x):=

œyDW (k(·);y|x)dy, H

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(4.3) (4.4)

A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

61

where GH = GH(·) and H are the same as (2.12-13). We attach the subscripts of H to the operator and the unknown function in order to emphasize that they depend on the parameter of H. See Assumptions 2.2 and 2.4 for U = U(·) and C = C(·). See Eq.(3.11) for c3.1. If we replace GH = GH(x±y) by 1 in Eq.(4.3), then Eq. (4.1) becomes exactly the same as the master equation prevoiusly studied in [26,29,30]. In Remark 4.4, we will advance the reason why Eq.(4.3) contains c3.1 as a factor. We consider the mixed problem for the master equation with Eq. (4.1) by using the following initial condition: fH(0,x) = f 0(x),

(4.5)

where f 0 = f 0(x) is a given function that satisfies the following conditions: f 0 = f 0(x)  C m(D), m > 0,

(4.6)

f 0(x) > 0 for each x  D,

(4.7)

||f 0(·)||L1(D) = J.

(4.8)

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See Section 2 for ||·||L1(D) and J. If a continuous function of (t,x)  [0,T]uD satisfies Eqs.(4.1) and (4.5) for each (t,x)  [0,T]uD, then we say that the function is a classical solution to the mixed problem, where T is a positive constant. Proposition 4.1. (i) For each H(0,1] the mixed problem for Eq.(4.1) with Eq.(4.5) has a unique classical solution fH = fH(t,x) such that fH = fH(t,x)  C 0,0([0,S2]uD),

(4.9)

where S2 is a positive constant independent of H(0,1]. (ii) For each H(0,1] the solution fH = fH(t,x) satisfies that fH(t,x) > 0 for each (t,x)  [0,S2]uD,

(4.10)

||fH(t,·)||L1(D) = J for each t  [0,S2],

(4.11)

||fH||C n,m([0,T]uD) < On,m(||f 0||C m(D),T) for each T  (0,S2] and n  ‰{0},

(4.12)

where On,m = On,m(·,·), n,m  ‰{0}, are nonnegative-valued smooth functions independent of H(0,1] and increase monotonically with each argument. Proof. We will prove the present proposition by following the line of [25, 26, 29, 30]. In the same way as in [25, Proposition 6.1] we can prove that the mixed problem has a unique classical solution in a time interval [0,S2], where S2 is a positive constant. We obtain Eq.(4.10) from Eq.(4.7) in the same way as in [25, Proposition 6.1]. Integrating both sides of the master

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Minoru Tabata and Nobuoki Eshima

equation of Eq. (4.1) with respect to xD, recalling Eq.(2.14), and Eqs.(4.3), (4.4), and making use of Eq.(4.10), we easily obtain ˜˜t)||fH(t,·)||L1(D) = 0 for each t  [0,S2].

(4.13)

This equality along with Eq.(4.8) imply Eq.(4.11). We need to prove that S2 is independent of H. By applying Assumption 2.2, Eqs. (3.12), and (3.13), and recalling that 0 < c3.2< 1 (see Assumption 3.1), we see that exp{U(z)} < c4.1z + c4.2 for each z > 0,

(4.14)

where c4.j, j = 1,2, are positive constants independent of z > 0. Applying this inequality and Eq.(4.10) to Eq.(4.3) and recalling that 0 < c3.1 < 1,

(4.15)

and that U = U(z) and C = C(r) are nonnegative-valued for each z,r > 0 (see Eq.(3.11) and Assumptions 2.2 and 2.4), we see that if t  [0,S2], then 0 < WH(fH(t,·);x|y) ”GH(x±y)(c4.1 fH(t,x) + c4.2) for each x,yD.

(4.16)

Noting that Eq.(4.4) is nonnegative-valued, and applying Eqs.(2.18), (4.10), and (4.16) to Eq.(4.1), we easily obtain

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˜˜t)fH(t,x) ”F(||fH(t,·)||Lf(D)),

(4.17)

where F = F(z):= c4.1z2 + c4.2z. Replace t by s in this inequality, integrate both sides with respect to s  [0,t], and replace fH(0,x) by ||f 0(·)||Lf(D) in the left-hand side (see Eq.(4.5)). By J = J(t) we denote the right-hand side of the inequality thus obtained. Noting that J = J(t) is independent of x, we can replace fH(t,x) by ||fH(t,·)||Lf(D) in the left-hand side. Hence we obtain ||fH(t,·)||Lf(D) < J(t) + ||f 0(·)||Lf(D). Applying this inequality to dJ(t)/dt = F(||fH(t,·)||Lf(D)), we obtain dJ(t)/dt < F(J(t) + ||f 0(·)||Lf(D)). Solving this differential inequality, we easily deduce that S2 is independent of H, and we obtain Eq.(4.12) with (n,m) = (0,0). Let us prove Eq.(4.12) when n = 0 and m > 1. By the periodic boundary condition we easily see that

œyDG (x+k±y)e H

±C(|x+k±y|)

v(y)dy =

œyDG (x±y)e H

±C(|x±y|)

v(y+k)dy.

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(4.18)

A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

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Making use of this equality, we see that

œyDG (x±y)e

(w/wxi)

H

±C(|x±y|)

v(y)dy=

œyDG (x±y)e H

±C(|x±y|)

(w/wyi)v(y)dy, i=1,2,

(4.19)

where xi denotes the i-th component of x, i = 1,2. Multiplying both sides of the master equation by Di, |i| = 1, and applying Assumption 2.2 and Eq.(4.19), we can obtain a system of linear integro-partial differential equations with respect to DifH, |i| = 1, (see Eq.(3.2) for Di). Applying Eq.(4.12) with (n,m) = (0,0) to the system, and performing calculations similar to, but easier than, those done in obtaining Eq.(4.12) with (n,m) = (0,0), we obtain Eq.(4.12) with (n,m) = (0,1). Multiplying both sides of the master equation by D i , |i| > 2, and iterating the same calculations as done earlier, we obtain Eq.(4.12) when n = 0 and m > 2. Let us prove Eq.(4.12) when n > 1 and m = 0. Applying Eq.(4.12) with (n,m) = (0,0) to the right-hand side of the master equation in Eq.(4.1), and performing calculations similar to those done in obtaining Eq.(4.17), we obtain Eq.(4.12) with (n,m) = (1,0). Multiplying both sides of the master equation in Eq.(4.1) by w/wt, and applying Eq.(4.12) with (n,m) = (1,0) to the righthand side, we can obtain Eq.(4.12) with (n,m) = (2,0). Iterating the same calculations, we obtain Eq.(4.12) when n > 3 and m = 0. Noting that

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||v||C n,m([0,T]uD) < ||v||C n,0([0,T]uD) + ||v||C 0,m([0,T]uD),

(4.20)

we obtain Eq.(4.12) for each n, m > 0. Remark 4.2. (i) We reasonably accept the assumption in Eq.(4.7), because the initial function f 0 = f 0(x) represents the density of agents at the initial time t = 0. If we do not impose Eq.(4.7) on the initial data, then we cannot obtain Eq.(4.10). If Eq.(4.10) is not obtained, then we can prove neither Eq.(4.11) nor Eq.(4.17), and we cannot deduce that S2 is independent of H. (ii) In proving the estimates in Eq.(4.16), we employ Eq.(3.12), as derived from Assumption 3.1. By accepting Eq.(4.16), we can prove Proposition 4.1. Hence, Assumption 3.1 implies not only that the Fokker-Planck equation is uniformly parabolic for each H(0,1] but also that the master equation has a unique classical solution in a time interval whose length is independent of H(0,1]. Proposition 4.3. The random variable r = r(t) converges to the solution fH = fH(t,x) of the mixed problem for the master equation in probability as follows: for each T > 0 and H  (0,1] there exists an integer NH,T > 0 such that if N > NH,T , then lim¨tp0limr(0)of 0(x)limRo’sup”t”S2P(r(t),ZN(fH(t,·))(·);T) = 0,

(4.21)

where S2 is the positive constant defined in Proposition 4.1, and the convergence r(0)of 0(x) is that in probability. Proof. By making use of Assumptions 2.3, 2.5, and 2.7, in exactly the same way as [10, Lemma 3.11] we can prove that if an agent is contained in a section dj, j  I, at a time t of the form Eq.(2.8), if the density of agents is equal to g = g(x)  IN,J at the time t, and if 't > 0 is sufficiently small, then the following equality holds for each i  I, where by IN,J we denote

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64

Minoru Tabata and Nobuoki Eshima

the set of all g = g(x)  IN such that ||g(·)||L1(D) = J: Pr{the agent moves from dj to di at the time t}

œyd G (y±X )dyexp{U(g(X ))±U(g(X ))±C(|X ± X |)}.

= 4c3.1't

H

j

i

j

i

j

(4.22)

i

See Assumption 2.4 for Xj, j  I. See Eq.(3.11) for c3.1. For this result, we can prove Proposition 4.3 in the same way [10, Theorem 5.1, (i)]. Remark 4.4. (i) We easily deduce that ZN(f 0(·))(x)  IN,J .

(4.23)

Hence, as with Eq. (3.27), we can make r(0) converge to f 0(x) in probability for Eq. (4.21). (ii) For the same reason as shown in [10, Eqs.(3.10) and (3.14)] the right-hand side of Eq.(4.22) contains 4c3.1 as a factor. Recalling that the area of D is equal to 4, we see that the right-hand side of Eq.(4.3) contains c3.1 as a factor for the same reason as given in [10, Eq.(4.5)]. The Fokker-Planck equation is sufficiently close to the master equation when applying Proposition 4.5. Proposition 4.5. If Eq.(4.6) holds for some m > 4 and Eq.(3.15) holds for some m > 2, then the following inequality holds for each H(0,1] and T  (0,S]:

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||fH ±gH||C1,2([0,T]uD) ” c4.3||f 0 ± g0||C2(D) + H4O1(||f 0||C4(D),T),

(4.24)

where S:= min{8S1,S2}, fH = fH(t,x) and gH = gH(t,x) are the solutions to the mixed problems for the master equation and the Fokker-Planck equation respectively; c4.3 is a positive constant independent of H, Sj, j = 1,2, are the positive constants defined in Propositions 3.2 and Proposition 4.1, and O1 = O1(·,·) is a nonnegative-valued function independent of H(0,1] and increases monotonically with each argument. Proof. In the next section, we transform the master equation in Eq.(4.1) into this equivalent: ˜fH(t,x ˜t = FH(fH(t,·))(x) + F(H,t,x),

(4.25)

where F = F(H,t,x) is a function of (H,t,x)  (0,1]u[0,S2)uD, as defined in the next section (see Eq.(5.7)). Subtract Eq.(4.25) from the Fokker-Planck equation in Eq.(3.6), consider that fH = fH(t,x) and gH = gH(t,x) contained in the coefficients are known functions, regard F = F(H,t,x) as a known function, and regard fH(t,x) ± gH(t,x) as an unknown function. Then we obtain a linear partial differential equation of parabolic type. Applying Propositions 4.1 and 3.2 to the linear equation thus obtained, making use of the following inequality (we obtain the inequality easily from Eqs.(2.15), (2.16), (3.11), (3.8), (2.13), and (2.1)): (4.26) 0 < D(H) < 1/8,

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A Stochastic Agent-Based Approach to the Fokker-3ODQFN(TXDWLRQ«

65

and performing the same calculations as [31, IV-V], we obtain the following inequality for each H(0,1] and T  (0,S]: ||fH ± gH||C1,2([0,T]uD) ”c4.3||f 0 ± g0||C2(D) + c4.4||F(H,·,·)||C0,0([0,T]uD),

(4.27)

where c4.j, j = 3,4, denote positive constants independent of H and T. Applying Eq.(4.12) with (n,m) = (0,4) and Lemma 4.6 to this inequality, we obtain the present proposition. Lemma 4.6. ||F(H,·,·)||C0,0([0,T]uD) ” H 4O2(||fH||C0,4([0,T]uD)) for each H  (0,1] and T  (0,S2], where O2 = O2(·) is a nonnegative-valued function independent of H(0,1] and increases monotonically with each argument, and S2 is the positive constant defined in Proposition 4.1. Proof of Theorem 3.3. By making use of O1 = O1(·,·) of Proposition 4.5, we define O = O(·,·) as follows:

O(r,s):= 2O1(r,s), r,s > 0,

(4.28)

Making use of Propositions 4.3 and 4.5 with

T:= (1/2)H 4O(||g0||C4(D),S),

(4.29)

f 0(x) = g0(x),

(4.30)

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we obtain Theorem 3.3.

5. THE TRANSFORM OF THE MASTER EQUATION AND PROOF OF LEMMA 4.6 Let us equivalently transform Eq.(4.1) into Eq.(4.25). We change the integral variable yD into the following new integral variable z in Eq.(4.1) (see Eqs.(4.2) and (4.4)): z:= x±y.

(5.1)

We see that z is contained in D, since the periodic boundary condition is imposed on D. Recalling Eq.(2.14), we can easily rewrite the right-hand side of the master equation as follows: MH(fH(t,·))(x) = c3.1

œzDG (z)e H

±C(|z|)

E(1;t,x,z)dz,

where we define E = E(r;t,x,z):= E(fH(t,x),fH(t,x±rz)), E = E(u,v):= ±uexp{U(v)±U(u)} + vexp{U(u)±U(v)},

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(5.2)

(5.3) (5.4)

66

Minoru Tabata and Nobuoki Eshima

and r denotes a variable such that r[0,1]. Applying Maclaurin's formula to E = E(r;t,x,z) with respect to the variable r, we see that E(r;t,x,z) = ™n=0,...,3(rn/n ^˜˜r)nE(r;t,x,z))|r=0}

œ0 0). In addition to the contribution due to the concentration gradient (Eq. (1)), the flux of particles will have a second contribution j t , called the transport term, due to the movement of the surface. Indeed, during a time dt while the surface shifts a distance dx, an additional number dn = c(x, t).dx of particles per unit of surface will diffuse, leading to a flux:

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j t  x, t

dn i dt

c  x,t

dx i dt

c  x, t v

(14)

Eq.(1) becomes then:

j  x, t

D

wc  x, t i  c  x, t v wx

(15)

If v is constantWKHVHFRQG)LFN¶Vlaw becomes:

wc  x, t wt

D

w 2c  x, t wx

2

v

wc  x, t wx

(16)

II.4. Trap-Limited Diffusion Let us suppose now that the diffusing (mobile) particles can be captured with a rate D in some traps whose concentration is NW. The concentration of particles diffusing into the sample (cm (x, t)) and that of those trapped (cW (x, t)) are governed by the following set of equations:

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80

Aomar Hadjadj

w 2c m  x , t wc IJ  x , t  D wt wx 2

wc m  x , t wt wc IJ  x , t wt

ª c x , t Į «1  IJ NIJ ¬

(17)

º

» c m x , t ¼

(18)

If the number of traps (NW) is very large, Eq. (17) becomes:

wc m  x , t wt

w 2c m  x , t

D

wx

2

 Į c m x , t

(19)

Before solving the partial differential Eq. (19), let us try to find the inverse Laplace transform of the following function F(x, V, K), with K homogenous to the inverse of a time:

§ 1 ı Į · exp ¨  x ¸ ı K © a2 ¹

F  x , ı K

(20)

Remarking that:

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V K

1

1

§ ¨ 1 1 1 ¨  ¨ D K V D D K V D D K   2 ¨ 2 2 2 © a a a a a2

1

a V K 2

2a2

a2

· ¸ ¸ ¸ ¸ ¹

(21)

F(x, V, K) can be rewritten in the form:

F x , V , K

1 2a

2

>

1 F  x , V , K  F  x , V , K D K a2

@

(22)

where:

F r (x , V , K)

§ V D · exp ¨  x ¸ V D D K © a2 ¹ r a2 a2 1

If we set:

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(23)

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

81

V

­ °°q ® °h °¯

a2 D K a2

and refer to Table 1, the inverse Laplace transform of F± (x, V, K) is :

f

r

x , t , K 1

ª a § x2 · D K 2 º ¸¸ # ¨¨  exp a .» « 2 2 S t 4 a t a ¹ © « » « · » § D K  D  K t ¸ . » exp   D IJ t «exp ¨ r x ¹ » © a2 « « » D  K t · § «erfc ¨ x r a » ¸ ¹ © 2a t a2 ¬« ¼»

(24)

Consequently, the inverse Laplace transform of F (x, V, K) will be:



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f x ,t , K

-1



D  K t · º ª § x § D K · a ¸» ¸ erfc ¨ «exp ¨  x 2 2 2a t © ¹ © ¹» a a 1 (25) exp   Kt « 2 « D  K t · » § x § D K · a ¸» ¸ erfc ¨ «  exp ¨ x © © 2a t ¹¼ a2 a2 ¹ ¬

/HW¶VFRPHEDFNWR(T ,WV/DSODFH transform gives:

w 2C m  x , ı § ı  Į · ¨ ¸C m  x , ı © D ¹ wx 2

(26)

As we deed it previously, we shall try to find solutions in the form of Eq. (5).First, the solutions of the characteristic equation in k are :

k

r

V D

(27)

D

Then, the boundary condition:

c m f, t

0 Ÿ C m f , V

0

rules that k must be negative. Consequently:

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82

Aomar Hadjadj

§ C m  x , ı C m  0 , ı exp ¨  x ©

ı Į · ¸ D ¹

(28)

III. EXPERIMENTAL CONDITIONS FOR FILM DEPOSITION AND HYDROGEN PLASMA TREATMENT The thin hydrogenated amorphous silicon (a-Si:H) films were deposited on glass substrates, in a conventional plasma enhanced chemical vapor deposition (PECVD) reactor, operating at 13.56 megahertz (MHz), schematically represented in Fig. 6 [33]. The capacitively-coupled radio-frequency (rf) glow discharge is confined in a cylindrical box between two asymmetric electrodes, a powered radiofrequency (rf) electrode and a grounded electrode. The deposition conditions were those of standard conditions (low silane gas pressure, low rf power and a moderate substrate temperature 230-250 °C). Doping is obtained by adding a doping gas (diborane or phosphine) to the silane. The deposition duration was adjusted to obtain ~ 200 nanometer (nm) -thick films. After deposition, the films were immediately exposed to hydrogen plasma in order to avoid their oxidation. Indeed, a thin native oxide layer acts as an efficient barrier against hydrogen diffusion. The monochamber reactor, designed for in situ studies, is equipped with a UltraVioletVisible modulated-phase ellipsometer to investigate the optical properties of the deposited films [34]. The fit of the spectroscopic ellisposmetry spectra, using an adapted multilayer optical model, allows us to determine the thickness and the composition of each layer.

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RF generator

a

13.56 MHz

Matching box

Photoelastic modulator

RF electrode

Gas

Plasma box Analyzer

Plasma Optical fiber

Polariser Shutter

Monochromator

Sample Xe lamp Filter

PM

Figure 6. Schematic description of the monochamber PECVD deposition reactor equipped with an in situ phase-modulated ellipsometer. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

83

IV. TRAP-LIMITED HYDROGEN DIFFUSING IN AMOURPHOUS SILICON When a hydrogenated amorphous silicon (a-Si:H) thin film is exposed to H2 plasma, two processes occur. When the walls of the reactor are clean, the hydrogen plasma results in the formation of a H-rich subsurface layer and the etching of the film surface. On the other hand, if the walls are coated beforehand with a-Si:H, the plasma results in the deposition of hydrogenated microcrystalline silicon (Pc-Si:H) by chemical transport. In this case, silicon is etched from the reactor walls by hydrogen and deposited on the heated substrate, leading to the formation of a Pc-Si:H film accompanied with an out-diffusion of hydrogen from the sample.

IV.1. Kinetics of Hydrogen Etching of Hydrogenated Amorphous Silicon As previously mentioned, the plasma treatment LV SHUIRUPHG LQ FRQGLWLRQV RI ³FOHDQ UHDFWRU ZDOOV´ 7KH NLQHWLFV of the hydrogen plasma-induced modifications, as determined from in situ spectroscopic ellipsometric measurements, is schematically represented in Fig. 7:

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a) A H-rich layer, with a thickness dH(t), starts forming right from the ignition of H2 plasma. b) Simultaneously, an etching of the film surface occurs. The etching rate (re) reaches its mean value when the H-rich sub-layer reaches its final thickness. c) The reduction of the film thickness goes on until the complete etching of the film.

H2 plasma

dH

(a) a-Si:H

(b) (c) H-rich layer

Figure 7. When a-Si:H film is exposed to H2 SODVPDLQ³FOHDQZDOOV´FRQGLWLRQVD+-rich layer forms accompanied with an etching of the film surface. The surface roughness is not represented for the sake of simplification. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

84

Aomar Hadjadj

The linear decrease in the film thickness indicates a constant etching rate (re) at a given temperature (Fig. 8). When T is raised from 100 to 250 °C re decreases from 3.30 to 1.33 nm.min1 (nanometer per minute).

Film thickness (nm)

200 150 100 T (°C) re (nm/min) 100 3.30 150 3.14 200 2.45 250 1.33

50 0 0

10 20 30 40 50 H2 plasma exposure (min)

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Figure 8. Time evolution of the thin a-Si:H film thickness during H2 plasma exposure at different temperatures. The lines are linear fits. The etching rates are indicated.

The observed modifications of a-Si:H, as a result of H2 plasma exposure, are due to atomic H impinging on the film surface and diffusing through it. We shall treat this effect within the one-dimensional diffusion of hydrogen into a semi-infinite material (x t 0) schematically represented in Fig. 9. The film surface is fixed at the coordinate x = 0 and the substrate extends to x = f. The etching effect leads to the shifting of the film surface at a rate re. The diffusing hydrogen can be captured and immobilized at traps with a concentration NW [12, 19, 20, 21, 26, 35]. Under these assumptions, the concentration of mobile hydrogen diffusing into the sample (cm(x, t)) and that of the trapped hydrogen (cW(x, t)) are governed by the following set of equations:

w 2 c m x , t

w c m  x , t w cIJ  x , t  wt wx

w c m x , t wt

DH

w c IJ x , t wt

ª c  x , t º c x , t  Į r c IJ x , t Į IJ «1  IJ N IJ »¼ m ¬

wx

2

 re

(29)

(30)

The capture rate DW can be linked to the density of traps NW and their capture radius RW by [36]:

DW

4 ʌ RW D H N IJ

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(31)

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

re t

H2 plasma

a-Si:H

85

Substrate

x=0

x

Figure 9. Schematic of H impinging on an etched surface of a-Si:H and diffusing through it. The film surface is fixed at the coordinate x = 0 and the substrate extends to x = f. We assume that etching and diffusion occur simultaneously.

where DH is the diffusivity of hydrogen and DW (Dr) is the capture (release) frequency of the traps. However, we have to emphasize the following simplifications we made while writing Eqs. (29) and (30): i.

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ii. iii.

iv.

We consider that etching and H diffusion start simultaneously, although we observed experimentally that etching needs few seconds particularly at high temperature. We do not take into account a probable time-dependence of the hydrogen diffusivity DH [19-21]. Defect density in a-Si:H is higher at the two film surfaces compared to bulk material. Moreover, H2 plasma leads to a higher density of trapping sites at the near-surface region compared to the bulk of the sample [37]. For the sake of simplification, we shall consider, for a given temperature, a uniform and constant density of traps NW. Amorphous solids contain several kinds of defects (vacancies, weak bonds, dangling bonds, etc) that can trap mobile H atoms and slow their diffusion. We shall not make any allusion to the microscopic nature of the trapping sites and consider only one type of traps.

The formation of the H-rich layer suggests that the trapping process is predominant (DW Dr). Moreover, to find an analytical solution cm(x, t) of Eq. (29), one has to drop the coupled term. This simplification amounts to underestimate the trapping rate DW. With these simplifications, Eqs. (29) and (30) become:

wc m  x , t wt

DH

w 2c m  x , t wx

2

 re

wc m  x , t  Į IJ c m x , t wx

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(32)

86

Aomar Hadjadj

C IJ x , t

ª N IJ «1  exp « «¬

§ ¨ ĮIJ ¨ N IJ ¨ ©

t

³C 0

m

·º ¸  x , t' dt' ¸ » ¸» ¹ »¼

(33)

At the beginning of the plasma exposure, the initial conditions for Eqs (32) and (33) are:

c m x , 0 c IJ x , 0

C0

x d0

0

x !0

0

x

(34)

and the boundary conditions are:

c m 0 , t C 0 c m f , t

0

c IJ f , t

0

t

(35)

The Laplace transform of Eq. (32) leads to:

w 2C m  x , ı wx

2



re wC m  x , ı § ı  Į IJ ¨ DH wx © DH

· ¸C m  x , ı ¹

0

(36)

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and a characteristic equation in k whose solutions are : 2

k

r re ı  ĮIJ  e r  2 2D H DH 4D H

(37)

Keeping only the negative solution for k, the Laplace transform of the solution of Eq. (4) will be then:

C m x , ı

ª C0 exp «  ı «¬

2 § r re ı  ĮIJ ¨ e   2 ¨ 2D H DH 4D H ©

· º ¸x » ¸ » ¹ ¼

Remarking that:

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(38)

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

1 ı DH

1 2

re

2

4D H

2

Į  IJ DH

1 § ¨ 2 2 re ı  ĮIJ re Į ¨    IJ ¨ 2 2 DH DH 4D H ¨ 4D H ¨ 1 ¨ 2 2 re ı  ĮIJ re Į ¨    IJ ¨ 2 2 DH DH 4D H 4D H ©

· ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹

87

(39)

and using the inverse Laplace transform (Table 1), one can easily arrive to the solution of the partial differential Eq. (32):

c m  x,t

C0 r § · exp ¨  e x ¸ 2 © 2D H ¹ 2 2 ª · § § re Į IJ ·¸ re Į x ¨ ¨   IJ  erfc DH t ¸ «exp x 2 2 ¸ ¨2 DH t ¨ DH ¸ DH « 4D H 4D H ¹ © ¹ © « 2 2 § § « re Į IJ ·¸ re Į x ¨ ¨  erfc DH t   IJ «  exp ¨  x 2 2 ¸ ¨ DH DH 2 DH t 4D H 4D H © ¹ © ¬«

º » » » ·» ¸» ¸ ¹ ¼»

(40)

10-1

101 0

t=

Count.s-1

103

cW (x , t) / cW (0 , t)

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100

Depth (nm) 50

10 s 30 s 1 min 2 min 5 min 10 min

0

20

40 Depth (nm)

60

80

Figure 10. Concentration profiles cW(x, t) at different times calculated with Eq. (40). The values used for the calculations are DH = 2.36 10-13 cm2s-1, re = 1.33 nm.min-1 and NW = 4.5 1017 cm-3. The inset shows a deuterium depth profile from SIMS resulting D-plasma treatment of a 200 nm-thick a-Si:H layer.

In Figs. 10 are plotted the concentration profiles cW(x, t) at different times calculated with the values of DH = 2.36 10-13 cm2s-1, re = 1.33 nm.min-1 and NW = 4.5 1017 cm-3, determined at Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

88

Aomar Hadjadj

T = 250 °C. Fig. 10 indicates that despite its complicated form, cW(x, t) exhibits really an exponential behavior. The exponential behavior of cW(x, t) has been confirmed by secondary-ion mass spectroscopy (SIMS) measurements [10, 29]. The inset of Fig. 10 shows a deuterium depth profile from SIMS resulting from a D-plasma treatment of a-Si:H at 250°C. In order to compare our calculated results with those derived from ellipsometry measurements, we define an effective diffusion length LD by:

c IJ x L D , t c IJ x 0 , t

1 e

(41)

dH (nm)

30

T (°C)

DH (cm2/s)

250

2.36 10 -13

200

8.70 10 -14

150

5.16 10 -14

100

3.0 10 -14

20

10

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0 0

100 200 H2 plasma exposure (s)

300

Figure 11. Time-evolution of the thickness of the H-rich layer (dH) at different etching temperatures (symbols). The value of DH used for the calculation (solid curves) is also indicated.

To correlate the excess of hydrogen in the H-rich layer, determined by ellipsometry, to the concentration of trapped hydrogen cW(x, t), we shall identify the mean diffusion length LD with the thickness dH. In this way we shall be able to determine both the diffusion coefficient DH and the capture rate DW (and consequently the density of traps NW) at each temperature. Indeed, one can determine these two parameters (DH and DW) by minimising the following quantity:

¦ j

ª «cIJ (d H , t)  ¬

1º e »¼

2

(42)

The summation j is over the whole (t, dH) experimental points. The time-dependence of dH and the mean diffusion length LD given by Eq. (41) are compared in Fig. 11 for the different etching temperatures. At each temperature, the corresponding DH is also indicated. A good agreement is found between the experimental data (ellipsometry) and the calculated Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

89

diffusion length. The diffusion length rapidly increases at the beginning of H-plasma treatment and reaches a plateau. The saturation value increases from 6.6 to 26.7 nm as temperature increases from 100 to 250°C.

IV.2. Hydrogen Evolution during Microcrystalline Silicon Deposition The solid phase transformation of a-Si:H to hydrogenated microcrystalline silicon (µcSi:H) in low temperature plasma conditions has been the subject of a large number of studies [38±45]. All the growth models of Pc-Si:H agree on the fact that hydrogen is an essential ingredient in the plasma enhanced chemical vapour deposition (PECVD) of such a material [38, 46±50]. However, its exact role is still the subject of broad controversies. The reasons preventing the extraction of an unambiguous relationship between the crystallites formation and the hydrogen evolution are related to the complexity of the overall PECVD process: growth via different Si precursors, hydrogen etching of weak Si±Si bonds, exothermic surface reactions of hydrogen, etc.

Pc-Si:H

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a-Si:H

GaAs

200 nm Figure 12. Cross-section transmission electron microscopy (TEM) image of a 190 nm-thick a-Si:H film deposited on a GaAs substrate and exposed to H2 plasma during 45 minutes. The treatment leads to the formation of a 65 nm-thick Pc-Si:H. The inset displays the electron diffraction patern of the crystallized layer.

When an a-Si:H thin film is exposed to hydrogen plasma, with the walls of the reactor previously coated with a-Si:H, the deposition of µc-Si:H by chemical transport occurs [5053]. In this last case, silicon is etched from the reactor walls by hydrogen and deposited on the heated substrate, leading to the formation of a µc-Si:H film. The cross-sectional transmission electron microscopy (TEM) image of Fig. 12 clearly shows the formation of a 65 nm-thick Pc-S:H layer. The electron diffraction pattern corresponding to the grown layer (inset of Fig. 12) is that of a microcrystalline silicon (Pc-Si:H) structure.

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Fig. 13 summarizes the kinetics of H-induced modifications in a-Si:H in conditions of chemical transport. This kinetics was supported by other ex situ characterization techniques [51-54]. The surface and subsurface modifications, due to atomic H impinging on the film and diffusing through it, lead to the formation of a H±rich layer accompanied with the etching of the film surface. These effects start at the ignition of the hydrogen plasma. As previously shown, the etching rate as well as the time needed for the complete formation of the H±rich layer depends on temperature. With the beginning of the microcrystalline layer deposition, the the H-rich layer vanishes.

H2 plasma

(a)

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Bulk a-Si:H

(b) H±rich layer

(c) Pc-Si:H

Figure 13. (a): From the ignition of the H2 plasma, a H±rich layer starts forming accompanied with an etching of the film. (b) and (c) : When Pc-Si:H deposition by chemical transport begins, an outdiffusion of hydrogen leads to the disappearance of the H±rich layer. The surface roughness is not represented for the sake of simplification.

The hydrogen out-diffusion is inconsistent with a normal diffusion process. Although this phenomenon is, without any doubt, intimately linked to the amorphous/microcrystalline phase-transformation of the a-Si:H network, the mechanism of such a relationship remains unsolved. The out-diffusion of hydrogen can be explained thanks to the role of the grown PcSi:H layer that gradually reduces the hydrogen supply from the plasma and then decreases the solubility of H in Si until it recovers the value of an untreated a-Si:H. The growth of the Pc-Si:H will consume most of H impinging atoms and then will reduce the amount of hydrogen reaching the Pc-Si:H/a-Si:H interface. Thus, once the Pc-Si:H layer starts to grow at t = T1, less and less hydrogen atoms reach the PcSi:H/a-Si:H interface. To express the fact that, after a time T hydrogen atoms cannot reach the PcSi:H/a-Si:H interface any more, we can use the following boundary condition:

t dT1

C0 c m 0 , t

C0

§ t T 1 · exp ¨  ¸ ș ¹ ©

t !T1

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(43)

Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

91

As the etching process stops with the growth of the PcSi:H layer, we shall neglect it even during the formation of the H-rich layer. Thus, the concentration of mobile hydrogen obeys the following equation:

wc m  x , t wt

DH

w 2c m  x , t wx

2

 Į IJ c m x , t

(44)

The concentration of trapped hydrogen remains still given by Eq. (33).

a) Before The Growth of the Pc-Si:H Layer ( T d T1) With the given initial (Eq. (43)) and boundary conditions (Eq. (35)) the Laplace transform of the solution of Eq. (44) will be:

C0 ı  ĮIJ § exp ¨¨  x ı DH ©

C m x , ı

· ¸¸ C 0 F  x , ı K ¹

0

(45)

and its inverse Laplace transform is :

c m (x , t)



C 0 f x , t , f

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where f x , t , K

-1

(46)

is the same function as Eq. (25).

b) During the Growth of the Pc-Si:H Layer ( T > T1) With the given initial (Eq. (43)) and boundary conditions (Eq. (35)) the Laplace transform of the solution of Eq. (44) will be:

C m x , ı

C0 ı  ĮIJ § exp ¨¨  x ı DH ©

ı  ĮIJ § · C0 ¸¸  exp   T 1ı exp ¨¨  x DH © ¹ ı

ı  ĮIJ § exp   T 1ı exp ¨¨  x  1 DH © ı  ș C0

· ¸¸ ¹

· ¸¸ ¹ (47)

According to Eq. (25) and Table 1, the inverse Laplace transform will be:

c m  x , t C 0 >f  x , t , f  f  x , t T 1 , f  f x , t T 1 , ș @

(48)

c) Hydrogen Out-Diffusion In Fig. 14 are plotted the concentration profiles cm(x, t) and ct(x, t) at different times obtained using the values of DH = 2.09 10-14 cm2s-1, DW = 1.35 10-1 s-1, C0/NW = 10, T1 = 5 min and T = 1 min. Before the deposition of the microcrystalline layer (t < T1), both cm and cW profiles rapidly extend and reach a saturation limit. Beyond the H-rich layer formation and

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during the crystallization process (t > T1), the sample starts to run out of mobile hydrogen. Consequently, the H trapped profile remains unchanged. The integrated amount of mobile (or trapped) hydrogen Qm,W within the whole film thickness can be defined by: f

Q m,IJ t

³c

m,IJ

 x , t dx

(49)

0

H2 plasma treatment

cm (x , t) / c m (0 , t)

10-1

10 s 20 s 1 min 5 min 7 min

10-3

10-5

cW(x , t) / N W

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10-1

10-3

10-5 0

20

40 Depth (nm)

60

Figure 14. Concentration profiles cm(x, t) (up) and cW(x, t) (down) at different H2 plasma treatment times. The values used for the calculations are: DH = 2.09 10-14 cm2s-1, DW = 1.35 10-1 s-1, C0/NW = 10, T1 = 5 min and T = 1 min.

Fig. 15 shows the time-dependence of the Qm/QW ratio, deduced from the hydrogen profiles of Fig. 14. Although Qm/QW gradually decreases for t < T1, the amount of mobile hydrogen remains higher than that trapped in the H±rich layer. Within this range of time, the capture process of hydrogen is the prevailing one. Beyond t = T1, the ratio Qm/QW undergoes a drastic fall. Such a sharp depletion in mobile hydrogen marks the end of the predominance of the hydrogen capture process. According to Figs. 14 and 15, the H±rich subsurface layer

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Trap-Limited Diffusion of Hydrogen in Amorphous Silicon Thin Films

93

starts to vanish for plasma exposure times higher than 10 min. This happens when Qm/QW < 0.1.

101

Qm / QW

100

10-1 T1

10-2 0

T2

5 10 H2 plasma exposure (min)

15

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Figure 15. Time-dependence of the ratio Qm/QW deduced from the hydrogen profiles of Fig. 14. The arrows indicate the end of the H-rich layer formation (T1) and the beginning of its disappearance (T2).

The growth of the Pc-Si:H layer prevents the H atoms coming form the plasma from reaching the Pc-Si:H/ a-Si:H interface. Consequently, by analogy with the gas dissolution in a liquid, the H pressure above the a-Si:H surface vanishes leading to a decrease in the H solubility in the a-Si:H network. During this period of time, the H reemission becomes predominant at the expense of the trapping process. In this case, Eq. (30) becomes:

wc IJ  x , t wt

Į r c IJ  x , t

(50)

and its integration leads to:

Q IJ t Q IJ T 2 exp > Į r t T 2 @

t tT 2

(51)

In Fig. 16 we compare calculated and experimental values of QW(t) at different H2 plasma exposure times. Both experimental and calculated values are normalized with respect to t = T2. The calculations were performed using: DH = 2.09 10-14 cm2s-1, DW = 1.35 10-1 s-1, C0/NW = 10, T1 = 5 min and T = 1 min, for the H trapping process, and Dr = 1.7 10-2 s-1 and T2 = 9 min for the H reemission process. In the case of the hydrogenation of intrinsic a-Si:H at 250 °C with a heated filament, An et al. have found a quite lower reemission rate (Dr < 2 10-7 s-1) upon extinguishing the heated filament [26]. Fig. 7 displays a good agreement between calculated and experimental results within the whole time-range.

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QW (t) / QW (T2)

1 0.8 0.6 0.4

Experimental H trapping

0.2

H reemission

0 0

5 10 H2 plasma exposure (min)

15

Figure 16. Experimental (symbols) and calculated (curves) integrated amount of trapped hydrogen versus the time of the H plasma exposure. The values used for the calculations are: DH = 2.09 10-14 cm2s-1, DW = 1.35 10-1 s-1, C0/NW = 10, T1 = 5 min and T = 1 min, for the trapping process (solid curve), and Dr = 1.7 10-2 s-1 and T2 = 9 min for the reemission process (dashed curve). The data are normalized with respect to their values at t = T2.

REFERENCES J. I. Pankove and N.M. Jonhson, Editors of ³+\GURJHQLQ6HPLFRQGXFWRUV´9RORI Semiconductors and Semimetals (1991), (Boston, Academic Press). [2] R. A. Street, in ³+\GURJHQDWHG$PRUSKRXV6LOLFRQ´ (1991), edited by R. W. Cahn, E. A. Davis and I. M. Ward (Cambridge, Cambridge University Press), p. 51 [3] W. Beyer and U. Zastrow, in ³$PRUSKRXV 6LOLFRQ 7HFKQRORJ\´ (1996) edited by M. Hock, E. A. Schiff, S. Wagner, R. Schropp and A. Matsuda (Pittsburgh, Materials Research Society), p. 463. [4] 1+1LFNHO(GLWRURI³+\GURJHQLQ6HPLFRQGXFWRUV,,´9RO of Semiconductors and Semimetals (1999), (Boston, Academic Press). [5] K. Zellama, P. Germain, S. Squelard, B. Bourdon, J. Fontenille and R. Danielou, , Phys. Rev. B 23 (1981) 6648. [6] W. Beyer and H. Wagner, J Appl. Phys. 53 (1982) 8745. [7] A. J. Franz, M. Mavrikakis and J. L. Gland, Phys. Rev. B 57 (1998) 3927. [8] R. Rizk, P. De Mierry, D. Ballutaud, M. Aucouturier and D. Mathiot, Phys. Rev. B 44 (1991) 6141. [9] W. B. Jackson and C. C. Tsai, Phys. Rev. B 45 (1992) 6564. [10] P. V. Santos and W. B. Jackson, Phys. Rev. B 46 (1992) 4595. [11] J. Shinar, R. Shinar, D. L. Williamson, S. Mitra, H. Kavak, and V. L. Dalal, Phys. Rev. B 60 (1999) 15875. [12] D. A. Tulchinsky, J. W. Corbett, J. T. Borenstein and S. J. Pearton, Phys. Rev. B 42 (1990) 11881.

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[13] H. M. Branz, R. Reedy, R. S. Crandall, H. Mahan, Y. Xu and B. P. Nelson, J. NonCryst. Solids 299-302 (2002) 191. [14] A. von Keudell and J. R. Abelson, J. Appl. Phys. 84 (1998) 489. [15] S. Agarwal, A. Takano, M .C .M. van de Sanden, D. Maroudas, and E. S. Aydil, J. Chem. Phys. 117 (2002) 10805. [16] M. Nakamura, T. Ohno, K. Miyata, N. Konishi and T. Suzuki, J. Appl. Phys. 65 (1989) 3061. [17] N. M.Johnson, J. Walker and K. S. Stevens, J. Appl. Phys. 69 (1991) 2631. [18] S. Zafar and E. A. Schiff, Phys. Rev. Lett. 66 (1991) 1493. [19] U. K. Das, T. Yasuda and S. Yamasaki, Phys. Rev. Lett. 85 (2000) 2324. [20] U. K. Das, T. Yasuda and S. Yamasaki, Phys. Rev. B 63 (2001) 245204. [21] S. Yamasaki, U.K. Das and T. Yasuda, J. Non-Cryst. Solids, 299-302 (2002) 185. [22] X. M. Tang, J. Weber, Y. Baer and F. Finger, Phys. Rev. B 41 (1990) 7945. [23] X. M. Tang, J. Weber, Y. Baer and F. Finger, Phys. Rev. B 42 (1990) 7277. [24] R. Shinar, J. Shinar, H. Jia and X. L. Wu, Phys. Rev. B 47 (1993) 9361. [25] J. P. Conde, K. K. Chan, J. M. Blum, M. Arienzo, P. A. Monteiro, J. A. Ferreira, V. Chu and N. Wyrsh, J. Appl. Phys. 73 (1993) 1826. [26] I. An, Y. M., Li, C. R. Wronski and R. W. Collins, Phys. Rev. B 48 (1993) 4464. [27] Y. H. Yang, M. Katiyar, G. F. Feng, N. Maley and J. R. Abelson, Appl. Phys. Lett. 65 (1994) 1769. [28] J. E. Gerbi and J. R. Abelson, J. Appl. Phys. 89 (2001) 1463. [29] A. Fontcuberta i Morral and P. Roca i Cabarrocas, J. Non-Cryst. Solids 299-302 (2002) 196. [30] F. Kaïl, A. Fonctcuberta i Morral, A. Hadjadj, P. Roca i Cabarrocas and A. Beorchia, Philos. Mag. 21 (2004) 595. [31] N. Pham, P. Roca i Cabarrocas, A. Hadjadj, A. Beorchia, F. Kaïl and L. Chahed, Philos. Mag. 88 (2008) 297. [32] J. Robertson, C. W. Chen, M. J. Powell and S. C. Deane, J. Non-Cryst. Solids 227-230 (1998) 138. [33] A. Hadadj, N. Pham, P. Roca i Cabarrocas and O. Jbara, J. Appl. Phys. 107 (2010) 083509. [34] R. M. Azzam and N. M. Bashara, ³Ellipsometry and Polarized Light´, North-Holland, Amsterdam (1997). [35] M. Kemp and H. M. Branz, Phys. Rev. B 47 (1993) 7067. [36] T. R. Waite, Phys. Rev. 107 (1957) 463. [37] J. P. Kleider, C. Longeaud and P. Roca i Cabarrocas, J. Appl. Phys. 72 (1992) 4727. [38] S. Veprek, Z. Iqbal, H. R. Ostwald and P. Webb, J. Phys. C 14 (1981) 295. [39] N. Layadi, P. Roca i Cabarroca, B. Drévillon and I. Solomon, Phys. Rev. B 52 (1995)5136. [40] P. Roca i Cabarroca, N. Layadi, B. Drévillon and I. Solomon, J. Non-Cryst. Solids 198200 (1996) 871. [41] S. Hamma and P. Roca i Cabarrocas, J. Appl. Phys. 81 (1997) 7282. [42] A. Matsuda, Thin Solid Films 337 (1999) 1. [43] J. Jang. S.O. Koh, T.G. Kim and S.C. Kim, Appl. Phys. Lett. 60 (1992) 2874. [44] M. Katiyar and J. R. Abelson, Mat. Sci. Eng. A 304-306 (2001) 349. [45] B. Lyka, E. Amanatides and D. Mataras, J. Non-Cryst. Solids 352 (2006) 1049.

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I. Solomon, B. Drévillon, H. Shirai and N. Layadi, J. Non-Cryst. Solids 164 (1993) 989. A. Matsuda and K. Tanaka, J. Non-Cryst. Solids 114 (1987) 1367. H. Shirai, D. Das, J. Hamma and I. Shimizu, Appl. Phys. Lett. 59 1096 (1991). M. Otobe, and S. Oda, J. Non-Cryst. Solids 164 (1993) 993. R.A. Street, Phys. Rev. B 43 (1991) 2454. F. Kaïl, A. Hadjadj and .P. Roca i Cabarrocas, Proceedings of 19th European photovoltaic Solar Energy Conference, Paris, June 7-11 (2004) FRANCE, Eds. W. Hoffmann, J.-L. Bal, H. Ossenbrink, W. Palz and P. Helm, p. 1367. [52] F. Kaïl, A. Hadjadj and P. Roca i Cabarrocas, Thin Solid Films 126-131 (2005) 487. [53] N. Pham, P. Roca i Cabarrocas, A. Hadjadj, A. Beorchia, F. Kaïl and L. Chahed, Philos. Mag. 88 (2008) 297. [54] A. Fontcuberta i Morral, P. Roca i Cabarrocas and C. Clerc, Phys. Rev. B 69 (2004) 125307-1.

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[46] [47] [48] [49] [50] [51]

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 ©2011 Nova Science Publishers, Inc.

Chapter 4

THE ROLE OF THE METHOD OF CHARACTERISTICS IN THE SOLUTION OF ESTIMATION AND CONTROL PROBLEMS FOR HYPERBOLIC PDE SYSTEMS Efrén Aguilar-Garnica*1 and Juan P. García-SandovalÁ2 1

Universidad Autónoma de Guadalajara, Zapopan, México 2 Universidad de Guadalajara, Guadalajara, México

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ABSTRACT In this chapter we summarize the use of the Method of Characteristics (MC) in the solution of estimation and control problems for processes whose model is given by FirstOrder Partial Differential Equations or Hyperbolic systems. First, a dynamical robust observer is built with the MC for a non-isothermal plug flow reactor whose state variables travel along one characteristic line. Then, this observer in its integral equation version is designed for a solid-waste anaerobic digestion process which is described by a set of variables that follow different characteristic lines. In both cases the aim of the observers is to estimate the evolution of certain specific variables which is necessary to monitor but whose measurement becomes difficult either because do not exists the suitable analytical techniques or because such measurement is relatively expensive. Finally, the chapter is closed with an analysis of a robust control law derived with the MC for a heat exchanger in which a time-variant characteristic line is involved. The performance of the observers and the control law is analyzed with simulation experiments obtaining satisfactory results.

* Corresponding author: Departamento de Química, Universidad Autónoma de Guadalajara, 1201 Av. Patria, 44100 Zapopan, México. Tel. +(55)-33-36-48-88-24x2220. E-mail: [email protected] Á 'HSDUWDPHQWR GH ,QJHQLHUtD 4XtPLFD 8QLYHUVLGDG GH *XDGDODMDUD  %OYG 0* %DUUDJin, 44430 Guadalajara, México. Tel. +(55)-13-78-59-00x7536. E-mail: [email protected] Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

98

Efrén Aguilar-Garnica and Juan P. García-Sandoval

NOMENCLATURE

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‫ܭ‬, ‫ܭ‬1 , ‫ܭ‬2 , ‫ܭ‬2 Ș , ‫ݒ‬, ‫ ݂ݒ‬, ‫ ݂ݒ‬, ‫ ܯݒ‬, ‫ݒ‬2 , ‫ܥ‬, ‫ܥ‬11 , ‫ܥ‬12 , ‫ܥ‬21 , ‫ܥ‬22 :Constant coefficients of the mathematical model ܹ, ܵ, ‫ܤ‬, ܴܺ , ܻ :Chemical species ‫ ܺܥ‬, ‫ ݅ݔܥ‬: Concentration of ܴܺ , inlet concentration of ܴܺ ‫ܣ‬, ‫ܦ‬, ‫ܧ‬, ‫ܪ‬, ߛ1 ሺ‫ݐ‬ሻ, ߛ2 ሺ‫ݐ‬ሻ: Elements of the proposed estimation scheme ‫ݎ‬, ݂, ݂‫ ݖ‬, ݂‫ ݐ‬: Size of the nonlinearities vector, nonlinearities vector and derivatives of the vector ‫ݏܭ‬, ‫݂ܭ‬, ‫݃ܭ‬, ‫ܮ‬, ܸܴ , ܷ‫ ݓ‬,‫ ݌ܥ‬, ݇10 , ݇1 , ݇݀ , ݂݉ , ݉݃ , ‫ܨ‬, ߩ݉ ,ߩ,ȟ‫ ܴܪ‬, ‫ݍ‬, ߛ, ܻܴ , ߯ , ܴത , ‫ ܣܧ‬, ‫ ܿܪ‬, ‫ ݏܣ‬, ߢ: Parameters of the mathematical models ܶ, ܶ݅ , ݆ܶ , ܶ‫ ܬ‬, ܶ‫ ݏ‬, ܾܶ , ܶ‫ ݐݑ݋‬, ܶ‫ ݎ‬:Temperatures ‫ݔ‬1 ሺ‫ݐ‬ሻ, ‫ݔ‬2 ሺ‫ݐ‬ሻ: State variables in the dynamical version of the proposed estimation scheme ‫ݖ‬, ‫ݖ‬0 , ‫ ݖ ݉ݖ‬, ܽ, ܾ: Space variable and fixed specific axial points. ߙ(‫)ݖ‬, ݃ܺ (‫)ݖ‬, ݃ܶ (‫)ݖ‬, ߙܹ (‫)ݖ‬, ߙܵ ሺ‫ݖ‬ሻ, ߙ‫ ܤ‬ሺ‫ݖ‬ሻ, ߙ1 , ߙ2 : Initial conditions ߚሺ‫ݐ‬ሻ, ‫ ܺܩ‬ሺ‫ݐ‬ሻ, ‫)ݐ( ܶܩ‬, ߚܹ (‫)ݐ‬, ߚܵ (‫)ݐ‬, ߚ‫)ݐ( ܤ‬, ߚ1 , ߚ2 :Boundary conditions ܺ, ܺ1 , ܺ2 , ܺሺܽ, ‫ݐ‬1 ሻ ǥ ܺ൫ܽ, ‫ ݐ ݊ݐ‬൯, ܺ0 , ܴܺ , ܺܵ : Different representations for the state vector ݊, ݊1 , ݊2 :Dimensions of the state vector. ‫)ݐ(ݑ‬, ‫ݑ‬1 , ‫ݑ‬2 : Process inputs and partitions ‫)ݐ( ܿݑ‬, ‫ ݏݑ‬, ‫)ݐ( ߬ܿݑ‬: Different representations for the input control ‫ݐ‬, ‫ݐ‬1 , ‫ݐ‬2 , ‫ݐ‬3 , ǥ , ‫ ݐ ݊ݐ‬, ‫ݐ‬0 : Time ans simulation time points Ȱሺ‫ݖ‬, ‫ݐ‬, ܺሻ, Ȳሺ‫ݖ‬, ‫ݐ‬, ܺሻ, ߶ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ, ߮ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ, ߰ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ, ȣሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ, ݃൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯, ‫ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯, ݄൫ܺሺ‫ܮ‬, ‫ݐ‬ሻ൯, ߶݆݅ (‫ݖ‬, ‫ݐ‬, ܺ), ߰݅ ሺ‫ݖ‬, ‫ݐ‬, ܺሻ: ‫ ݇݅ܮ‬, ܰ݇ Nonlinear functions of the state variables ߣ, ߬, ߬ ‫ כ‬, ߬(‫ ) ݈ݐ‬:Constant, integration time along the characteristic line, residence time ෡ i , ߜ݆݅ : Parameters, matrices and vectors of the method of characteristics ߣ݅ , ߪ, ߪ݅ , ȟi , ߦ݅ , Ȧ, ‫ ݅ܯ‬, M Ȟ, ‫݌‬, ‫ ܱ݌‬, ‫ ݂݌‬, ‫ ݅݌‬: State variable and parameters in the numerical method to simulate the second study case ߳ሺ‫ܮ‬, ‫ݐ‬ሻ, ߰‫ ܱܥ‬, ܾ1 (‫)ݐ‬,ܾ2 (‫)ݐ‬, ߜܿ‫ ݋‬ሺܾ2݁ ሻ: Estimation error dynamics and parameters involved in this dynamics ‫ݓ‬: Auxiliary variable in the proposed estimation scheme ෡ : Vectors and surfaces ܰ, ܸ, π, π ‫ݕ‬, ‫ ݏݕ‬, ‫ ݎݕ‬: Output variable, output variable in steady state, reference ߠ, Ȟc , ݇ܿ , ߟሺ‫ݐ‬ሻ, ݁‫ ݎ‬ሺ‫ݐ‬ሻ, ߱1 ሺ‫ݐ‬ሻ, ߱2 ሺ‫ݐ‬ሻ: Elements of the proposed robust controller ݂ ܸ‫ ܮ‬, ‫ݒ‬ሺ‫ݐ‬ሻ, ȯ‫ݐ‬െ‫ ݐ‬0 (ܺ0 ) : Lyapunov function, integral function and flow vector field of function ݂ ܲ: Certain point in the solution VXUIDFHRID)23'(¶V

ABBREVIATIONS Finite Differences Method (FDM) Orthogonal Collocation Method (OCM). Method of Characteristics (MC) )LUVW2UGHU3DUWLDO'LIIHUHQWLDO(TXDWLRQV)23'(¶V 2UGLQDU\'LIIHUHQWLDO(TXDWLRQV2'(¶V 3DUWLDO'LIIHUHQWLDO(TXDWLRQV3'(¶V

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99

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1. INTRODUCTION Several estimation and control techniques have been proposed in the last two decades for distributed parameter systems (i.e. mathematical models in Partial Differential Equations 3'(¶V  7KHVH WHFKQLTXHV KDYH EHHQ XVXDOO\ GHULYHG ZLWK D PHWKRGRORJ\ called early lumping which consists basically in the reduction of the infinite dimensional model to a set of Ordinary Differential Equations 2'(¶V by using modal (spectral) decomposition methods followed by the design of the estimation and/or control schemes. Nevertheless the aforementioned methodology cannot be followed if the distributed parameter system is given E\ D VHW RI )LUVW 2UGHU 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV )23'(¶V  or hyperbolic systems (Christofides, 2001). This is mainly because the decomposition methods (e.g. Finite Differences, Orthogonal Collocation) are not able to accurately capture the dynamic behavior RI WKH )23'(¶V DQG DV D FRQVHTXHQFH RQH FDQ REWDLQ HUURQHRXV FRQFOXVLRQV FRQFHUQLQJ stability properties of the open-loop and/or closed-loop system. This important restriction has contributed to GHOD\ WKH UHVHDUFKRQ HVWLPDWLRQ DQGFRQWURO GHVLJQIRU )23'(¶V XQWLO VRPH alternatives to the early lumping approach were suggested. For instance, Christofides and Daoutidis 1996; 1998a,b; Christofides, 2001; Dubljevic et al. 2005 have solved the control SUREOHPIRU)23'(¶VXVLQJFRQFHSWVIURPWKHJHRPHWULFFRQWUROWKHRU\ without reducing the original model. Specifically, they have developed distributed feedback controllers in order to enforce output tracking and guarantee the overall stability of closed-loops systems. However, they focused on the cases where the manipulated input, the controlled output and the measured output are distributed in space. Furthermore, Gundepudi and Friedly (1998) have shown that geometric tools for input/output linearizing feedback control were not suitable to control FOPDE¶V systems due to the non-stationary response of process outputs that could induce oscillatory responses. More recently, the Method of Characteristics (MC) has been considered to solve control problems in hyperbolic systems. For instance, Hanczyc and Palazoglu (1995) use this method combined with nonlinear control techniques to transform the PDE¶V system into a class of ODE¶V models. On the other hand, Gundepudi and Friedly (1998) proposed a discrete control scheme with variable time intervals based on the MC to regulate temperature in a heat exchange process and the concentration in a nonisothermal reactor by manipulating the characteristic flow velocity. Although they showed that the control output was a function of time alone, the application of such a controller yielded sluggish responses due to the updating step at each residence time. Other approaches were derived for particular FOPDE¶V systems to attenuate undesired oscillations or to suppress the effect of inlet disturbances. This is the case of the work conducted by Shang et al. (2005) where a continuous controller based on the MC is proposed such that the closed-loop response of the process output moved towards the set-point; however, this controller requires several measurements along the axial directions in order to regulate the output only at a specific point. Another contribution related with the design of control laws IRU)23'(¶VV\VWHPVE\XVLQJWKH0&LVWKHRQHGHVFULEed by GarcíaSandoval et al. (2008). In this contribution a framework for the synthesis of robust controllers that are able to handle explicit time varying uncertain variables and unmodeled dynamics for )2'3(¶V V\VWHP ZDs developed. Furthermore, the MC has also been considered to solve certain specific state estimation problems as it is shown in the contribution of AguilarGarnica et al. (2010) where a robust estimation tool for plug flow reactors is proposed. In all

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these works, the MC has been used since it is able to generate an exact representation in 2'(¶VRUin integral equations) for the )23'(¶V model that are valid at a certain fixed axial point and from which the controllers and estimators are build. This is possible because the )23'(¶V GHVFULEH WKHEHKDYLRURI VWDWH YDULDEOHVWUDYHOOLQJ DORQJ FKDUDFWHULVWLF OLQHV LQ WKH positive z direction with certain velocity and the MC provides a precise framework for analyzing such equations (Rhee et al. 2000). Thus, the MC offers the possibility of handling 2'(¶V ZLWKRXW GHVSLVLQJ WKH GLVWULEXWHG QDWXUH RI WKH PRGHO ,Q RWKHU ZRUGV WKH 0& provides the simplicity of the early lumping approach but considering the infinite dimensional nature of the model. The aim of this chapter is to describe structural properties of the MC including its role in the solution of estimation and control problems for certain processes whose model is given in )23'(¶VDQGLWLVRUJDQL]HGDVIROORZV,QWKHIROORZLQJ VHFWLRQ basic concepts of the MC are introduced and then applied to generate dynamical simulations for the processes that are being analyzed. Besides, in the third section, it is described the methodology to design a robust state observer (either in a dynamical form or as integral equations) considering the properties of the MC. Then, this observer is designed for a non-isothermal plug flow reactor in which all the state variables travel along the same characteristic line and for which the dynamical version of the robust estimator is implemented. Later, this same estimator, but in its integral equations version, is designed for a solid-waste anaerobic digestion process whose main characteristic is that all the state variables involved in the process follow different characteristic lines. In section number four, it is described a methodology to design a robust control law regarding fundamental properties of the MC and then such law is implemented to fulfill certain regulation task in a heat exchanger. Since the manipulated variable is the volumetric flow and the characteristic line depends on it, this case is appropriate to analyze the application of the MC for time-variant characteristic lines. The chapter is closed with a conclusion section.

2. METHOD OF CHARACTERISTICS: BASIC CONCEPTS In the observation and control problems presented in this chapter, we deal with onespatial dimension dynamical systems modeled by a finite number of coupled quasi-linear, )23'(¶VZLWKWKHfollowing form ܺ‫ ݐ‬+ ĭሺ‫ݖ‬, ‫ݐ‬, ܺሻܺ‫ = ݖ‬Ȍ(‫ݖ‬, ‫ݐ‬, ܺ)

(2.1)

with the initial condition ܺ(‫ݖ‬, 0) = ߙ(‫)ݖ‬

(2.2)

and the boundary condition ܺ(ܽ, ‫)ݐ(ߚ = )ݐ‬

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(2.3)

The Role of the Method of Characteristics in the Solution of Estimation«

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where ܺሺ‫ݖ‬, ‫ݐ‬ሻ = [ܺ1 ሺ‫ݖ‬, ‫ݐ‬ሻ ‫ ݊ܺ ڮ‬ሺ‫ݖ‬, ‫ݐ‬ሻ]ܶ denotes the vector of state variables. ܺ(‫ݖ‬, ‫א )ݐ‬ ࣢ ݊ [ሺܽ, ܾሻ, Ը݊ ], with ࣢ ݊ ሾሺܽ, ܾሻ, Ը݊ ሿ being the infinite-dimensional Hilbert space of ݊dimensional-like vector functions defined on the interval [ܽ, ܾ], ‫ܽ[ א ݖ‬, ܾ] ‫ ؿ‬Ը and ‫[ א ݐ‬0, ’) denote position and time. ܺ‫ ݐ‬denotes the partial derivative of the state vector with respect to time, while ܺ‫ ݖ‬is the partial derivative with respect to space. In addition, the vectorial function Ȍ ‫ܽ[ ׷‬, ܾ] × [0, ’) × Ը݊ ՜ Ը݊ , and the matrix ĭ ‫ܽ[ ׷‬, ܾ] × [0, ’) × Ը݊ ՜ Ը݊×݊ , are continuous and differentiable. It is also important to state that ĭ is not necessarily an invertible matrix however, to guarantee that system (2.1) is hyperbolic we establish that matrix Ȱ has only real eigenvalues. In addition, ߚ(‫ )ݐ‬is a column vector which is a sufficiently smooth function of time and ߙሺ‫ݖ‬ሻ ‫[ ݊ ࣢ א‬ሺܽ, ܾሻ, Ը݊ ] where ࣢ ݊ [ሺܽ, ܾሻ, Ը݊ ] is the infinite dimensional Hilbert Space of ݊-dimensional-like vector functions defined on the interval [ܽ, ܾ]. In order to analyze the behavior of system (2.1) we first consider a single quasi-linear )23'(¶VZLWKWKHJHQHUDOIRUP ߶ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ‫ ݖݔ‬+ ߮ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ‫ݖ(߰ = ݐݔ‬, ‫ݐ‬, ‫)ݔ‬

(2.4)

෡ ‫ ؿ‬Ը3 , such that where ߶, ߮ and ߰ are continuously differentiable functions of (‫ݖ‬, ‫ݐ‬, ‫ א )ݔ‬π 2 2 ߶ + ߮ ് 0. Although the solution RI )23'(¶V   FDQ EH H[SUHVVHG LPSOLFLWO\ LH Ĭሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ = 0), or explicitly (i.e. ‫݂ = ݔ‬ሺ‫ݖ‬, ‫ݐ‬ሻ), it can be interpreted as a surface in Ը3 as shown in Figure 2.1. In fact, equation (2.4) can be seen as a vectorial product

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ሺ߶

߮

‫ݖݔ‬ ߰ሻ ቆ ‫ ݐݔ‬ቇ = 0 െ1

෡ of the solution surface, vector ܰ = ሺ‫ ݐݔ ݖݔ‬െ1ሻ is its where, for each point (‫ݖ‬, ‫ݐ‬, ‫ א )ݔ‬π normal vector. Hence, the previous equation means that the vector ܸ = ሺ߶ ߮ ߰ሻ is orthogonal to the normal vector and therefore, it is a tangent vector of the solution surface (see Figure 2.1). Without loss of generality, we can state that a solution of equation (2.4) in a domain (‫ݖ‬, ‫ א )ݐ‬π of Ը2 is a function ‫݂ = ݔ‬ሺ‫ݖ‬, ‫ݐ‬ሻ such that the following two conditions are satisfied: i. ii.

෡ of the functions For every (‫ݖ‬, ‫ א )ݐ‬π, the point (‫ݖ‬, ‫ݐ‬, ݂(‫ݖ‬, ‫ ))ݐ‬belongs to the domain π ߶, ߮ and ߰. When ‫݂ = ݔ‬ሺ‫ݖ‬, ‫ݐ‬ሻ is substituted in (2.4), the resulting equation is an identity in ‫ݖ‬, ‫ݐ‬ for all (‫ݖ‬, ‫ א )ݐ‬π.

If we apply the total derivative to solution ‫݂ = ݔ‬ሺ‫ݖ‬, ‫ݐ‬ሻ we get ݀‫ ݖ݀ ݖ݂ = ݔ‬+ ݂‫ݐ݀ ݐ‬ and considering condition (ii) one arrives to

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Efrén Aguilar-Garnica and Juan P. García-Sandoval

)LJXUH6ROXWLRQVXUIDFHRI)23'(¶VDQGnormal and tangent vector for the point P of the solution.

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Therefore comparing both equations it is straightforward that ݀‫߶ = ݖ‬ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ ݀‫߮ = ݐ‬ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ ݀‫߰ = ݔ‬ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ

(2.5a) (2.5b) (2.5c)

Equations (2.5) can be expressed in several forms, for example, if ߶ ് 0, then ݀‫ݐ‬ ݀‫ݖ‬

=

߮ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ ߶ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

and

݀‫ݔ‬ ݀‫ݖ‬

=

߰ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ ߶ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

(2.6)

or in a similar way, if ߮ ് 0 equations (2.5) produce ݀‫ݖ‬ ݀‫ݐ‬

=

߶ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ ߮ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

and

݀‫ݔ‬ ݀‫ݐ‬

=

߰ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ ߮ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

(2.7)

Another usual form to rewrite equations (2.6) and (2.7) is ݀‫ݖ‬ ߶ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

=

݀‫ݐ‬ ߮ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

=

݀‫ݔ‬ ߰ ሺ‫ݖ‬,‫ݐ‬,‫ݔ‬ሻ

(2.8)

The solution of equations (2.6) or equation (2.7) produces the surface solution which is obtained along the axis ‫ ݖ‬or ‫ݐ‬, respectively. It is also possible to obtain a parametric solution ‫ݖ = ݖ‬ሺߦሻ, ‫ )ߦ(ݐ = ݐ‬and ‫ )ߦ(ݔ = ݔ‬with parameter ߦ by rewriting equations (2.5) as Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Role of the Method of Characteristics in the Solution of Estimation« ݀‫ݖ‬ ݀ߦ ݀‫ݐ‬ ݀ߦ ݀‫ݔ‬ ݀ߦ

103

= ߶ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ

(2.9a)

= ߮ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ

(2.9b)

= ߰ሺ‫ݖ‬, ‫ݐ‬, ‫ݔ‬ሻ

(2.9c)

in this case ߦ is called the characteristic direction. To complete the solution of equation (2.4) it is necessary some conditions or initial data. In most of the cases these conditions establish the behavior of ‫ ݔ‬along an initial curve ‫ܥ‬1 as is described in Figure 2.1.

2.1. Dynamic Systems with One Characteristic Line System (2.1) has only one characteristic line when Ȱሺ‫ݖ‬, ‫ݐ‬, ܺሻ = ߶ሺ‫ݖ‬, ‫ݐ‬, ܺሻ‫ ݊ܫ‬holds, where ߶ሺ‫ݖ‬, ‫ݐ‬, ܺሻ is a scalar function, while ‫ ݊ܫ‬is the identity matrix of dimension ݊ (i.e. matrix Ȱ has only one eigenvalue repeated ݊ times). In this case, using the method of characteristics, the solution can be expressed in a similar form than equations (2.7) as ݀‫ݖ‬ ݀‫ݐ‬ ݀ܺ ݀‫ݐ‬

= ߶ሺ‫ݖ‬, ‫ݐ‬, ܺሻ, ‫ݖ‬ሺ‫ݐ‬0 ሻ = ‫ݖ‬0

(2.10a)

= Ȍሺ‫ݖ‬, ‫ݐ‬, ܺሻ, ܺሺ‫ݐ‬0 ሻ = ܺ0

(2.10b)

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However the solution is now a set of surfaces ܺ݅ = ݂݅ (‫ݖ‬, ‫)ݐ‬, for ݅ = 1,2, ǥ , ݊, with the same characteristic lines. Since we have both initial and boundary conditions the initial data must have two domains as follows ܺ0 = ߙ(‫ݖ‬0 ) if ‫ݐ‬0 = 0 ܺ0 = ߚ(‫ݐ‬0 ) if ‫ݖ‬0 = ܽ

(2.11) (2.12)

In some cases system (2.10) can be solved analytically for each ሺ‫ݖ‬0 , ‫ݐ‬0 ሻ as in the first study case. However, it is also possible to apply numerical methods to solve the problem along each characteristic line as it is demonstrated in the second study case. In particular, if the function ߶ does not depends on ܺ, the solution of (2.10a) represents the characteristic line in the plane (‫ݖ‬, ‫)ݐ‬.

2.1.1. Study Case 1: Tubular Heat-Exchanger Let us consider a tubular heat-exchanger with a heating jacket described by ܶ‫ ݐ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ ݖܶݒ‬ሺ‫ݖ‬, ‫ݐ‬ሻ = ‫ ݆ܶ( ܿܪ‬െ ܶሺ‫ݖ‬, ‫ݐ‬ሻ) ܶሺ0, ‫ݐ‬ሻ = ܾܶ (‫)ݐ‬ ܶሺ‫ݖ‬, 0ሻ = ܶ݅ (‫)ݖ‬

(2.13)

here, ܶ denotes the temperature of the fluid flowing through the tube and ݆ܶ denotes the temperature of the surrounding jacket which is supposed to be constant, ‫ ݒ‬is the flow velocity and ‫ ܿܪ‬is a constant related with the heat convection. In addition, the initial condition for ܶ is Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

104

Efrén Aguilar-Garnica and Juan P. García-Sandoval

ܶ݅ (‫)ݖ‬, while the boundary condition ܾܶ (‫ )ݐ‬may be a function of time. For this system we expect two different regions. The first one is related to the initial condition and when the necessary enough fresh fluid has entered to the system to replace the initial mass of the fluid, the behavior at this axial point will be dominated by the boundary condition. If the method of characteristics is used, one can find the solution of the ODE ݀‫ݖ‬ ݀‫ݐ‬ ݀ܶ ݀‫ݐ‬

= ‫ݒ‬, ‫ݖ‬ሺ‫ݐ‬0 ሻ = ‫ݖ‬0 = ‫ ݆ܶ( ܿܪ‬െ ܶ) ,ܶሺ‫ݐ‬0 ሻ = ܶ0

which correspond to equations (2.10) for this particular system. These are ‫ݖ = ݖ‬0 + ‫ ݐ(ݒ‬െ ‫ݐ‬0 ) ܶ = ൫ܶ0 െ ݆ܶ ൯݁ െ‫ ܿܪ‬ሺ‫ݐ‬െ‫ ݐ‬0 ሻ + ݆ܶ As it is expected, this solution is composed by two regions, when ‫ݐ‬0 = 0 the system is dominated by the initial condition, therefore ܶ0 = ܶ݅ ሺ‫ݖ‬0 ሻ, and for each (‫ݖ‬, ‫ )ݐ‬its respective characteristic line starts in ሺ‫ݖ‬0 , ‫ݐ‬0 ሻ = ሺ‫ ݖ‬െ ‫ݐݒ‬, 0ሻ leading to solution ܶሺ‫ݖ‬, ‫ݐ‬ሻ = ൫ܶ݅ ሺ‫ ݖ‬െ ‫ݐݒ‬ሻ െ ݆ܶ ൯݁ െ‫ ݐ ܿܪ‬+ ݆ܶ , for ‫ ݖ‬൒ ‫ݐݒ‬.

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On the other hand, when ‫ ݐݒ < ݖ‬the initial fluid is no longer in ሺ‫ݖ‬, ‫ݐ‬ሻ and the system is dominated by the boundary condition, therefore for each (‫ݖ‬, ‫ )ݐ‬its characteristic lines start in ሺ‫ݖ‬0 , ‫ݐ‬0 ሻ = ሺ0, ‫ ݐ‬െ ‫ݖ‬/‫ݒ‬ሻ and the condition must be ܶ0 = ܾܶ ሺ‫ݐ‬0 ሻ, while the solution becomes ܶሺ‫ݖ‬, ‫ݐ‬ሻ = ൫ܾܶ ሺ‫ ݐ‬െ ‫ݖ‬/‫ݒ‬ሻ െ ݆ܶ ൯݁ െ‫ݖ ܿܪ‬/‫ ݒ‬+ ݆ܶ , for ‫ݐݒ < ݖ‬. The boundary between both regions is given by ‫ݐݒ = ݖ‬, this means that this boundary travels along the space with velocity ‫ ݒ‬and it is possible to have discontinuities in the solution when the initial and boundary conditions are different. Figure 2.2 depicts the solution of the heat exchanger for ‫ = ܿܪ‬1 s-1, ‫ = ݒ‬2 m/s, ݆ܶ = 100ºC, ܶ݅ ሺ‫ݖ‬ሻ = 20 + 10 sin(ߨ‫ )ݐ‬and ܾܶ ሺ‫ݐ‬ሻ = 35 + 5 tanh(30‫ ݐ‬െ 15) and it exhibits a discontinuity precisely at ‫ݐݒ = ݖ‬. The characteristic lines, given by equations ‫ݖ = ݖ‬0 + ‫ ݐ(ݒ‬െ ‫ݐ‬0 ) for the each pair ሺ‫ݖ‬0 , ‫ݐ‬0 ሻ, are also plotted in Figure 2.2, physically this characteristic lines represents the flow lines of the fluid along time and space.

2.1.2. Study Case 2: Non-Isothermal Plug Flow Reactor Let us now consider a non-isothermal plug flow reactor with length ‫ܮ‬, volume ܸܴ , constant inlet stream to the reactor ‫ ݂ݒ‬and lateral feed per unit of volume ‫ ܨ‬where the following first-order reaction in liquid phase is carried out ܴܺ ՜ ܻ. If the volume of the reactor, densities an heat capacities of the involved chemical species are constants, if the heat losses, diffusion and dispersion are negligible, if the lateral flow rate is significantly smaller than the inlet stream (i.e. ‫ ) ݂ݒ ا ܨ‬and under conditions of perfect radial mixing, the

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mathematical model describing the behavior of ܴܺ and the temperature of the reactor ܶ is given by (Christofides and Daoutidis, 1998a)

Figure 2.2. Solution surface for a heat-exchanger. ߲‫ݖ( ܺܥ‬,‫)ݐ‬ ߲‫ݐ‬ ߲ܶ (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬

= െ‫݂ݒ‬

= െ‫݂ݒ‬

߲‫ݖ( ܺܥ‬,‫)ݐ‬

߲ܶ (‫ݖ‬,‫)ݐ‬ ߲‫ݖ‬

߲‫ݖ‬

െ

െ ݇10 ݁

ȟ‫ ݇ ܴܪ‬10 ߩ ‫݌ܥ‬

݁

‫ܣܧ‬ ܴ ܶሺ‫ݖ‬,‫ݐ‬ሻ

െഥ

‫ܧ‬ െഥ ‫ܣ‬ ܴ ܶሺ‫ݖ‬,‫ݐ‬ሻ

‫ ܺܥ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܨ‬ሺ‫ ݅ݔܥ‬െ ‫ݖ( ܺܥ‬, ‫)ݐ‬ሻ

‫ ܺܥ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܨ‬ሺܶ݅ െ ܶ(‫ݖ‬, ‫)ݐ‬ሻ +

ܷ‫ݓ‬ ܸܴ ‫݌ܥ‬

൫ܶ‫ ܬ‬െ ܶ(‫ݖ‬, ‫)ݐ‬൯

‫ ܺܥ‬ሺ‫ݖ‬, 0ሻ = ݃ܺ (‫ܶ ; )ݖ‬ሺ‫ݖ‬, 0ሻ = ݃ܶ (‫)ݖ‬ ‫ ܺܥ‬ሺ0, ‫ݐ‬ሻ = ‫ܶ ; )ݐ( ܺܩ‬ሺ0, ‫ݐ‬ሻ = ‫)ݐ( ܶܩ‬

(2.14) (2.15) (2.16) (2.17)

In the previous equations ‫ݖ( ܺܥ‬, ‫ )ݐ‬and ܶ(‫ݖ‬, ‫ )ݐ‬denote the concentration of ܴܺ and the temperature in the reactor, ݇10 is the pre-exponential factor, ‫ ܣܧ‬is the activation energy, ܴത is the ideal gases constant, ‫ ݅ݔܥ‬and ܶ݅ represent the concentration and the temperature of the lateral inlet stream, ܶ‫ ܬ‬is the wall temperature, ȟ‫ ܴܪ‬is the enthalpy of the reaction, ߩ and ‫݌ܥ‬ denote the density and heat capacity of the reacting liquid and ܷ‫ ݓ‬is the overall heat transfer coefficient. According with the boundary conditions of the model, a solution stream consisting of pure ܴܺ at concentration ‫ )ݐ( ܺܩ‬with temperature ‫ )ݐ( ܶܩ‬enters towards the reactor whose initial conditions are given by ݃ܺ (‫ )ݖ‬and ݃ܶ (‫)ݖ‬. Note that (2.14) and (2.15) exactly match the mathematical model given by (2.1) if ܺሺ‫ݖ‬, ‫ݐ‬ሻ = ሾ‫ݖ( ܺܥ‬, ‫ݖ(ܶ )ݐ‬, ‫)ݐ‬ሿܶ , Ȱሺ‫ݖ‬, ‫ݐ‬, ܺሻ = ‫ܫ ݂ݒ‬2 and

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106

Efrén Aguilar-Garnica and Juan P. García-Sandoval ‫ܧ‬ െത ‫ܣ‬ ܴ ܶሺ‫ݖ‬,‫ݐ‬ሻ ‫ ܥ‬ሺ‫ݖ‬, ‫ݐ‬ሻ ܺ

+ ‫ܨ‬ሺ‫ ݅ݔܥ‬െ ‫ݖ( ܺܥ‬, ‫)ݐ‬ሻ ൪ Ȳ(‫ݖ‬, ‫ݐ‬, ܺ) = ൦ ǻ‫݇ ܴܪ‬10 െ ‫ܣܧ‬ ܷ‫ݓ‬ െ ݁ ܴത ܶሺ‫ݖ‬,‫ݐ‬ሻ ‫ ܺܥ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܨ‬ሺܶ݅ െ ܶ(‫ݖ‬, ‫)ݐ‬ሻ + ൫ܶ‫ ܬ‬െ ܶ(‫ݖ‬, ‫)ݐ‬൯ ߩ‫݌ܥ‬ ܸߩ‫݌ܥ‬ െ݇10 ݁

In addition, (2.16) and (2.17) can be expressed in the form of (2.2) and (2.3) respectively, as follows: ܺሺ‫ݖ‬, 0ሻ = ሾ݃ܺ (‫)ݖ( ܶ݃ )ݖ‬ሿܶ = ߙ(‫)ݖ‬, ܺሺ0, ‫ݐ‬ሻ = ሾ‫)ݐ( ܶܩ )ݐ( ܺܩ‬ሿܶ = ߚ(‫)ݐ‬. Note that this model is more complex than the model of the heat exchanger and therefore is not possible to obtain an analytical solution for (2.10). Thus, it is necessary to generate a solution for such equation by means of numerical methods. According to García-Sandoval et al. (2008) this can be done with a methodology in which for each instant ‫ݐ‬1 , ‫ݐ‬2 , ‫ݐ‬3 , ǥ , ‫ ݐ ݊ݐ‬the solution of equations (2.14)-(2.15) at a certain specific fixed axial point ‫ ܽ = ݖ‬is given by the set of point ܺሺܽ, ‫ݐ‬1 ሻ, ܺሺܽ, ‫ݐ‬2 ሻ, ܺሺܽ, ‫ݐ‬3 ሻ, ǥ , ܺ൫ܽ, ‫ ݐ ݊ݐ‬൯ that are obtained when the differential equation ݀ī ݀‫݌‬

= Ȍሺ‫ݖ‬, ‫ ݌‬+ ‫ ݈ݐ‬െ ߬ሺ‫ ݈ݐ‬ሻ, īሻ

(2.18)

is integrated from ‫݌‬0 = 0 to ‫߬ = ݂݌‬ሺ‫ ݈ݐ‬ሻ where ߬ሺ‫ ݈ݐ‬ሻ = ൜

‫݈ݐ‬ ܽ/‫݂ݒ‬

if ‫ܽ < ݈ݐ‬/‫݂ݒ‬
with the if ‫ ݈ݐ‬൒ ܽ/‫݂ݒ‬

following conditions

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īሺ0ሻ = ߙ(ܽ െ ‫) ݈ݐ ݂ݒ‬ īሺ0ሻ = ߚ(‫ ݈ݐ‬െ ܽ/‫) ݂ݒ‬

݂݅ ݂݅

‫ܽ < ݈ݐ‬/‫݂ݒ‬ ‫ ݈ݐ‬൒ ܽ/‫݂ݒ‬

The solution at ‫ ܽ = ݖ‬for certain instant ‫ = ݈( ݈ݐ‬1,2,3 ǥ ݊‫ ) ݐ‬is ܺ(ܽ, ‫ =) ݈ݐ‬Ȟ൫‫ ݂݌‬൯.This method can also be explained in a graphical form considering the array shown in Figure 2.3. In this figure ‫ ݐ ݊ݐ‬and ‫ ݖ ݉ݖ‬represent the final simulation time and the length of the plug flow reactor (i.e. ‫)ܮ = ݖ ݉ݖ‬, respectively. In addition, each point is a node that represents the behavior of the state variable at ‫ ܽ = ݖ‬and ‫ ݈ݐ = ݐ‬: ܺ(ܽ, ‫) ݈ݐ‬. The collection of triangular nodes denotes the characteristic line that follows a particle of fluid which is initially within the reactor at ‫ݖ‬3 leaving the reaction system at ‫ݐ‬3 . The set of circular nodes corresponds to the FKDUDFWHULVWLFOLQHWUDFHGE\³IUHVKIOXLG´ZKLFKLVWKHYHU\ILUVWSRUWLRQRIWKHIOXLGJRYHUQHG by the boundary condition leaving the reactor at ‫= ݐ‬

‫ݖ ݉ݖ‬ ‫݂ݒ‬

=

‫ܮ‬ ‫݂ݒ‬

. Besides, the square nodes show

the trajectory of a particle that departs at ‫ݐ‬3 and whose behavior is influenced by the boundary condition. Furthermore, the dotted line denotes the initial condition of the fluid whereas the dashed line represents its boundary condition. For instance, if it is desired to know the value of ܺ(ܽ = ‫ܮ = ݖ ݉ݖ‬, ‫ݐ‬3 ) one needs to integrate equation (2.18) from 0 to ‫ݐ‬3 (since ‫ݐ‬3 < ‫ܮ‬/‫) ݂ݒ‬ with the initial condition Ȟሺ0ሻ = ߙ(‫ ܮ‬െ ‫ݐ ݂ݒ‬3 ) and then ܺ(‫ܮ‬, ‫ݐ‬3 )= Ȟሺ‫ݐ‬3 ሻ. In this case ܺ(‫ܮ‬, ‫ݐ‬3 ) is clearly influenced by the initial condition of the infinite dimensional model. On the other hand, if one is interested in the behavior of the state variables when ‫ ݐ‬൒ ‫ܮ‬/‫ ݂ݒ‬is necessary to integrate along the characteristic line (i.e. integrate (2.18)) with an initial condition influenced by the boundary condition.The aforementioned methodology was applied in simulations for an arbitrary number of 40 divisions along the ‫ݖ‬-axis (i.e. 41 positions along the ‫ݖ‬-axis including ‫ݖ‬1 = 0 and ‫ݖ‬41 = ‫ )ܮ‬and an equal number of divisions along the ‫ݐ‬±axis (i.e.

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݊‫ ) ݖ݉ = ݐ‬. These simulations are presented in Figure 2.4 and Figure 2.5 and they were carried out with MATLAB by using the command ode45 (Runge Kutta 4th order, five steps) considering that ‫ = ܮ‬10݉, ‫ = ܨ‬0.1݉3 ݄െ1 , ‫ = ݂ݒ‬2݄݉െ1 (note that it can be considered that ‫ ) ݂ݒ ا ܨ‬along with the boundary conditions, the initial conditions and the parameters obtained by Christofides and Daoutidis (1998a) that are shown in Table 2.1.

Figure 2.3. Graphical scheme considered in the simulation of the infinite dimensional mathematical models.

Table 2.1. Parameters, boundary conditions and initial conditions for the isothermal plug flow reactor system.

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Parameter ݇10 ‫ܣܧ‬ ܴത ‫ܮ‬ ‫݅ݔܥ‬ ܶ݅ ܶ‫ܬ‬ ǻ‫ܴܪ‬ ߩ ‫݌ܥ‬ ܷ‫ݓ‬ ܸܴ Initial condition ݃ܺ ሺ‫ݖ‬ሻ ݃ܶ (‫)ݖ‬ Boundary condition ‫)ݐ( ܺܩ‬ ‫)ݐ( ܶܩ‬

Units h-1 kcalkmol-1 kcalkmol-1K-1 m kmolm-3 K K kcalkmol-1 kgm-3 kcalkg-1K-1 kcalh-1K-1 m3 Units kmolm-3 kmolm-3 Units kmolm-3 kmolm-3

Value 5x1012 2x104 1.987 10 0.1 300 310 35480 90 0.0231 500 1 Function 0.5 300 Function ‫ݐ‬ 0.5 sin ൬ + 2൰ + 0.6 2 0.1625‫ ݐ‬+ 300

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Efrén Aguilar-Garnica and Juan P. García-Sandoval

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Figure 2.4. Dynamical and axial behavior of ܶ(‫ݖ‬, ‫ )ݐ‬within the nonisothermal plug flow reactor considering 41 positions along the z-axis.

Figure 2.5. Dynamical and axial behavior of ‫ݖ( ܺܥ‬, ‫ )ݐ‬within the nonisothermal plug flow reactor considering 41 positions along the z-axis.

2.2. Dynamic Systems with Multiple Characteristic Lines The dynamic system (2.1) has multiple characteristic lines when matrix ĭ has at least two different eigenvalues. In fact, the number of characteristic lines associated to the system is equal to the number of non equal eigenvalues of matrix ĭ, not matter its multiplicity. Each equation of system (2.1) can be written as ࣦ݅ =

߲ܺ ݅ ߲‫ݐ‬

+ σ݆݊=1 ߶݆݅ (‫ݖ‬, ‫ݐ‬, ܺ)

߲ܺ ݆ ߲‫ݖ‬

െ ߰݅ ሺ‫ݖ‬, ‫ݐ‬, ܺሻ = 0

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(2.19)

The Role of the Method of Characteristics in the Solution of Estimation«

109

where ߶݆݅ is the element (݅, ݆) of matrix ĭ, while ߰݅ is the element ݅ of vector Ȍ. Now, let us consider the linear combination: ࣦ = σ݊݅=1 ߣ݅ ࣦ݅ = 0

(2.20)

The derivatives of the dependent variables in ࣦ = 0 are in the same direction ߪ = σ݊݅=1

ߣ ݅ ߶ ݆݅ (‫ݖ‬,‫ݐ‬,ܺ) ߣ݆

݀‫ݖ‬ ݀‫ݐ‬

for ݆ = 1,2, ǥ , ݊

= ߪ if: (2.21)

Suppose that at a point (‫ݐ‬, ‫ݖ‬, ܺ) on the solution surface ߣ݅ , ݅ = 1,2, ǥ , ݊, are chosen to satisfy these equations; then: σ݊݅=1(߶݆݅ െ ߜ݆݅ ߪ)ߣ݅ = 0

(2.22)

where ߜ݆݅ = 0 for ݅ ് ݆ and ߜ݅݅ = 1. This set of equations has nontrivial solution ሺߣ݅ , ݅ = 1,2, ǥ , ݊ሻ if:

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(߶11 െ ߪ) ߶12 ‫ڮ‬ ߶21 (߶22 െ ߪ) ‫ڰ‬ ተ ‫ڭ‬ ‫ڰ‬ ‫ڰ‬ ߶݊1 ߶݊2 ‫ڮ‬

߶1݊ ߶2݊ ተ=0 ‫ڭ‬ (߶݊݊ െ ߪ)

(2.23)

This may be written as a ݊-order polynomial whose roots ߪ݆ , ݆ = 1,2, ǥ , ݉ are precisely the eigenvalues of matrix ĭ; notice that the number of roots is ݉ ൑ ݊, since some roots may be repeated. If, for a moment, the solution surfaces ݆ܺ ሺ‫ݐ‬, ‫ݖ‬ሻ, ݆ = 1,2, ǥ , ݊, are assumed as known then, the families of curves ǻ = ሼǻ1 , ǥ , ǻ݉ ሽ could be drawn in the ‫ ݖ‬െ ‫ ݐ‬plane satisfying at each point: ǻ݆ :

݀‫ݖ‬ ݀‫ݐ‬

= ߪ݆ , for ݆ = 1,2, ǥ , ݉. The name of these curves is

characteristic curves or simply characteristics (see Figure 2.6). Furthermore, if each one of these curves are represented by the following characteristic parameters ߦ݆ , ݆ = 1,2, ǥ , ݉, then the former equations can be rewritten as: ߲‫ݖ‬ ߲ߦ ݆

where

െ ߪ݆ ߲ ߲ߦ ݆

߲‫ݐ‬ ߲ߦ ݆

= 0, for ݆ = 1,2, ǥ , ݉

(2.24)

, denotes differentiation with respect to ߦ݆ , along the characteristic ǻ݆ . In general,

we have: ߲ ߲ߦ ݆

=

߲‫ݐ‬ ߲ߦ ݆

߲

߲

߲‫ݐ‬

߲‫ݖ‬

ቀ + ߪ݆

ቁ=

߲‫ݖ‬ ߲ߦ ݆

൬

1 ߲

ߪ ݆ ߲‫ݐ‬

+

߲ ߲‫ݖ‬

൰, for ݆ = 1,2, ǥ , ݉

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(2.25)

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Efrén Aguilar-Garnica and Juan P. García-Sandoval

Figure 2.6. Characteristic curves intersection at point P for systems of quasi-linear hyperbolic )23'(¶V.

Now, if ߪ = ߪ݇ in (2.22) and these equations are multiplied by σ݊݅=1 ߣ݅ (߶݆݅ െ ߜ݆݅ ߪ݇ )

߲‫ݐ‬ ߲ߦ ݇

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߲ܺ ݅ ߲‫ݐ‬

+ σ݆݊=1 ߣ݆ ߶݆݅

:

= 0, for ݆, ݇ = 1,2, ǥ , ݉

Similarly, multiplying equation (2.20) by σ݊݅=1 ቀߣ݅

߲‫ݐ‬ ߲ߦ ݇

߲ܺ ݅ ߲‫ݖ‬

ቁ

߲‫ݐ‬ ߲ߦ ݇

߲‫ݐ‬ ߲ߦ ݇

(2.26)

gives:

െ σ݊݅=1 ߣ݅ ߰݅

߲‫ݐ‬ ߲ߦ ݇

= 0, for ݇ = 1,2, ǥ , ݉

and using equation (2.21) with ߪ = ߪ݇ to eliminate the terms σ݆݊=1 ߣ݆ ߶݆݅ yields σ݊݅=1 ߣ݅ ቂቀ

߲ܺ ݅ ߲‫ݐ‬

+ ߪ݇

߲ܺ ݅ ߲‫ݖ‬

ቁ െ ߰݅ ቃ

߲‫ݐ‬ ߲ߦ ݇

= 0, for ݇ = 1,2, ǥ , ݉

Equation (2.25) can now be used to express the derivatives with respect to ߦ݇ ; then σ݊݅=1 ߣ݅ ቀ

߲ܺ ݅ ߲ߦ ݇

െ ߰݅

߲‫ݐ‬ ߲ߦ ݇

ቁ = 0, for ݇ = 1,2, ǥ , ݉

If we had multiplied equation (2.20) by

߲‫ݖ‬ ߲ߦ ݇

and eliminated the terms σ݆݊=1 ߣ݆ ߶݆݅ by using

(2.21), we would have had the equation σ݊݅=1 ߣ݅ ቀ

߲ܺ ݅ ߲ߦ ݇

െ ߰݅

߲‫ݖ‬ ߲ߦ ݇

(2.27)

ቁ = 0, for ݇ = 1,2, ǥ , ݉

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The Role of the Method of Characteristics in the Solution of Estimation«

111

which can be used as well instead (2.27). Considering now (2.27) along with (2.26) as a set of equations with the form ‫ ݇ܯ‬ȁ = 0, for ݇ = 1,2, ǥ , ݉

(2.28)

to be solved for ȁ = (ߣ1 , ǥ , ߣ݉ ), where

‫ۇ‬ ‫ۈ‬ ‫ۈ = ݇ܯ‬ ‫ۈ‬ ‫ۈ‬

(߶11 െ ߪ݇ )

߲‫ݐ‬ ߲ߦ݇

‫ڭ‬

߲‫ݐ‬ ߲ߦ݇ ߲ܺ1 ߲‫ݐ‬ െ ߰1 ‫݇ߦ߲ ۉ‬ ߲ߦ݇ ߶݉ ,1

‫ڮ‬

߶1,݉ െ1

‫ڰ‬ ‫ڮ‬ ‫ڮ‬

߲‫ݐ‬ ߲ߦ݇

‫ڭ‬ ߶݉ ,݉ െ1

߶1,݉

߲‫ݐ‬ ߲ߦ݇

‫ۊ‬ ‫ۋ‬ ߲‫ۋ ݐ‬ (߶݉ ,݉ െ ߪ݇ ) ߲ߦ݇ ‫ۋ‬ ‫ۋ‬ ߲ܺ݉ ߲‫ݐ‬ െ ߰݉ ߲ߦ݇ ߲ߦ݇ ‫ی‬ ‫ڭ‬

߲‫ݐ‬ ߲ߦ݇

߲ܺ݉ െ1 ߲‫ݐ‬ െ ߰݉ െ1 ߲ߦ݇ ߲ߦ݇

‫ ݇ܯ‬is a (݉ + 1) × ݉ matrix, however one row is linearly dependent and can be eliminated to ෡݇ ȁ = 0, which has not trivial obtain a ݉ × ݉ homogeneous algebraic system with the form ‫ܯ‬ ෡݇ vanishes. Thus solution if the determinant of ‫ܯ‬ ෡݇ ห = 0, for ݇ = 1,2, ǥ , ݉ ห‫ܯ‬

(2.29)

෡݇ ห, the set of equations (2.29) has the After the computation of the determinants ห‫ܯ‬ structure

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σ݊݅=1 ‫݇݅ܮ‬

߲ܺ ݅ ߲ߦ ݇

+ ܰ݇

߲‫ݐ‬ ߲ߦ ݇

= 0, for ݇ = 1,2, ǥ , ݉

(2.30)

෡݇ ห. where ‫ ݇݅ܮ‬and ܰ݇ are functions of ߶ , ߪ and ߰ that are obtained when one computes ห‫ܯ‬ Equation (2.30) together with equation (2.24) are called the characteristic differential equations. They will be satisfied on the solution surfaces ݆ܺ ሺ‫ݐ‬, ‫ݖ‬ሻ, ݆ = 1,2, ǥ , ݊. The solution of the characteristic differential equations (2.30) and (2.24) are usually obtained by means of discrete methods defining nodes where the characteristic curves converge.

2.2.1. Study Case 3: Solid-Waste Anaerobic Digestion Now, let us consider a third study case related with the solid-waste anaerobic digestion process carried out within a bioreactor with length ‫ܮ‬. This process has been studied by Vavilin et al. (2003a) under the following assumptions. First, the polymer hydrolysis/ acidogenesis and acetogenesis/methanogenesis are two possible rate-limiting steps of the overall process. Second, these steps are being inhibited by an intermediate product and finally, all transformation reactions involved in the conversion of Volatile Fatty Acids 9)$¶V WRPHWKDQHFDQEHJDWKHUHGWRJHWKHUDVDVLQJOHVWHS8QGHUWKHVHFRQVLGHUDWLRQVWKH\ have proposed and validated a mathematical model involving five state variables: the solid waste FRQFHQWUDWLRQWKH9)$¶VFRQFHQWUDWLRQWKHPHWKDQRJHQLFELRPDVV concentration, the methane concentration and the sodium concentration. However, it is straightforward to verify

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112

Efrén Aguilar-Garnica and Juan P. García-Sandoval

that the equations related to methane and sodium concentrations are decoupled from the rest. Besides, one can compute the Peclet number with the reported parameters obtaining relatively high values. As a consequence, the mathematical model proposed by et al. (2003a) can be studied as a set of equations describing the behavior of three state variables neglecting diffusive phenomena with the following structure ߲ܹ ሺ‫ݖ‬,‫ݐ‬ሻ ߲‫ݐ‬ ߲ܵ (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬ ߲‫ݖ( ܤ‬,‫)ݐ‬ ߲‫ݐ‬

= െ݇1 ܹሺ‫ݖ‬, ‫ݐ‬ሻ‫ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯

= െ‫ݍ‬

߲ܵ (‫ݖ‬,‫)ݐ‬

= െ‫ߛݍ‬

+ ߯݇1 ܹሺ‫ݖ‬, ‫ݐ‬ሻ‫ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯ െ ߩ݉ ݃൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯

߲‫ݖ‬ ߲‫ݖ( ܤ‬,‫)ݐ‬ ߲‫ݖ‬

(2.31)

+ ܻ‫݃ ݉ߩ ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯

ܵሺ‫ݖ‬,‫ݐ‬ሻ‫ݖ(ܤ‬,‫)ݐ‬ ‫ݏܭ‬+ܵ(‫ݖ‬,‫)ݐ‬

ܵሺ‫ݖ‬,‫ݐ‬ሻ‫ݖ(ܤ‬,‫)ݐ‬

(2.32)

‫ݏܭ‬+ܵ(‫ݖ‬,‫)ݐ‬

െ ݇݀ ‫ݖ(ܤ‬, ‫)ݐ‬

(2.33)

with the following conditions ܹሺ‫ݖ‬, 0ሻ = ߙܹ (‫ ; )ݖ‬ሺ‫ݖ‬, 0ሻ = ߙܵ (‫ܤ ; )ݖ‬ሺ‫ݖ‬, 0ሻ = ߙ‫)ݖ( ܤ‬ ܹሺ0, ‫ݐ‬ሻ = ߚܹ (‫ ; )ݐ‬ሺ0, ‫ݐ‬ሻ = ߚܵ (‫ܤ ; )ݐ‬ሺ0, ‫ݐ‬ሻ = ߚ‫)ݐ( ܤ‬

(2.34) (2.35)

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where ‫ݖ(ܤ‬, ‫)ݐ‬, ܵ(‫ݖ‬, ‫ )ݐ‬and ܹ(‫ݖ‬, ‫ )ݐ‬denote the methanogenic biomass concentration, the 9)$¶V FRQFHQWUDWLRQ DQG WKH VROLG ZDVWH concentration, respectively. In addition, ‫ ݍ‬is the volumetric liquid flow rate per unit surface area, ߛ is the fraction of biomass transferred by liquid flow, ݇1 is the first-order hydrolysis rate constant, ݇݀ is the specific biomass decay coefficient, ܻ‫ ݎ‬is the biomass yield coefficient whereas ߯ is a stoichiometric coefficient. Furthermore, ߩ݉ LV WKH PD[LPXP VSHFLILF UDWH RI 9)$¶V XWLOL]DWLRQ DQG ‫ ݏܭ‬is the halfsaturation FRQVWDQW IRU 9)$¶V XWLOL]DWLRQ %HVLGHV ‫ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯ and ݃(ܵ(‫ݖ‬, ‫ ))ݐ‬are dimensionless non-linear functions describing non-LRQL]HG 9)$¶V LQKLELWLRQ of polymer hydrolysis/acidogenesis and acetoghenesis/methanogenesis, respectively. These functions have the following structure ‫ݎ‬൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯ = ቀ(1 + ቀ ቀ(1 + ቀ

െ1 ܵ(‫ݖ‬,‫݃ ݉ )ݐ‬ ‫݃ܭ‬

ቁ

ቁ

െ1 ܵ(‫ݖ‬,‫݂ ݉ )ݐ‬ ‫݂ܭ‬

ቁ

ቁ

and ݃൫ܵሺ‫ݖ‬, ‫ݐ‬ሻ൯ =

where ݂݉ , ݉݃ , ‫ ݂ܭ‬and ‫ ݃ܭ‬are inhibition degree indexes and inhibition

constants, respectively. At this point it is important to remark that in some cases, sodium can be considered as inhibitor. Nevertheless, in this work we assume that the alkalinity and 9)$¶VSURGXFWLRQDUHLQEDODQFHLHWKHSURFHVVLVVWDEOH DQGWKHQVRGLXPDVLQKLELWRUFDQ be excluded from the model (2.27)-(2.29) (Björnsson et al. 2001). Model (2.31)-(2.33) has the structure of dynamic system (2.1) with 0 ܺ = (ܹ, ܵ, ‫)ܤ‬, ĭ = ൭0 0

0 ‫ݍ‬ 0

െ݇1 ܹ‫ݎ‬ሺܵሻ 0 ܵ‫ܤ‬ 0 ൱ and Ȍ = ‫݇߯ۇ‬1 ܹ‫ݎ‬ሺܵሻ െ ߩ݉ ݃ሺܵሻ ‫ݏܭ‬+ܵ‫ۊ‬ ܵ‫ܤ‬ ‫ߛݍ‬ ‫݃ ݉ߩ ݎܻ ۉ‬ሺܵሻ ‫ݏܭ‬+ܵ(‫ݖ‬,‫ )ݐ‬െ ݇݀ ‫ی ܤ‬

Therefore, according to (2.23) the characteristic polynomial is െߪ อ0 0

0 ‫ݍ‬െߪ 0

0 0 อ = െߪሺ‫ ݍ‬െ ߪሻሺ‫ ߛݍ‬െ ߪሻ ‫ ߛݍ‬െ ߪ

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113

Note that the roots of this equation are given by ߪ1 = 0, ߪ2 = ‫ ݍ‬and ߪ3 = ‫ߛݍ‬. This means that there exist three characteristic trajectories and the characteristic differential equations can be obtained by replacing the values of ߪ1 , ߪ2 and ߪ3 ĭ and Ȍ in the linear system (2.28) with

‫ۇ‬ ‫ۈ‬ ‫ۈ‬ ‫ۈ = ݇ܯ‬ ‫ۈ‬ ‫ۈ‬

െߪ݇

߲‫ݐ‬ ߲ߦ݇

0

0

0

߲‫ݐ‬ (‫ ݍ‬െ ߪ݇ ) ߲ߦ݇

0

0

߲ܹ ߲‫ݐ‬ െ ߰1 ߲ߦ݇ ‫݇ߦ߲ۉ‬

߲ܵ ߲‫ݐ‬ െ ߰2 ߲ߦ݇ ߲ߦ݇

‫ۊ‬ ‫ۋ‬ 0 ‫ۋ‬ ߲‫ۋ ݐ‬ (‫ ߛݍ‬െ ߪ݇ ) ‫ۋ‬ ߲ߦ݇ ‫ۋ‬ ߲‫ܤ‬ ߲‫ݐ‬ െ ߰3 ߲ߦ݇ ߲ߦ݇ ‫ی‬

Notice that the first row of ‫ܯ‬1 , the second row of ‫ܯ‬2 and the third row of ‫ܯ‬3 are equal to ෡݇ are zero when ߪ1 , ߪ2 and ߪ3 are respectively replaced. Then matrices ‫ܯ‬ ‫ݍ‬

0 ‫ۇ‬ ෡ ‫ܯ‬1 = ‫ۈ‬ ‫ۈ‬ ‫ۈ‬

0

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߲‫ݐ‬ ‫ߛݍ‬ ߲ߦ1

‫ۊ‬ ‫ۋ‬ ‫ۋ‬ ‫ۋ‬

߲ܵ ߲‫ݐ‬ െ ߰2 ߲ߦ1 ߲ߦ1

߲‫ܤ‬ ߲‫ݐ‬ െ ߰3 ߲ߦ1 ߲ߦ1 ‫ی‬

߲‫ݐ‬ ߲ߦ2

0

0

0

0

െ‫ݍ‬

߲ܹ ߲‫ݐ‬ െ ߰1 ‫ߦ߲ ۉ‬2 ߲ߦ2 െ‫ߛݍ‬

‫ۇ‬ ෡3 = ‫ۈ‬ ‫ܯ‬ ‫ۈ‬ ‫ۈ‬

0

0

߲ܹ ߲‫ݐ‬ െ ߰1 ‫ߦ߲ ۉ‬1 ߲ߦ1

‫ۇ‬ ෡2 = ‫ۈ‬ ‫ܯ‬ ‫ۈ‬ ‫ۈ‬

߲‫ݐ‬ ߲ߦ1

߲‫ݐ‬ ߲ߦ3

0

߲ܹ ߲‫ݐ‬ െ ߰1 ‫ߦ߲ ۉ‬3 ߲ߦ3

߲ܵ ߲‫ݐ‬ െ ߰2 ߲ߦ2 ߲ߦ2

‫ۊ‬ ߲‫ۋ ݐ‬ (ߛ െ 1)‫ݍ‬ ߲ߦ2 ‫ۋ‬ ‫ۋ‬ ߲‫ܤ‬ ߲‫ݐ‬ െ ߰3 ߲ߦ2 ߲ߦ2 ‫ی‬

0 ߲‫ݐ‬ ߲ߦ3 ߲ܵ ߲‫ݐ‬ െ ߰2 ߲ߦ3 ߲ߦ3

0

(1 െ ߛ)‫ݍ‬

0

‫ۊ‬ ‫ۋ‬ ‫ۋ‬ ‫ۋ‬

߲‫ܤ‬ ߲‫ݐ‬ െ ߰3 ߲ߦ3 ߲ߦ3 ‫ی‬

whose respective determinants are ෡1 ห = ห‫ܯ‬ ෡2 ห = ห‫ܯ‬ ෡3 ห = ห‫ܯ‬

߲ܹ ߲‫ݐ‬ + ݇1 ܹ‫ݎ‬ሺܵሻ =0 ߲ߦ1 ߲ߦ1 ߲ܵ ߲ߦ2

െ ቀ߯݇1 ܹ‫ݎ‬ሺܵሻ െ ߩ݉ ݃ሺܵሻ

ܵ‫ܤ‬

ቁ

߲‫ݐ‬

‫ݏܭ‬+ܵ ߲ߦ2

=0

߲‫ܤ‬ ߲‫ݐ‬ ܵ‫ܤ‬ െ ൬ܻ‫݃ ݉ߩ ݎ‬ሺܵሻ െ ݇݀ ‫ܤ‬൰ =0 ߲ߦ3 ߲ߦ3 ‫ ݏܭ‬+ ܵ(‫ݖ‬, ‫)ݐ‬

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(2.36)

114

Efrén Aguilar-Garnica and Juan P. García-Sandoval

Equations (2.36) are particular cases of (2.30). On the other hand, for this system equations (2.24) becomes ߲‫ݖ‬ =0 ߲ߦ1 ߲‫ݖ‬ ߲ߦ2

െ‫ݍ‬

߲‫ݖ‬ ߲ߦ3

߲‫ݐ‬ ߲ߦ2

െ ߛ‫ݍ‬

=0 ߲‫ݐ‬

߲ߦ3

(2.37)

=0

It is important to note that in equations (2.36) each state variable is associated with a different characteristic curve. According to the MC, the state variables described by equations (2.31)-(2.33) travel along three characteristic lines, which are the solution of (2.37) for given initial conditions, i.e.

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Characteristic 1 (related to ܹ) ǻ1 : ‫ݖ‬1 െ ‫ݖ‬1,0 = ߦ1 െ ߦ1,0 Characteristic 2 (related to ܵ) ǻ2 : ‫ݖ‬2 െ ‫ݖ‬2,0 = ‫ݐ(ݍ‬2 െ ‫ݐ‬2,0 ) Characteristic 3 (related to ‫ )ܤ‬ǻ3 : ‫ݖ‬3 െ ‫ݖ‬3,0 = ߛ‫ݐ(ݍ‬3 െ ‫ݐ‬3,0 )

(2.38a) (2.38b) (2.38c)

It is not possible to simulate this model with the method described by García-Sandoval et al. (2008) since it can be used if there exists only one characteristic line. For this particular case, it is proposed a simulation method inspired in Figure 2.7 where the axial position varies along the x-axis whereas the y-axis represents time variations. In this diagram, the characteristic lines depicting the solid waste concentration are the horizontal dashed-dot lines, WKHFKDUDFWHULVWLFOLQHVRIWKH9)$¶VFRQFHQWUDWLRQDUHWKHGRWWHGOLQHVZLWKDVORSHHTXDOWR‫ݍ‬ (dotted lines) and the biomass concentration can be represented by the characteristic continuous lines with a slope equal to ߛ‫ݍ‬. The grid generated in Figure 2.7 is a combination of triangles with the structure depicted in Figure 2.8. In this figure: ‫( ݅݌‬i=1...7) represent certain ‫ ݖ‬െ ‫ ݐ‬positions. For instance, the characteristic line for ܹ(‫ݖ‬, ‫ )ݐ‬is fully vertical according to (2.38a) and, as it is described by (2.38b) and (2.38c), the characteristic lines for ܵ(‫ݖ‬, ‫ )ݐ‬and ‫ݖ(ܤ‬, ‫ )ݐ‬have slopes given by ‫ ݍ‬and ߛ‫ݍ‬, respectively. Furthermore, in Figure 2.8 the symbol ‫݌‬4 represent the point where these three lines converge whereas ‫݌‬1 , ‫݌‬2 and ‫݌‬3 denote the ³GHSDUWLQJ SRLQWV´ IRU ܹ(‫ݖ‬, ‫)ݐ‬, ܵ(‫ݖ‬, ‫ )ݐ‬and ‫ݖ(ܤ‬, ‫)ݐ‬, respectively. Furthermore, replacing these points in (2.37): Characteristic 1 (related to ܹ) ǻ1 : ‫݌ݖ‬4 = ‫݌ݖ‬1 Characteristic 2 (related to ܵ) ǻ2 : ‫ ݌ݖ‬4 = ‫݌ݖ‬3 + ‫ݍ‬൫‫ ݌ݐ‬4 െ ‫݌ݐ‬3 ൯ Characteristic 3 (related to ‫ )ܤ‬ǻ3 : ‫݌ݖ‬4 = ‫݌ݖ‬2 + ‫ߛݍ‬൫‫݌ݐ‬4 െ ‫݌ݐ‬2 ൯ In addition, consider that

߲‫ݖ‬ ߲ߦ ݅

݀ߦ݅ = ‫ ݌ݖ‬4 െ ‫ ݅ ݌ݖ‬.For ݅ = 1,2,3, and

߲‫ݐ‬ ݀ߦ = ‫݌ݐ‬4 െ ‫݅ ݌ݐ‬ ߲ߦ݅ ݅ ߲ܹ (‫ݖ‬,‫)ݐ‬ ߲ߦ1

݀ߦ1 = ܹ‫݌‬4 െ ܹ‫݌‬1 ;

߲ܵ (‫ݖ‬,‫)ݐ‬ ߲ߦ2

݀ߦ2 = ܵ‫݌‬4 െ ܵ‫݌‬3 ;

߲‫ݖ( ܤ‬,‫)ݐ‬ ߲ߦ3

݀ߦ3 = ‫݌ܤ‬4 െ ‫݌ܤ‬2

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The Role of the Method of Characteristics in the Solution of Estimation«

115

where ݀ߦ1 , ݀ߦ2 and ݀ߦ3 represent differential variations on ߦ1 , ߦ2 and ߦ3 , respectively. Finally, if equations (2.36) are multiplied by these differential variations and regarding the former equations one arrives to ܹ‫݌‬4 = ܹ‫݌‬1 െ ݇1 ܹ‫݌‬1 ‫ݎ‬൫ܵ‫݌‬1 ൯(‫݌ݐ‬4 െ ‫݌ݐ‬1 ) ܵ‫݌‬4 = ܵ‫ ݌‬3 + ൤߯݇1 ܹ‫ ݌‬3 ‫ݎ‬൫ܵ‫ ݌‬3 ൯ െ ߩ݉ ݃൫ܵ‫݌‬3 ൯ ‫݌ܤ‬4 = ‫݌ܤ‬2 + ൤ܻ‫݃ ݉ߩ ݎ‬൫ܵ‫݌‬2 ൯

ܵ‫ ݌‬2 ‫ ݌ܤ‬2 ‫ݏܭ‬+ܵ‫ ݌‬2

(2.39) ܵ‫ ݌‬3 ‫ ݌ܤ‬3 ‫ݏܭ‬+ܵ‫ ݌‬3

൨ (‫݌ݐ‬4 െ ‫݌ݐ‬3 )

െ ݇݀ ‫݌ܤ‬2 ൨ (‫݌ݐ‬4 െ ‫݌ݐ‬2 )

(2.40) (2.41)

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here the subscripts ‫݌‬1 , ‫݌‬2 , ‫݌‬3 and ‫݌‬4 in both the state variables and time, are considered to represent their values at such ‫ ݖ‬െ ‫ ݐ‬positions. Note that ܹ‫݌‬4 is computed with (2.39) from values at ‫݌‬1 (i.e. to the left of ‫݌‬4 along the characteristic line for the solid waste), ܵ‫݌‬4 is obtained with (2.40) from information at ‫݌‬3 (i.e. over ‫݌‬4 along the characteristic line for the substrate concentration) whereas ‫݌ܤ‬4 is generated with (2.41) from values along the characteristic line for the biomass concentration that are located before ‫݌‬4 . Once these variables are obtained, WKH\FDQEHFRQVLGHUHGDV³starting points´WRFRPSXWHWKHLUQXPHULFDO values at ‫݌‬5 , ‫݌‬6 and ‫݌‬7 .The results of this procedure are shown in Figure 2.9 for a period of 160d using 46 positions along the ‫ݖ‬-axis and were obtained with the parameters depicted in Vavilin et al. (2003a, 2003b) and the boundary and initial conditions that are shown in Table 2.2.

Figure 2.7. Scheme of the characteristic lines involved in the solid-waste anerobic digestion process. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

116

Efrén Aguilar-Garnica and Juan P. García-Sandoval

Figure 2.8. Graphical iteration scheme required to simulate the solid waste anaerobic digestion process.

Table 2.2. Parameters, boundary conditions and initial conditions for the solid waste anaerobic digestion process.

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Parameter ‫ݍ‬ ߛ ݇݀ ݇1 ߯ ܻ‫ݎ‬ ߩ݉ ‫ݏܭ‬ ݂݉ = ݉݃ ‫݂ܭ‬ ‫݃ܭ‬ ‫ܮ‬ Initial condition ߙܹ (‫)ݖ‬ ߙܵ (‫)ݖ‬ ߙ‫)ݖ( ܤ‬ Boundary condition ߚܹ (‫)ݐ‬ ߚܵ (‫)ݐ‬ ߚ‫)ݐ( ܤ‬

Units md-1 d-1 d-1 d-1 gL-1 gL-1 gL-1 m Units gL-1 gL-1 gL-1 Units gL-1 gL-1 gL-1

Value 1 0.1 0.001 0.011 0.48 0.12 0.31 1.2 3 16 10 10 Function 250 15(1 ± 0.08) 4(1 + 0.05z) Function 250 14 3

At this point it is important to say that the boundary and initial conditions were arbitrarily selected. Nevertheless, they can be changed without modifying the simulation methodology which is described in this paper. In order to understand the behaviour of biomass and substrate concentrations depicted in Figure 2.9, it is necessary to observe again Figure 2.7. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

117

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The Role of the Method of Characteristics in the Solution of Estimation«

.

Figure 2.9. Dynamical and axial behavior of ܹ(‫ݖ‬, ‫)ݐ‬, ܵ(‫ݖ‬, ‫ )ݐ‬and ‫ݖ(ܤ‬, ‫ )ݐ‬in the solid waste anaerobic digestion process.

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118

Efrén Aguilar-Garnica and Juan P. García-Sandoval

In this Figure, the continuous line that departs from ‫ = ݐ‬0 and ‫ = ݖ‬0 represent the very first portion of biomass JRYHUQHGE\WKHERXQGDU\FRQGLWLRQLH³IUHVKIOXLG´RIELRPDVV OHDYLQJ the reactor at ‫ = ߬ = ݐ‬100݀. The continuous lines below this one are influenced by the initial condition whereas the continuous lines departing from ‫ = ݖ‬0 and ‫ > ݐ‬0 are governed by the boundary condition. As a consequence, one expects a change in the dynamical behaviour of the biomass concentration just at ‫߬ = ݐ‬. This change is clearly observed in Figure 2.9. In what concerns the substrate concentration one also expects a variation but at ‫ = כ ߬ = ݐ‬10݀ since WKLVLVWKHWLPHDWZKLFKWKH³IUHVKIOXLG´RIVXEVWUDWHOHDYHVWKHUHDFWRU6XFKYDULDWLRQLVDOVR observed in Figure 2.9.

3. THE ROLE OF THE MC IN THE SOLUTION OF ESTIMATION PROBLEMS 3.1. General Dynamical Model Let us consider certain process carried out within a plug flow reactor whose mathematical model LVJLYHQE\WKHIROORZLQJVHWRIWKH)23'(¶V ߲ܺ (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬

= െ‫ܯݒ‬

߲ܺ (‫ݖ‬,‫)ݐ‬ ߲‫ݖ‬

+ ‫݂ܭ‬ሺܺሻ + ‫ܺܥ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫)ݐ(ݑ‬

(3.1)

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which is a particular case of (2.1). Furthermore, this model has the following initial and boundary conditions: ܺሺ‫ݖ‬, 0ሻ = ߙ(‫)ݖ‬ ܺሺ0, ‫ݐ‬ሻ = ߚ(‫)ݐ‬

(3.2) (3.3)

In the previous equations, ܺሺ‫ݖ‬, ‫ݐ‬ሻ ‫ א‬Ը݊ denotes the state vector, ݂ሺܺሻ ‫ א‬Ը‫ ݎ‬represents the nonlinearities vector, ‫ א ܭ‬Ը݊‫ ݎݔ‬is a matrix of known coefficients (e.g. stoichiometric or yield coefficients), ‫ א ܥ‬Ը݊‫ ݊ݔ‬is the state matrix with known elements and ‫ א )ݐ(ݑ‬Ը݊ is a vector gathering the process inputs (e.g. mass and/or energy feeding rate vector) and/or other time varying functions (e.g. gaseous outflow rate). Besides, ‫ ݐ‬represents the time variable, ‫ݖ‬ (‫ א ݖ‬ሾ0, ‫ܮ‬ሿ) is the axial position, ‫ ܮ‬is the reactor length and ‫ א ܯݒ‬Ը݊‫ ݊ݔ‬is considered as a positive and known matrix related with the superficial velocities of the state variables. In addition, ߚ(‫ )ݐ‬and ߙሺ‫ݖ‬ሻ were defined at the beginning of section two. Once the general dynamical model was described one needs to introduce a couple of assumptions: Assumption 3.1 The elements of matrix ‫ ܭ‬are constants and known. Furthermore, it is possible to compute the rank of matrix ‫ ܭ‬such as ‫݇ = ܭ݇݊ܽݎ‬. Assumption 3.2 There are ݊1 state variables that are gathered in ܺ1 (‫ݖ‬, ‫ܺ( )ݐ‬1 (‫ݖ‬, ‫ א )ݐ‬Ը݊ 1 ) which are difficult to measure and ݊2 (݊2 = ݊ െ ݊1 ) state variables that are grouped in ܺ2 (‫ݖ‬, ‫ܺ( )ݐ‬2 (‫ݖ‬, ‫ א )ݐ‬Ը݊ 2 ) that can be measured. Besides ݊2 ൒ ݇.

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119

Considering Assumption 3.2, the general dynamical model described in (3.1) can be split as: ߲ܺ1 (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬ ߲ܺ2 (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬

߲ܺ1 (‫ݖ‬,‫)ݐ‬

= െ‫ݒ‬1 ‫ ݊ܫ‬1 = െ‫ݒ‬2

߲‫ݖ‬ ߲ܺ2 (‫ݖ‬,‫)ݐ‬ ߲‫ݖ‬

+ ‫ܭ‬1 ݂ሺܺሻ + ‫ܥ‬11 ܺ1 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܥ‬12 ܺ2 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ݑ‬1 (‫)ݐ‬

+ ‫ܭ‬2 ݂ሺܺሻ + ‫ܥ‬21 ܺ1 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܥ‬22 ܺ2 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ݑ‬2 (‫)ݐ‬

(3.4) (3.5)

with the following initial and boundary conditions: ܺ1 ሺ‫ݖ‬, 0ሻ = ߙ1 ሺ‫ݖ‬ሻ ܺ2 ሺ‫ݖ‬, 0ሻ = ߙ2 (‫)ݖ‬ ܺ1 ሺ0, ‫ݐ‬ሻ = ߚ1 (‫)ݐ‬ ܺ2 ሺ0, ‫ݐ‬ሻ = ߚ2 (‫)ݐ‬

(3.6) (3.7) (3.8) (3.9)

where ߙ1 (‫ א )ݖ‬Ը݊ 1 , ߙ2 ሺ‫ݖ‬ሻ ‫ א‬Ը݊ 2 , ߚ1 (‫ א )ݐ‬Ը݊ 1 , ߚ2 (‫ א )ݐ‬Ը݊ 2 , ‫ܭ‬1 ‫ א‬Ը݊ 1 ‫ ݎݔ‬, ‫ܭ‬2 ‫ א‬Ը݊ 2 ‫ ݎݔ‬, ‫ܥ‬11 ‫ א‬Ը݊ 1 ‫ ݊ݔ‬1 , ‫ܥ‬12 ‫ א‬Ը݊ 1 ‫ ݊ݔ‬2 , ‫ܥ‬21 ‫ א‬Ը݊ 2 ‫ ݊ݔ‬1 , ‫ܥ‬22 ‫ א‬Ը݊ 2 ‫ ݊ݔ‬2 , ‫ݑ‬1 (‫ א )ݐ‬Ը݊ 1 and ‫ݑ‬2 (‫ א )ݐ‬Ը݊ 2 are the corresponding partitions of ߙሺ‫ݖ‬ሻ, ߚሺ‫ݐ‬ሻ, ‫ ܭ‬, ‫ ܥ‬and ‫ )ݐ(ݑ‬. Furthermore ‫ݒ‬1 is a positive constant, ‫ ݊ܫ‬1 is the identity matrix of dimension ݊1 and ‫ݒ‬2 ‫ א‬Ը݊ 2 ‫ ݊ݔ‬2 is a matrix of constant ‫ݒ‬1 ‫ ݊ܫ‬1 0 elements. Thus, ‫ ܯݒ‬can be expressed as ‫ = ܯݒ‬൬ ൰. 0 ‫ݒ‬2

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3.2. Design of a Robust State Observer The structure of the proposed state observer is derived by partially following the methodology to design an asymptotic observer for infinite dimensional models. This methodology states that if Assumption 3.1 and 3.2 hold it is possible to compute a matrix ‫ܪ‬ such that (Dochain and Vanrolleghem, 2001): ‫ܭ‬1 + ‫ܭܪ‬2 = 0

(3.10)

this equation is fulfilled when ‫ = ܪ‬െ‫ܭ‬1 ‫ܭ‬2 Ș where ‫ܭ‬2 Ș is the generalized pseudo-inverse of ‫ܭ‬2 ( ‫ܭ‬2 ‫ܭ‬2 Ș ‫ܭ‬2 = ‫ܭ‬2 ). Note that when ‫ܭ‬2 is a square matrix (i.e. ݊2 = ݇) then ‫ܭ‬2 Ș = ‫ܭ‬2 െ1 .Once ‫ ܪ‬is obtained then it is possible to define a linear combination of the process state variables as follows (Dochain, 2000): ‫ݖ(ݓ‬, ‫ܺ = )ݐ‬1 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܺܪ‬2 ሺ‫ݖ‬, ‫ݐ‬ሻ where ‫ݖ(ݓ‬, ‫ א )ݐ‬Ը݊ 1 is an auxiliary variable. The dynamics of this variable for processes carried out in plug flow reactors whose model is described by (3.1) with partitions (3.4) and (3.5) is robust against the nonlinearities vector ݂ሺܺሻ because (3.10) holds and it is given by: ߲‫ݖ( ݓ‬,‫)ݐ‬ ߲‫ݐ‬

= െ‫ݒ‬1 ‫ ݊ܫ‬1

߲‫ݖ( ݓ‬,‫)ݐ‬ ߲‫ݖ‬

+ ‫ݓܣ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݖ‬,‫ݐ‬ሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

(3.11)

where ‫ܥ = ܣ‬11 + ‫ܥܪ‬21 , ‫ܥ = ܦ‬12 + ‫ܥܪ‬22 െ ‫ ܪܣ‬and ‫ݒ = ܧ‬1 ‫ ݊ܫ‬1 ‫ ܪ‬െ ‫ݒܪ‬2 . In addition, the initial and boundary conditions for the previous equation remain as: ‫ݓ‬ሺ‫ݖ‬, 0ሻ = ܺ1 ሺ‫ݖ‬, 0ሻ + ‫ܺܪ‬2 ሺ‫ݖ‬, 0ሻ = ߙ1 (‫ )ݖ‬+ ‫ߙܪ‬2 (‫)ݖ‬ ‫ݓ‬ሺ0, ‫ݐ‬ሻ = ܺ1 ሺ0, ‫ݐ‬ሻ + ‫ܺܪ‬2 ሺ0, ‫ݐ‬ሻ = ߚ1 (‫ )ݐ‬+ ‫ߚܪ‬2 (‫)ݐ‬

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(3.12) (3.13)

120

Efrén Aguilar-Garnica and Juan P. García-Sandoval

The next step of the methodology would be the transformation of (3.11) into a set of ordinary differential equations by means of diverse numerical well-known methods such as the Finite Differences Method (FDM) or the Orthogonal Collocation Method (OCM). Nevertheless, there are certain drawbacks associated with the use of both methods. For example, if the FDM is considered to derive the lumped model, then it may be required a considerable amount of discretization points if it is desired to capture the dynamic behavior of the distributed parameter model (Chirstofides, 2001). On the other hand, although the OCM is able to reduce the number of the discretization points and some efforts have been made to generate such points in an optimal form (Lefevre et al. 2000), the OCM is considered as an approximation method since the convenient selection of a weight on the error function is still an open problem. Regarding these facts what we propose is an alternative in this step of the methodology. Specifically we suggest to represent (3.11)-(3.13) with the MC (Rhee et al. 2000). This method is able to generate an exact solution for ‫ݖ(ݓ‬, ‫( )ݐ‬instead the approximate solutions that offer both the FDM and the OCM) at certain specific axial point. In order to apply the MC, it is necessary to define ߬ = ‫ܮ‬/‫ݒ‬1 as the residence time for the unmeasured variables. Then, the application of the method at ‫ ܮ = ݖ‬generates the set of integral equations presented below (see Appendix A): If ‫ ߬ < ݐ‬at ‫ܮ = ݖ‬ ‫ݐ‬

‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ݁ ‫ݓ ݐܣ‬ሺ‫ݒ‬1 (߬ െ ‫)ݐ‬,0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ

‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ ߲‫ݖ‬

+‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

቉ ݀ߣ

(3.14)

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If ‫ ߬ > ݐ‬at ‫ܮ = ݖ‬ ‫ݐ‬

‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ݁ ‫ݓ ߬ ܣ‬ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ݐ׬‬െ ߬ ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ

‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ ߲‫ݖ‬

+‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

቉ ݀ߣ

(3.15)

If ‫ ݐ‬൑ ߬ along ‫ݒ = ݖ‬1 ‫ݐ‬ ߲ܺ ሺ‫ߣ ݒ‬,ߣሻ

‫ ܺܦ‬ሺ‫ߣ ݒ‬, ߣሻ + ‫ ܧ‬2 1 ‫ݐ‬ ߲‫ݖ‬ ‫ݓ‬ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ = ݁ ‫ݓ ݐ ܣ‬ሺ0,0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ 2 1 +‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

቉ ݀ߣ

(3.16)

In equation (3.14) the term ‫ݒ(ݓ‬1 ሺ߬ െ ‫ݐ‬ሻ) is the initial condition for ‫ ݓ‬along the characteristic lines of the unmeasured state variables and in equation (3.15) the term ‫(ݓ‬0, ‫ ݐ‬െ ߬) denotes the boundary condition of ‫ ݓ‬along these same lines. On the other hand, the variable ‫ݒ(ݓ‬1 ‫ݐ‬, ‫ )ݐ‬described by equation (3.16) gathers certain state variables that travel along the characteristic line ‫ݒ = ݖ‬1 ‫ ݐ‬whose departing point is ‫(ݓ‬0,0) is the first one influenced by the boundary condition. In other words, along this characteristic line is travelling the first portion of ³IUHVKIOXLG´RIWKHXQPHDVXUHGVWDWHYDULDEOHVWKDWZLOOUHSODFH the initial mass of the fluid.Furthermore, it can be demonstrated (see Appendix B) that the dynamics of these equations have the following structure: If ‫ ݐ‬൑ ߬

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The Role of the Method of Characteristics in the Solution of Estimation« ߲ܺ2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ1 ሺ‫ݐ‬ሻ ߲‫ݖ‬ ߲ܺ ሺ‫ݐ ݒ‬,‫ݐ‬ሻ ‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ + ‫ ܧ‬2 1 + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

121

‫ݔ‬ሶ 1 ሺ‫ݐ‬ሻ = ‫ݔܣ‬1 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲‫ݖ‬

(3.17)

‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬1 ሺ‫ݐ‬ሻ ‫ݓ‬ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ = ‫ݔ‬2 ሺ‫ݐ‬ሻ ܺ1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ If ‫߬ > ݐ‬

‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

‫ݔ‬ሶ1 ሺ‫ݐ‬ሻ ߲ܺ 2 ሺ‫ܮ‬,‫ݐ‬ሻ ߲‫ݖ‬

=0

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ2 (‫)ݐ‬

(3.18)

‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬2 ሺ‫ݐ‬ሻ ܺ1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ with the following initial conditions ‫ݔ‬1 ሺ0ሻ = ߙ1 ሺ‫ܮ‬ሻ + ‫ߙܪ‬2 ሺ‫ܮ‬ሻ and ‫ݔ‬2 ሺ0ሻ = ߚ1 ሺ‫ܮ‬ሻ + ‫ߚܪ‬2 ሺ‫ܮ‬ሻ. Besides ݀‫ݓ‬ሺ‫ݖ‬, 0ሻ ቉ ߛ1 ሺ‫ݐ‬ሻ = െ‫ݒ‬1 ݁ ‫ ݐܣ‬ቈ ݀‫ݖ‬ ‫ݒ=ݖ‬

1 ሺ߬െ‫ݐ‬ሻ

‫ݐ‬

െ ‫ݒ‬1 න ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൝‫ ܦ‬ቈ

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0

߲ܺ2 ሺ‫ݖ‬, ߣሻ ቉ ߲‫ݖ‬ ‫ݒ=ݖ‬

1

߲ 2 ܺ2 ሺ‫ݖ‬, ߣሻ +‫ܧ‬ቈ ቉ ߲‫ ݖ‬2 ሺ߬+ߣെ‫ݐ‬ሻ ‫ݒ=ݖ‬

ൡ ݀ߣ

1 ሺ߬+ߣെ‫ݐ‬ሻ

݀‫ݓ‬ሺ0, ‫ݐ‬ሻ ቉ െ ‫ݓ ߬ܣ ݁ܣ‬ሺ0, ‫ ݐ‬െ ߬ሻ ߛ2 ሺ‫ݐ‬ሻ = ݁ ‫ ߬ܣ‬ቈ ݀‫ݐ‬ ‫ݐ=ݐ‬െ߬ െ ݁ ‫ ߬ܣ‬ቈ‫ܺܦ‬2 ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ܧ‬ ‫ݐ‬

െ ‫ݒ‬1 න ‫ݐ‬െ߬

݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൝‫ ܦ‬ቈ

߲ܺ2 ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ݑ‬1 ሺ‫ ݐ‬െ ߬ሻ + ‫ݑܪ‬2 ሺ‫ ݐ‬െ ߬ሻ቉ ߲‫ݖ‬

߲ܺ2 ሺ‫ݖ‬, ߣሻ ቉ ߲‫ݖ‬ ‫ݒ=ݖ‬

1

߲ 2 ܺ2 ሺ‫ݖ‬, ߣሻ +‫ܧ‬ቈ ቉ ߲‫ ݖ‬2 ሺ߬+ߣെ‫ݐ‬ሻ ‫ݒ=ݖ‬

ൡ ݀ߣ

1 ሺ߬+ߣെ‫ݐ‬ሻ

Assumption 3.3 ߙ2 ሺ‫ݖ‬ሻ and ߚ2 ሺ‫ݐ‬ሻ are known whereas ߙ1 ሺ‫ݖ‬ሻ and ߚ1 ሺ‫ݐ‬ሻ are unknown. As a consequence of the previous assumption, ‫ݓ‬ሺ0, ‫ݐ‬ሻ, ‫ݓ‬ሺ‫ݖ‬, 0ሻ, ߛ1 ሺ‫ݐ‬ሻ, ߛ2 ሺ‫ݐ‬ሻ, ‫ݔ‬1 ሺ0ሻ and ‫ݔ‬2 ሺ0ሻ are unknown. Remark 3.1 It was assumed that ߙ2 ሺ‫ݖ‬ሻ and ߚ2 ሺ‫ݐ‬ሻ are known because they are related with measured variables. In contrast ߙ1 ሺ‫ݖ‬ሻ and ߚ1 ሺ‫ݐ‬ሻ are considered as unknown since they represent the initial and the boundary conditions of the unmeasured variables. Assumption 3.4 It is possible to have an estimate of the unknown terms ߙ1 ሺ‫ݖ‬ሻ and ߚ1 ሺ‫ݐ‬ሻ. and 1 These estimates are labeled as: 1

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122

Efrén Aguilar-Garnica and Juan P. García-Sandoval

According to the Assumption 3.4, it is possible to generate estimates for ‫ݓ‬ሺ0, ‫ݐ‬ሻ, ‫ݓ‬ሺ‫ݖ‬, 0ሻ, ෝሺ0, ‫ݐ‬ሻ, ‫ݓ‬ ෝሺ‫ݖ‬, 0ሻ, ߛො1 ሺ‫ݐ‬ሻ, ߛො2 ሺ‫ݐ‬ሻ, ‫ݔ‬ො1 ሺ0ሻ and ߛ1 ሺ‫ݐ‬ሻ, ߛ2 ሺ‫ݐ‬ሻ, ‫ݔ‬1 ሺ0ሻ and ‫ݔ‬2 ሺ0ሻ that are given by ‫ݓ‬ ‫ݔ‬ො2 ሺ0ሻ. On the other hand let us define ܾ1 ሺ‫ݐ‬ሻ = ߛො1 ሺ‫ݐ‬ሻ െ ߛ1 ሺ‫ݐ‬ሻ and ܾ2 ሺ‫ݐ‬ሻ = ߛො2 ሺ‫ݐ‬ሻ െ ߛ2 ሺ‫ݐ‬ሻ. Assumption 3.5 The terms ܾ1 ሺ‫ݐ‬ሻ and ܾ2 ሺ‫ݐ‬ሻ remains bounded. Remark 3.2 The differences between ߛො݅ ሺ‫ݐ‬ሻ and ߛ݅ ሺ‫ݐ‬ሻ (i=1,2) can be relatively high because the initial and the boundary conditions for the unmeasured variables are unknown. In spite of this fact, the differences remain bounded and therefore assumption 3.5 holds. Considering the structure of (3.14)-(3.16) along with the estimates ‫ݓ‬ ෝሺ0, ‫ݐ‬ሻ, ‫ݓ‬ ෝሺ‫ݖ‬, 0ሻ, one arrives to the following set of integral equations If ‫ ݐ‬൑ ߬ ‫ݓ‬ ෝሺ‫ܮ‬, ‫ݐ‬ሻ ෝሺ‫ݒ‬1 (߬ െ ‫)ݐ‬,0ሻ = ݁ ‫ݓ ݐܣ‬ ‫ݐ‬

+ න ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ 0

‫ܧ‬

+

߲ܺ2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ ൨ ݀ߣ ߲‫ݖ‬

ෝሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ ܺ෠1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬

(3.19)

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If ‫߬ > ݐ‬ ‫ݓ‬ ෝሺ‫ܮ‬, ‫ݐ‬ሻ ෝሺ0, ‫ ݐ‬െ ߬ሻ = ݁‫ݓ߬ ܣ‬ ‫ݐ‬

+න

݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ

‫ݐ‬െ ߬

‫ܧ‬

+

߲ܺ2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ ൨ ݀ߣ ߲‫ݖ‬ (3.20)

ෝሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ ܺ෠1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ The previous equations can also be represented as a set of ordinary differential equations according to the structure of (3.17)-(3.18): If ‫ ݐ‬൑ ߬ ߲ܺ2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛො1 ሺ‫ݐ‬ሻ ߲‫ݖ‬ ߲ܺ ሺ‫ݐ ݒ‬,‫ݐ‬ሻ ‫ݔ‬ොሶ2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬ො2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ + ‫ ܧ‬2 1 + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ ‫ݔ‬ොሶ1 ሺ‫ݐ‬ሻ = ‫ݔܣ‬ො1 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲‫ݖ‬

‫ݓ‬ ෝሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬ො1 ሺ‫ݐ‬ሻ ෠1

2

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(3.21)

The Role of the Method of Characteristics in the Solution of Estimation«

123

If ‫߬ > ݐ‬ ሶ

‫ݔ‬ොሶ2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬ො2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

‫ݔ‬ො1 ሺ‫ݐ‬ሻ ߲ܺ 2 ሺ‫ܮ‬,‫ݐ‬ሻ ߲‫ݖ‬

=0

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ+ߛො2 (‫)ݐ‬

(3.22)

‫ݓ‬ ෝሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬ො2 ሺ‫ݐ‬ሻ ෝሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ ܺ෠1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ with the following initial conditions ‫ݔ‬ො1 ሺ0ሻ = ߙො1 ሺ‫ܮ‬ሻ + ‫ߙܪ‬2 ሺ‫ܮ‬ሻ and ‫ݔ‬ො2 ሺ0ሻ = ߚመ1 ሺ‫ܮ‬ሻ + ‫ߚܪ‬2 ሺ‫ܮ‬ሻ. Proposition 3.1 If matriz ‫ ܣ‬is Hurwitz and if Assumption 3.5 is fulfilled then (3.19)(3.20) describe the structure of a stable robust state observer in integral equations and (3.21)(3.22) represent the dynamical form of such observer. The proof of such proposition is depicted in Appendix C. The implementation of the proposed observer via simulation runs for the second and third study cases that were described in second section is presented in the following subsections.

3.3. Robust State Observer Design for a Non-Isothermal Plug Flow Reactor In practice, the temperature of the reactor can be measured by reliable on-line sensors. Thus, it is desired to design a robust state observer that provides estimates of the concentration of ܴܺ from measurements of ܶ (i.e. ܺ1 ሺ‫ݖ‬, ‫ݐ‬ሻ = ‫ܮ( ܺܥ‬, ‫ )ݐ‬and ܺ2 ሺ‫ݖ‬, ‫ݐ‬ሻ = ܶ(‫ݖ‬, ‫))ݐ‬. In this case, the mathematical model (2.14)-(2.17) can be represented by (3.4)-(3.9)

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with ‫ݒ‬1 = ‫ ݂ݒ‬, ‫ܰܫ‬1 = 1, ‫ݒ‬2 = ‫ ݂ݒ‬, ‫ܭ‬1 = െ1, ‫ܭ‬2 = െ െ ൬‫ ܨ‬+

ܷ‫ݓ‬ ܸܴ ߩ‫݌ܥ‬

൰, ‫ݑ‬1 ሺ‫ݐ‬ሻ = ‫ ݅ݔܥܨ‬, ‫ݑ‬2 ሺ‫ݐ‬ሻ = ‫ ݅ܶܨ‬+

ȟ‫ܴܪ‬ ߩ ‫݌ܥ‬

ܷ‫ݓ‬ ܶ, ܸߩ ‫ܬ ݌ܥ‬

, ‫ܥ‬11 = െ‫ܨ‬, ‫ܥ‬12 = ‫ܥ‬21 = 0, ‫ܥ‬22 = ߙ1 ሺ‫ݖ‬ሻ = ݃ܺ (‫)ݖ‬, ߙ2 ሺ‫ݖ‬ሻ = ݃ܶ (‫ )ݖ‬,

ߚ1 ሺ‫ݖ‬ሻ = ‫ )ݐ( ܺܩ‬and ߚ2 ሺ‫ݐ‬ሻ = ‫ )ݐ( ܶܩ‬. With these elements, one can compute the following terms: ‫ = ܪ‬െ

ߩ ‫݌ܥ‬ ȟ‫ܴܪ‬

, ‫ = ܣ‬െ‫ܨ‬, ‫= ܦ‬

ܷ‫ݓ‬ ȟ‫ܴܸ ܴܪ‬

and ‫ = ܧ‬0. The dynamical version of the proposed

robust observer is generated when one include the previous terms in (3.21)-(3.22). It was decided to test this version of the observer in the present study case whereas the observer in integral equations is analyzed in the second study case.

State Observer Simulation The performance of the proposed estimation scheme is obtained when equation (3.21)(3.22) are integrated in MATLAB with the command ode45. All the measurements required by this estimator were obtained from the simulation of the model. Besides, the elements ߛො1 ሺ‫ݐ‬ሻ and ߛො2 ሺ‫ݐ‬ሻ were obtained by selecting arbitrary approximations for ‫ݓ‬ሺ‫ݖ‬, 0ሻ, ‫ݓ‬ሺ0, ‫ݐ‬ሻ and ‫ݓ‬ሺ0, ‫ ݐ‬െ ߬ሻ and the integrals that conform these elements were obtained as follows. First, the derivative terms within these integrals were computed with finite differences and then, the integral was computed by using the well-NQRZQ 6LPSVRQ¶V  UXOH &RQVWDQWLQLGHV and Mostoufi, 1999). The simulation results for the proposed observer with different initial conditions are depicted in Figure 3.1. In this figure the continuous line denote the ³UHDO´ evolution of the estimated variable (that was obtained from the model simulation, see Figure

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124

Efrén Aguilar-Garnica and Juan P. García-Sandoval

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2.5) whereas the dash-dotted lines represent the performance of the state observer. As it was expected (see Appendix C), the estimate does not converge to the current state but it is able to recognize the trajectory that the estimated state follows. At this point it is important to remark that a relative low numerical value for ‫ ݂ݒ‬is being considered in order to observe how the estimator features when ‫ ݐ‬൑ ߬ = 5݄. Otherwise ߬ would have a lower value and then, the performance of the observer would be difficult to track when ‫ ݐ‬൑ ߬. Finally note that a discontinuity arises just at ‫߬ = ݐ‬. This is because at this moment the infinite dimensional model switches from the region influenced by the initial conditions to the region which depends on the boundary conditions. The proposed monitoring tool is able to handle such discontinuity providing satisfactory monitoring results.

Figure 3.1. Behavior of the robust state observer at ‫ ܮ = ݖ‬with different initial conditions.

3.4. Robust State Observer Design for a Solid- Waste Anaerobic Digestion In this case it is desired to estimate the concentration of methanogenic biomass from measurements of solid waste FRQFHQWUDWLRQDQG9)$¶VFRQFHQWUDWLRQLHܺ1 ሺ‫ݖ‬, ‫ݐ‬ሻ = ‫ݖ(ܤ‬, ‫)ݐ‬, ܺ2 ሺ‫ݖ‬, ‫ݐ‬ሻ = ሾܹ(‫ݖ‬, ‫ݖ(ܵ )ݐ‬, ‫)ݐ‬ሿܶ ). In this case, the mathematical model (2.31)-(2.35) can be represented by (3.4)-(3.9) with ‫ݒ‬1 = ‫ߛݍ‬, ‫ܰܫ‬1 = 1, ‫ܭ‬1 = ሾ0 ܻ‫ ݎ‬ሿ, ‫ܥ‬11 = െ݇݀ , ‫ܥ‬12 = 0 െ݇ 0 0 0 0 ሾ0 0ሿ, ‫ݒ‬2 = ൤ ൨, ‫ܭ‬2 = ൤ 1 ൨, ‫ܥ‬21 = 0, ‫ܥ‬22 = ቂ ቃ, ߙ1 ሺ‫ݖ‬ሻ = ߙ‫)ݖ( ܤ‬, ߙ2 ሺ‫ݖ‬ሻ = 0 ‫ݍ‬ ߯݇1 െ1 0 0 ሾߙܹ (‫)ݖ( ܵߙ )ݖ‬ሿܶ , ߚ1 ሺ‫ݐ‬ሻ = ߚ‫ )ݐ( ܤ‬, ߚ2 ሺ‫ݐ‬ሻ = ሾߚܹ (‫)ݐ( ܵߚ )ݐ‬ሿܶ . With these elements, one can compute the following terms: ‫ = ܪ‬ሾܻ߯‫ ݎܻ ݎ‬ሿ, ‫ = ܣ‬െ݇݀ , ‫ = ܦ‬ሾ݇݀ ߯ ܻ‫ ݎܻ ݀݇ ݎ‬ሿ and ‫ = ܧ‬ሾ‫ ݎܻߛݍ ݎܻ ߯ߛݍ‬െ ‫ ݎܻݍ‬ሿ . Note that in contrast with the observer for the non isothermal plug flow reactor, in this observer the matrix ‫ ܧ‬is non trivial. This implies that if it is desired to implement the dynamical version of the observer, it is necessary to compute a first and a second derivatives of the measured variables vector (i.e. ቂ ቂ

߲ 2 ܺ2 ሺ‫ݖ‬,ߣሻ ߲‫ ݖ‬2

ቃ

߲ܺ2 ሺ‫ݖ‬,ߣሻ ߲‫ݖ‬

ቃ

‫ݒ=ݖ‬1 ሺ߬+ߣെ‫ݐ‬ሻ

and

) since they are needed to generate ߛො1 ሺ‫ݐ‬ሻ and ߛො2 ሺ‫ݐ‬ሻ. On the other hand, if

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The Role of the Method of Characteristics in the Solution of Estimation«

125

it is desired to implement the integral equations version of the proposed observer one needs to compute only the first derivative. This reduces the number of computations that are needed to carry out the estimation and as a consequence, the integral equations version of the proposed observer is recommended instead its dynamical form. The structure of such integral equations version is obtained when one replaces ‫ܪ‬, ‫ܣ‬, ‫ ܦ‬and ‫ ܧ‬in (3.19)-(3.20).

State Observer Simulation The performance of the proposed estimation scheme for the solid waste anaerobic digestion process is depicted in Figure 3.2 by means of a dotted line whereas the continuous OLQHGHQRWHVWKH³real´evolution of the biomass concentration at ‫ ܮ = ݖ‬which is provided by the simulations shown in Figure 2.9. The dots in Figure 3.2 were obtained with (3.19) and ‫ܤ‬෠ ሺ‫ݖ‬, 0ሻ = ‫ܤ‬ሺ‫ݖ‬, 0ሻ െ 0.1 if ‫ ݐ‬൑ ߬ and with (3.20) and ‫ܤ‬෠ሺ0, ‫ݐ‬ሻ = ‫ܤ‬ሺ0, ‫ݐ‬ሻ െ 0.1 if ‫߬ > ݐ‬. Note that both (3.19) and (3.20) require the knowledge of integral terms whose arguments involves ߲ܺ ሺ‫ ݒ‬ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ . The first term was obtained directly from the model ܺ2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ and 2 1 ߲‫ݖ‬

simulations whereas the second one was computed with finite differences. Then, the

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areas enclosed by the function ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ ߲‫ݖ‬

ቃ were

computed. Since these computations are approximations, they cause the zig-zag behaviour of the observer. Nevertheless the proposed monitoring scheme reconstructs the estimated state satisfactorily even when the infinite dimensional model switches from the region influenced by the initial conditions to the region dominated by the boundary conditions Finally, it is important to comment that if it is desired to implement the proposed observer it would be necessary to know the behavior of the process inputs. In many industrial processes, it is very difficult to accomplish with this restriction and therefore the proposed observer could not be used. In such a case, it is recommended to handle an algorithm known as interval observer which is described in the work of Aguilar-Garnica et al. (2010) for the general mathematical model (3.1)-(3.3).

Figure 3.2. Performance of the robust state observer in integral equations at

.

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126

Efrén Aguilar-Garnica and Juan P. García-Sandoval

4. THE ROLE OF THE MC IN THE SOLUTION OF CONTROL PROBLEMS 4.1. General Dynamical Model Let us consider a convective spatially distributed system described by the following )23'(¶V ߲ܺ (‫ݖ‬,‫)ݐ‬ ߲‫ݐ‬

= െߢ‫ ܿݑ‬ሺ‫ݐ‬ሻ

߲ܺ ሺ‫ݖ‬,‫ݐ‬ሻ ߲‫ݖ‬

+ ݂(ܺሺ‫ݖ‬, ‫ݐ‬ሻ)

‫ݕ‬ሺ‫ݐ‬ሻ = ݄(ܺሺ‫ܮ‬, ‫ݐ‬ሻ)

(4.1) (4.2)

with the boundary and initial conditions ܺሺ‫ݖ‬, 0ሻ = ߙ(‫)ݖ‬ ܺሺ0, ‫ݐ‬ሻ = ߚ(‫)ݐ‬

(4.3) (4.4)

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where ܺ(‫ݖ‬, ‫)ݐ‬, denotes the state vector, ܺ(‫ݖ‬, ‫[ ݊ ࣢ א )ݐ‬ሺ0, ‫ܮ‬ሻ, Ը݊ ], with ࣢ ݊ [ሺ0, ‫ܮ‬ሻ, Ը݊ ] being the infinite dimensional Hilbert space of ݊-dimensional-like vector functions defined on the interval [0, ‫]ܮ‬. ‫[ א ݖ‬0, ‫ ؿ ]ܮ‬Ը and ‫[ א ݐ‬0, λ] denote the axial position and time, respectively. ߢ > 0 is a scalar which may be uncertain while the manipulated variable is the flow function ‫ ܿݑ‬ሺ‫ݐ‬ሻ ‫ܽ[ א‬, ܾ] ‫ ؿ‬Ը+. It is assumed that ݂(ܺ) is a sufficiently smooth vector function which can contain uncertainties, ߙ(‫[ ݊ ࣢ א )ݖ‬ሺ0, ‫ܮ‬ሻ, Ը݊ ] and ߚ(‫ )ݐ‬is a column vector which is a sufficiently smooth function of time. The output variable described by equation (4.2) is a function of the state variable ܺ at the boundary ‫ܮ = ݖ‬.

4.2. Design of a Robust Control Law First, let us state the following control problem. Boundary control problem: The proposed control problem is to regulate the output (4.2) around a constant reference ‫ ݎݕ‬. It is necessary to assume that the convective spatially distributed system (4.1)-(4.4) is carried out at steady state conditions (Gundepudi and Friedly, 1998). For this purpose, let us consider that the boundary conditions (4.4) and the input, are constant; then, under steady state conditions the model is reduced to ߢ‫ݏݑ‬

݀ܺ‫ ݏ‬ሺ‫ݖ‬ሻ ݀‫ݖ‬

= ݂(ܺ‫ ݏ‬ሺ‫ݖ‬ሻ)

‫ ݏܺ(݄ = ݏݕ‬ሺ‫ܮ‬ሻ)

(4.5a) (4.5b)

where ܺ‫ ݏ‬ሺ‫ݖ‬ሻ denotes the state vector at steady state and ‫ ݏݑ‬the corresponding constant input. Equations (4.5) together with the constant condition ܺ‫ ݏ‬ሺ0ሻ = ߚ represent a boundary problem from which it is possible to determinate the steady state input.

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The Role of the Method of Characteristics in the Solution of Estimation«

127

Notice that system (4.1)-(4.4) is a semi-linear partial differential equation with only one characteristic line (time variant) and is a particular case of (2.1)-(2.3) with ĭ = ߢ‫ ܿݑ‬ሺ‫ݐ‬ሻ‫ ݊ܫ‬, Ȍ = ݂(ܺሺ‫ݖ‬, ‫ݐ‬ሻ), ܽ = 0 and ܾ = ‫ܮ‬, therefore according with equations (2.10)-(2.12) its solution FDQREWDLQHGE\VROYLQJWKHIROORZLQJ2'(¶V ݀‫ݖ‬ ݀‫ݐ‬ ݀ܺ ݀‫ݐ‬

= ߢ‫ ܿݑ‬ሺ‫ݐ‬ሻ , ‫ݖ‬ሺ‫ݐ‬0 ሻ = ‫ݖ‬0

(4.6)

= ݂൫ܺሺ‫ݖ‬, ‫ݐ‬ሻ൯ , ܺሺ‫ݐ‬0 ሻ = ܺ0

(4.7)

that comply with the conditions: ܺ0 = ߙሺ‫ݖ‬0 ሻ if ‫ݐ‬0 = 0 ܺ0 = ߚሺ‫ݐ‬0 ሻ if ‫ݖ‬0 = 0

(4.8) (4.9)

The solutions of (4.6) and (4.7) are ‫ݖ = ݖ‬0 + ‫ݒ‬ሺ‫ݐ‬ሻ െ ‫ݒ‬ሺ‫ݐ‬0 ሻ ݂ ܺ = Ȋ‫ݐ‬െ‫ ݐ‬0 (ܺ0 ) ‫ݐ‬

(4.10) (4.11) ݂

where ‫ݒ‬ሺ‫ݐ‬ሻ = ߢ ‫׬‬0 ‫ ܿݑ‬ሺߣሻ݀ߣ and Ȋ‫ݐ‬െ‫ ݐ‬0 (ܺ0 ) is the flow vector field of function ݂ satisfying ߲

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߲‫ݐ‬

݂

݂

ቀȊ‫ݐ‬െ‫ ݐ‬0 ሺ‫ݔ‬ሻቁ = ݂ ቀȊ‫ݐ‬െ‫ ݐ‬0 ሺ‫ݔ‬ሻቁ

for a given fixed ‫( ݔ‬Isidori, 1995). Applying the condition (4.8) to (4.10) and (4.11) we obtain ݂ that ‫ݖ‬0 = ‫ ݖ‬െ ‫ݒ‬ሺ‫ݐ‬ሻ and ܺ = Ȋ‫ݖ(ߙ( ݐ‬0 )) and because the initial condition (4.3) is defined in the domain [0, ‫ ]ܮ‬then we obtain the solution ݂

ܺሺ‫ݖ‬, ‫ݐ‬ሻ = Ȋ‫ߙ( ݐ‬ሺ‫ ݖ‬െ ‫)ݐ(ݒ‬ሻ) if ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫ ݖ‬൑ ‫ ܮ‬+ ‫)ݐ(ݒ‬

(4.12)

In a similar way, applying condition (4.9) in (4.10) and (4.11) we obtain the solution ݂

ܺሺ‫ݖ‬, ‫ݐ‬ሻ = Ȋ߬ ‫ߚ( )ݐ( ݖ‬ሺ‫ ݐ‬െ ߬‫)ݐ( ݖ‬ሻ) if ‫)ݐ(ݒ < ݖ‬

(4.13)

where, for a given pair ሺ‫ݖ‬, ‫ݐ‬ሻ, ߬‫ ݖ‬ሺ‫ݐ‬ሻ = ‫ ݐ‬െ ‫ݐ‬0 LVD³UHVLGHQFH WLPH´ZKLFKLVWKHVROXWLRQ of the following equation ‫ݐ‬

‫ݐ׬ ߢ = ݖ‬െ߬

‫ ݖ‬ሺ‫ݐ‬ሻ

‫ ܿݑ‬ሺߣሻ݀ߣ

(4.14)

We are interested in the control of the output (4.2) which is a function of the state vector at the specify point ‫ܮ = ݖ‬, therefore it is important to analyze the behavior of (4.12) and (4.13) at ‫ܮ = ݖ‬.

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128

Efrén Aguilar-Garnica and Juan P. García-Sandoval

4.2.1. Dynamical Representation for a Particular Axial Point Let us define ‫ݐ‬1 as the time at which, for the specific axial point ‫ܮ = ݖ‬, the transition between solution (4.12) and solution (4.13) occurs. Then, for a given input ‫)ݐ( ܿݑ‬, ‫ݐ‬1 satisfies ‫ݐ‬ the equation ‫׬ = ܮ‬0 1 ‫ ܿݑ‬ሺߣሻ݀ߣ and solutions (4.12) and (4.13) at ‫ ܮ = ݖ‬are ݂

ܺሺ‫ܮ‬, ‫ݐ‬ሻ = Ȋ‫ߙ( ݐ‬ሺ‫ ܮ‬െ ‫)ݐ(ݒ‬ሻ) If ‫ ݐ‬൑ ‫ݐ‬1 (or equivalently ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫)ܮ‬ ܺሺ‫ܮ‬, ‫ݐ‬ሻ =

݂ Ȋ߬ ‫ߚ( )ݐ( ܮ‬ሺ‫ݐ‬

(4.15)

െ ߬‫)ݐ( ܮ‬ሻ) If ‫ݐ > ݐ‬1 (or equivalently ‫ݒ‬ሺ‫ݐ‬ሻ > ‫)ܮ‬

where ߬‫ )ݐ( ܮ‬is only function of time and can be obtained from (4.14) with ‫ܮ = ݖ‬. In fact, the dynamic behavior of ߬‫ )ݐ( ܮ‬can be also obtained by taking the time derivative of (4.14) to yield ݀߬ ‫)ݐ( ܮ‬ ݀‫ݐ‬

=1െ

‫)ݐ( ܿ ݑ‬

(4.16)

‫ݐ( ܿ ݑ‬െ߬ ‫ ܮ‬ሺ‫ݐ‬ሻ)

with the initial condition ߬‫ ܮ‬ሺ‫ݐ‬1 ሻ = ‫ݐ‬1 . From now on, to simplify the notation, let us define ‫ ݐ( ܿݑ = )ݐ( ߬ܿݑ‬െ ߬‫ ܮ‬ሺ‫ݐ‬ሻ). The following lemma formally states the dynamic state representation of ‫ܮ(ܺ = )ݐ(ݔ‬, ‫)ݐ‬. /HPPD  /HW XV FRQVLGHU WKH V\VWHP GHVFULEHG E\ WKH )23'(¶V   WRJHWKHU ZLWK conditions (4.3) and (4.4). The dynamic behavior in a fixed axial point, ‫ܮ = ݖ‬, can be described by a hybrid system with two commutable ODE's subsystems of the form

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Subsystem 1: ‫ݔ‬ሶ ሺ‫ݐ‬ሻ = ݂൫‫ݔ‬1 ሺ‫ݐ‬ሻ൯ െ ߱1 ሺ‫ݐ‬ሻ‫)ݐ( ܿݑ‬ ‫ ۓ‬1 ۖ‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = ݂൫‫ݔ‬2 ሺ‫ݐ‬ሻ൯ when ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫ݒ ܮ‬ሶ ሺ‫ݐ‬ሻ = ߢ‫)ݐ( ܿݑ‬ ‫۔‬ ۖ ߬ሶ ሺ‫ݐ‬ሻ = 1 ‫ݔ ە‬ሺ‫ݐ‬ሻ = ‫ݔ‬1 ሺ‫ݐ‬ሻ

‫ݔ‬1 ሺ0ሻ = ߙ(‫)ܮ‬ ‫ݔ‬2 ሺ0ሻ = ߚ(0)
‫ݒ‬ሺ0ሻ = 0 ߬ሺ0ሻ = 0

(4.17a)

Subsystem 2: ‫ ݑ‬ሺ‫ݐ‬ሻ

when ‫ݒ‬ሺ‫ݐ‬ሻ > ‫ܮ‬

‫ ݑ‬ሺ‫ݐ‬ሻ

ܿ ܿ ‫ݔۓ‬ሶ 2 ሺ‫ݐ‬ሻ = ݂൫‫ݔ‬2 ሺ‫ݐ‬ሻ൯ ቀ1 െ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻቁ + ߱2 ሺ‫ݐ‬ሻ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻ ۖ ‫ݒ‬ሶ ሺ‫ݐ‬ሻ = ߢ‫)ݐ( ݑ‬ ܿ


‫߬ ۔‬ሶ ሺ‫ݐ‬ሻ = 1 െ ‫ ܿ ݑ‬ሺ‫ݐ‬ሻ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻ ۖ ሺ‫ݐ‬ሻ ‫ݔ ە‬ሺ‫ݐ‬ሻ = ‫ݔ‬2

‫ݕ‬ሺ‫ݐ‬ሻ = ݄൫‫ݔ‬ሺ‫ݐ‬ሻ൯

(4.17c)

where

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(4.17b)

The Role of the Method of Characteristics in the Solution of Estimation« ݀ߙ

݂ ߱1 ሺ‫ݐ‬ሻ = ߢ
൫Ȋ‫ ݐ‬൯‫ څ‬ቚ

ߙ൫‫ܮ‬െ‫ݒ‬ሺ‫ݐ‬ሻ൯

݂ ߱2 ሺ‫ݐ‬ሻ = ߢ
൫Ȋ߬ሺ‫ݐ‬ሻ ൯ ቚ

൬
ቚ

‫ ߚ څ‬൫‫ݐ‬െ߬ሺ‫ݐ‬ሻ൯

and ሺ‫ܯ‬ሻ‫= څ‬

߲‫ܯ‬ ߲‫ݔ‬

݀‫ܮ ݖ‬െ‫ݒ‬ሺ‫ݐ‬ሻ ݀ߚ ൬
ቚ

൰

݀‫ݐ ݐ‬െ߬ሺ‫ݐ‬ሻ

129 (4.18a)

൰

(4.18b)

.

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Remark 4.1. The commutation between subsystems (4.17a) and (4.17b) happens only once, at ‫ݐ = ݐ‬1 when ‫ݒ‬ሺ‫ݐ‬1 ሻ = ‫ܮ‬. On the one hand, the state, ‫ݔ‬1 ‫ א‬Ը݊ in subsystem (4.17a) represents ܺሺ‫ܮ‬, ‫ݐ‬ሻ, while in subsystem (4.17b) it does not appear because it is no longer meaningful. On the other hand, ‫ݔ‬2 ‫ א‬Ը݊ in subsystem (4.17a) represents the evolution of an element which is initially at the boundary ‫ = ݖ‬0, flows through the system and in a given instant ‫ݐ‬, lies in ‫)ݐ(ݒ = ݖ‬, while in subsystem (4.17b) represents ܺሺ‫ܮ‬, ‫ݐ‬ሻ. ‫ א ݒ‬Ը+ is only necessary when subsystem (4.17a) is activated and defines the instant where the commutation occurs. Equation ߬ሶ ሺ‫ݐ‬ሻ = 1 in (4.17a) guarantees that ߬ሺ‫ݐ‬1 ሻ = ‫ݐ‬1 at the commutation instant and therefore ߬ሺ‫ݐ‬ሻ = ߬‫ )ݐ( ܮ‬for ‫ ݐ‬൒ ‫ݐ‬1 since the dynamics of ߬ሺ‫ݐ‬ሻ in subsystem (4.17b) are identical to (4.16). From (4.18), also notice that if the initial and boundary conditions are constants, then ߱1 ሺ‫ݐ‬ሻ = 0 and ߱2 ሺ‫ݐ‬ሻ = 0, respectively. The previous lemma can be easily verified (see García Sandoval et al. 2008) by taking the time derivatives of equations (4.15). System (4.17) is an exact representation of ܺሺ‫ܮ‬, ‫ݐ‬ሻ, therefore thank to the MC it has been possible to describe the dynamics of a FOPDE¶V system with one characteristic at a particular axial point by means of a ODE¶V system. Then, to solve the proposed boundary control problem we can use system (4.17) instead of the original system (4.1)-(4.4). In the following subsection, we apply the previous methodology to design a robust controller for the first study case.

4.3. Robust Control Law Design for a Tubular Heat-Exchanger Let us consider the heat-exchanger (2.13) presented in section 2, ܶ‫ ݐ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ ݖܶ)ݐ( ܿݑ ݏܣ‬ሺ‫ݖ‬, ‫ݐ‬ሻ = ‫ ݆ܶ( ܿܪ‬െ ܶሺ‫ݖ‬, ‫ݐ‬ሻ) ܶሺ0, ‫ݐ‬ሻ = ܾܶ (‫)ݐ‬ ܶሺ‫ݖ‬, 0ሻ = ܶ݅ (‫)ݖ‬

(4.19a) (4.19b) (4.19c)

where we have replaced the constant flow velocity ‫ݒ‬, for the product of the variable volumetric flow, ‫ ܿݑ‬ሺ‫ݐ‬ሻ, with the transversal area to the flow direction, ‫ ݏܣ‬. Notice that system (4.19) is a particular case of (4.1)-(4.4) with ܺሺ‫ݖ‬, ‫ݐ‬ሻ = ܶ(‫ݖ‬, ‫)ݐ‬, ߢ = ‫ ݏܣ‬and ݂ሺܺሻ = ‫ ݆ܶ( ܿܪ‬െ ܶ), therefore using (4.12) and (4.13) it is possible to obtain its solution ܶሺ‫ݖ‬, ‫ݐ‬ሻ = ݆ܶ + ൣܶ݅ ൫‫ ܮ‬െ ‫ݒ‬ሺ‫ݐ‬ሻ൯ െ ݆ܶ ൧݁ െ‫ ݐ ܿܪ‬if ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫ ݖ‬൑ ‫ ܮ‬+ ‫)ݐ(ݒ‬ ܶሺ‫ݖ‬, ‫ݐ‬ሻ = ݆ܶ + ൣܾܶ ൫‫ ݐ‬െ ߬‫ ݖ‬ሺ‫ݐ‬ሻ൯ െ ݆ܶ ൧݁ െ‫ )ݐ( ݖ ߬ ܿܪ‬if ‫)ݐ(ݒ < ݖ‬ ‫ݐ‬

‫ݐ‬

where ‫ݒ‬ሺ‫ݐ‬ሻ = ‫׬ ݏܣ‬0 ‫ ܿݑ‬ሺߣሻ݀ߣ and ߬‫ )ݐ( ݖ‬is the solution of equation ‫ݐ׬ = ݖ‬െ߬

‫ ݖ‬ሺ‫ݐ‬ሻ

‫ ܿݑ‬ሺߣሻ݀ߣ.

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130

Efrén Aguilar-Garnica and Juan P. García-Sandoval In addition, the hybrid representation (4.17) for the heat-exchanger is Subsystem 1: ‫ݔ‬ሶ ሺ‫ݐ‬ሻ = െ‫ ܿܪ‬൫‫ݔ‬1 ሺ‫ݐ‬ሻ െ ݆ܶ ൯ െ ߱1 ሺ‫ݐ‬ሻ‫)ݐ( ܿݑ‬ ‫ ۓ‬1 ۖ ‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = െ‫ ܿܪ‬൫‫ݔ‬2 ሺ‫ݐ‬ሻ െ ݆ܶ ൯ when ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫ݒ ܮ‬ሶ ሺ‫ݐ‬ሻ = ߢ‫)ݐ( ܿݑ‬ ‫۔‬ ۖ ߬ሶ ሺ‫ݐ‬ሻ = 1 ‫ܶە‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬1 ሺ‫ݐ‬ሻ

‫ݔ‬1 ሺ0ሻ = ܶ݅ (‫)ܮ‬ ‫ݔ‬2 ሺ0ሻ = ܾܶ (0)
‫ݒ‬ሺ0ሻ = 0 ߬ሺ0ሻ = 0

(4.20a)

Subsystem 2: ‫ ݑ‬ሺ‫ݐ‬ሻ

when ‫ݒ‬ሺ‫ݐ‬ሻ > ‫ܮ‬

‫ ݑ‬ሺ‫ݐ‬ሻ

ܿ ܿ ‫ݔ ۓ‬ሶ 2 ሺ‫ݐ‬ሻ = െ‫ ܿܪ‬൫‫ݔ‬2 ሺ‫ݐ‬ሻ െ ݆ܶ ൯ ቀ1 െ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻቁ + ߱2 ሺ‫ݐ‬ሻ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻ ۖ ‫ݒ‬ሶ ሺ‫ݐ‬ሻ = ߢ‫)ݐ( ݑ‬ ܿ


‫߬ ۔‬ሶ ሺ‫ݐ‬ሻ = 1 െ ‫ ܿ ݑ‬ሺ‫ݐ‬ሻ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻ ۖ ‫ܶ ە‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬2 ሺ‫ݐ‬ሻ

(4.20b)

where ߱1 ሺ‫ݐ‬ሻ = ‫ ݏܣ‬൬


݀ܶ݅

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߱2 ሺ‫ݐ‬ሻ = ‫ ݏܣ‬൬


ቚ

݀‫ܮ ݖ‬െ‫ݒ‬ሺ‫ݐ‬ሻ ܾ݀ܶ

ቚ

൰ ݁ െ‫ݐ ܿܪ‬

݀‫ݐ ݐ‬െ߬ ‫)ݐ( ܮ‬

(4.21a)

൰ ݁ െ‫ ܮ ߬ ܿܪ‬ሺ‫ݐ‬ሻ

(4.21b)

Now let assume that the boundary and initial conditions (4.19b) and (4.19c) are constants and define that the control objective consists in controlling the exit temperature, ܶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ = ܶ(‫ܮ‬, ‫)ݐ‬, around the constant reference ܶ‫ ݎ‬. Then the dynamic behavior of ܶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ according with (4.20) is ܶሶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ = െ‫ ܿܪ‬൫ܶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ െ ݆ܶ ൯ if‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫ܮ‬ ܶሶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ = െ‫ ܿܪ‬൫ܶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ െ ݆ܶ ൯ ቀ1 െ

‫ ܿ ݑ‬ሺ‫ݐ‬ሻ ቁ ‫ ߬ܿ ݑ‬ሺ‫ݐ‬ሻ

(4.22) if ‫ݒ‬ሺ‫ݐ‬ሻ > ‫ܮ‬

(4.23)

and the regulation error is ݁‫ ݎ‬ሺ‫ݐ‬ሻ = ܶ‫ ݐݑ݋‬ሺ‫ݐ‬ሻ െ ܶ‫ݎ‬

(4.24)

Notice that when ‫ݒ‬ሺ‫ݐ‬ሻ ൑ ‫( ܮ‬i.e. when not all the fluid initially within the heat-exchanger has leaved the system) it is not possible to control ܶ‫ ݐݑ݋‬since the input ‫ݑ‬ሺ‫ݐ‬ሻ is not present in (4.22) and we can only reduce the time at which the fresh fluid that entered at ‫ = ݐ‬0 leaves the heat-exchanger by increasing the volumetric flow, ‫)ݐ( ܿݑ‬. After this had happened (i.e. when ‫ݒ‬ሺ‫ݐ‬ሻ > ‫ )ܮ‬it is possible to control the temperature since ‫ ܿݑ‬ሺ‫ݐ‬ሻ is present in (4.23). Therefore, the boundary control problem of system (4.19) reduces to the control of system (4.23) in order to guarantee that lim‫ݐ‬՜’ ݁‫ ݎ‬ሺ‫ݐ‬ሻ = 0. When the heat-exchanger works at the steady state and is controlled, according to (4.5), satisfies the equations ‫ݏݑ ݏܣ‬

݀ܶ‫ ݏ‬ሺ‫ݖ‬ሻ ݀‫ݖ‬

= െ‫ ܿܪ‬൫ܶ‫ ݏ‬ሺ‫ݖ‬ሻ െ ݆ܶ ൯

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(4.25a)

The Role of the Method of Characteristics in the Solution of Estimation« ܶ‫ ݏܶ = ݎ‬ሺ‫ܮ‬ሻ

131 (4.25b)

The solution of equation (4.25a) applying the boundary condition ܶ‫ ݏ‬ሺ0ሻ = ܾܶ is ሺ‫ݖ‬ሻ ܶ‫ݏ‬ = ݆ܶ + ൫ܾܶ െ ݆ܶ ൯݁ െ‫ݖ ܿܪ‬/‫ ݏ ݑ ݏܣ‬, and using (4.25b) it is possible to determinate the steady state volumetric flow: ‫= ݏݑ‬

‫ܮ ܿܪ‬ ‫ ݏ ܣ‬ln ቆ

ܶ ܾ െܶ ݆ ܶ ‫ ݎ‬െܶ ݆

(4.26) ቇ

The heat-exchanger can work as a heater or as a chiller; for each case we have the following: Heater: When we heat the fluid it holds that ݆ܶ > ܶ(‫ݖ‬, ‫ )ݐ‬all along the heat-exchanger, therefore ݆ܶ > ܶ‫ ݐݑ݋‬and sign ቀ‫ ܿܪ‬൫݆ܶ െ ܶ‫ ݐݑ݋‬൯ቁ > 0. If the error (4.24) is positive (i.e. ܶ‫ ) ݎܶ > ݐݑ݋‬we are heating excessively and therefore the volumetric flow is smaller (or equivalently the resident time is bigger) than the necessary to obtain the desirable steady state (‫) ݏݑ < ܿݑ‬. Now, if the error (4.24) is negative (i.e. ܶ‫ ) ݎܶ < ݐݑ݋‬we are not heating enough and therefore the volumetric flow is bigger (or equivalently the resident time is smaller) than the necessary to obtain the desirable steady state (‫) ݏݑ > ܿݑ‬. Chiller: When we chill the fluid it holds that ݆ܶ < ܶ(‫ݖ‬, ‫ )ݐ‬all along the heat-exchanger,

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therefore ݆ܶ < ܶ‫ ݐݑ݋‬and sign ቀ‫ ܿܪ‬൫݆ܶ െ ܶ‫ ݐݑ݋‬൯ቁ < 0. If the error (4.24) is negative (i.e. ܶ‫ ) ݎܶ < ݐݑ݋‬we are cooling excessively and therefore the volumetric flow is smaller than the necessary to obtain the desirable steady state (‫) ݏݑ < ܿݑ‬. Now, if the error (4.24) is positive (i.e. ܶ‫ ) ݎܶ > ݐݑ݋‬we are not cooling enough and therefore the volumetric flow is bigger than the necessary to obtain the desirable steady state (‫) ݏݑ > ܿݑ‬. From the previous reasoning we obtain the Table 4.1, and from this table we infer the following relation: ߠ signሺ݁‫ ݎ‬ሻsignሺ‫ ܿݑ‬െ ‫ ݏݑ‬ሻ = െ1‫ ് ݎ݁׊‬0, ‫ ݏݑ ് ܿݑ׊‬,

(4.27)

where ߠ = sign ቀ‫ ܿܪ‬൫݆ܶ െ ܶ‫ ݐݑ݋‬൯ቁ (i.e. ߠ = 1 for the heater case and ߠ = െ1 for the Chiller case). Using this information a controller which solves the boundary control problem (GarcíaSandoval et al. 2008) is ߟሶ ሺ‫ݐ‬ሻ = ߠīc ݇ܿ ȁߟሺ‫ݐ‬ሻȁ݁‫ ݎ‬ሺ‫ݐ‬ሻ + ߠīc ȁ݁‫ݎ‬ሶ ሺ‫ݐ‬ሻȁsign൫݁‫ ݎ‬ሺ‫ݐ‬ሻ൯ ‫ ܿݑ‬ሺ‫ݐ‬ሻ = ߟሺ‫ݐ‬ሻ + ߠ݇ܿ ݁‫ ݎ‬ሺ‫ݐ‬ሻ

(4.28) (4.29)

where the control parameters are ݇ܿ > 0 and īc > ݇ܿ . To prove that controller (4.28)-(4.29) solves the problem we define the Lyapunov function ‫ܮ‬

ߛܿ

ܿ

2

‫ݎ‬

2

ܿ

ܿ

‫ݏ‬

ܿ ‫ݎ‬

ߛܿ +ߜ ܿ

ܿ

‫ݏ‬

2

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(4.30)

132

Efrén Aguilar-Garnica and Juan P. García-Sandoval

where ߛܿ and ߜܿ are positive constants. The time derivative of (4.30) is ܸ‫ܮ‬ሶ = െߛܿ ߠ݇݁‫ߟ ݎ‬ሶ + ߛܿ ሺ‫ ܿݑ‬െ ‫ ݏݑ‬ሻߟሶ + ߜܿ ሺ݇ܿ ߠ݁‫ݎ‬ሶ + ߟሶ ሻሺ‫ ܿݑ‬െ ‫ ݏݑ‬ሻ

(4.31)

Equation (4.28) can be also rewritten as ߟሶ ሺ‫ݐ‬ሻ = ߠīc ሺ݇ܿ ȁߟ݁‫ ݎ‬ȁ + ȁ݁‫ݎ‬ሶ ȁሻsignሺ݁‫ ݎ‬ሻ therefore ߠ݁‫ߟ ݎ‬ሶ ൒ 0. Form this inequality we conclude that the first term of (4.31) is negative definite and using (4.27) we also see that the second term of (4.31) is also negative definite. Finally, if īc > ݇ܿ the third term of (4.31) is also negative definite, concluding that ܸ‫ܮ‬ሶ < 0 for ݁‫ ് ݎ‬0 and ‫ ݏݑ ് ܿݑ‬, guaranteeing that lim‫ݐ‬՜’ ݁‫ = )ݐ( ݎ‬0. Table 4.1. Control action required to stabilize the heat-exchanger. Error Action Heater Cooler

ࢋ࢘ Positive Negative Positive Negative

Relation for the instantaneous (࢛ࢉ ) Action needed for ࢋ࢘ ՜ ૙ and the steady-state (࢛࢙ ) volumetric Resident time Volumetric flow flow Decrease Increase ‫ ܿݑ‬is smaller than ‫ݏݑ‬ Increase Decrease ‫ ܿݑ‬is bigger than ‫ݏݑ‬ Increase Decrease ‫ ܿݑ‬is bigger than ‫ݏݑ‬ Decrease Increase ‫ ܿݑ‬is smaller than ‫ݏݑ‬

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Robust Control Law Simulation To analyze the performance of controller (4.28)-(4.29) we carried out a simulation of system (4.19) in a cooling operation using the following parameters for the model: ‫ = ܮ‬10 cm, ‫ = ݏܣ‬1 cm2, ‫ = ܿܪ‬1.5 s-1, ݆ܶ = 10ºC and ܶ݅ = 35ºC, while the parameters for the controller are Ȟc = 0.35, ݇ܿ = 0.32 and ߠ = െ1. Figures 4.1 and 4.2 present the simulation results.

Figure 4.1. Heat-exchanger control: Spatial temperature behavior. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Role of the Method of Characteristics in the Solution of Estimation«

133

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At the beginning the initial temperature is equal to 35ºC, the input temperature is also equal to 35ºC and the reference is equal to 20ºC, therefore the controller reduces the volumetric flow in order to increase the resident time and decrease the output temperature. Then at time ‫ = ݐ‬1 s the transition between the initial and boundary condition takes place making the system controllable and the error becomes zero approximately at ‫ = ݐ‬3 s. At time ‫ = ݐ‬8 s we decrease the reference 5ºC and controller gets this new reference in approximately 3s. Finally, at ‫ = ݐ‬12s there is an increment in the input temperature of 10ºC. However, the controller rapidly modifies the volumetric flow attenuating the output temperature deviation. As it can be seen, the proposed controller stabilizes the system and is robust to uncertainties in the inflow temperature. It is also important to remark that it is only necessary to measure the output temperature and it is not necessary to measure the inflow temperature and/or intermediate temperature in the heat-exchanger.

Figure 4.2. Heat-exchanger control: (a) Output temperature and reference (b) Volumetric flow. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

134

Efrén Aguilar-Garnica and Juan P. García-Sandoval

5. CONCLUSION In this chapter it was shown the importance of the Method of Characteristics in the solution of certain estimation problems for processes whose mathematical model is given by First-Order Partial Differential Equations. Specifically, the MC was applied to derive a dynamical version and an integral equations version for a robust state observer. The dynamical version of this observer was implemented through simulation runs for an irreversible chemical reaction process carried out within a non-isothermal plug flow reactor. In this case, the estimation task of such observer was to generate estimates for the concentration of certain reagent from temperature measurements. Even when the estimate does not converge to the current state, it is able to recognize its trajectory. Then, the integral equations version of the proposed observer was tested for a solid-waste anaerobic digestion process where it was desired to estimate the evolution of the biomass concentration from measurements of solid waste and volatile fatty acids concentrations. In this particular case the observer also provides satisfactory estimation results but it has a zig-zag behavior as a consequence of the approximations that are required to implement it. In both processes, the proposed observer is able to handle the discontinuity caused when the process switches from the region influenced by the initial conditions towards the region dominated by the boundary conditions. Additionally, it was described the role of the MC in the design of a robust control law for a heat exchanger where the control variable was the volumetric flow. It was demonstrated via numerical simulations that the proposed controller yields a robust response despite its simplicity.

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ACKNOWLEDGMENT The authors gratefully acknowledge CONACyT for the support that made this study possible.

REFERENCES Aguilar-Garnica, E.; Sandoval-García, J.P.; González-Figueredo, C. A robust monitoring tool for distributed parameter plug flow reactors. Computers and Chemical Engineering 2010, article in press. Björnsson, L.; Murto, M.; Jantsch, T.G.; Mattiasson, B. Evaluation of new methods for the monitoring of alkalinity, dissolved hydrogen and the microbial community in anaerobic digestion. Water Research 2001, 35 (12), 2833-2840. Christofides, P.D. Nonlinear and robust control of PDE systems: methods and application to transport reaction processes; Birkhäuser: Boston, 2001. Christofides, P.D.; Daoutidis, P. Feedback control of hyperbolic PDE systems. A.I.Ch.E. Journal 1996, 42 (11), 3063-3086. Christofides, P.D.; Daoutidis, P. Robust control of hyperbolic PDE systems. Chemical Engineering Science 1998a, 53 (1), 85-105.

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

The Role of the Method of Characteristics in the Solution of Estimation«

135

Christofides, P.D.; Daoutidis, P. Distributed output feedback control of twotime-scale hyperbolic PDE systems. International Journal of Applied Mathematics and Computer Science 1998b, 8, 713-732. Constantinides, A.; Mostoufi, N. Numerical methods for chemical engineers with MATLAB applications; Prentice Hall PTR: New Jersey, 1999. Dochain, D. State observers for tubular reactors with unknown kinetics. Journal of Process Control 2000, 10, 259-268. Dochain, D.; Vanrolleghem,P. Dynamical modeling and estimation in wastewater treatment processes; IWA Publishing: London, 2001. Dubljevic, S.; Mhaskar, P.; El-Farra, N.H.; Christofides, P.D. Predictive control of transportreaction processes. Computers & Chemical Engineering 2005, 29, 2335-2345. García-Sandoval, J. P.; González-Álvarez, V.; Pelayo-Ortiz, C. Robust continuous control of convective spatially distributed systems. Chemical Engineering Science 2008, 63, 43734385. Gundepudi, P.K.; Friedly, J.C. Velocity control of partial differential equation systems with single characteristic variable. Chemical Engineering Science 1998, 53 (24), 4055-4072. Hanczyc, E.M.; Palazoglu, A. Sliding mode control of nonlinear distributed parameter chemical processes. I&EC Research 1995, 34, 557-566. Isidori, A. Nonlinear Control System; Springer: London, 1995. Lefèvre, L.; Dochain,D.; Feyo de Azevedo, S.; Magnus, A. Optimal selection of orthogonal polynomials applied to the integration of chemical reactors equations by collocation methods. Computers and Chemical Engineering 2000, 24, 2571-2588. Rhee, H.K.; Aris, R.; Amundson, N.R. First-Order Partial Differential Equations; Dover: New York, 2000; Vol. 1 Shang, H.; Forbes, J.F.; Guay, M. Feedback control of hyperbolic distributed parameter systems. Chemical Engineering Science 2005, 60 (4), 969-980. Vavilin, V.A.; Rytov, S.V.; Lokshina,L.Y.; Pavlostathis, S.G. ; Barlaz,M.A. Distributed model of solid waste anaerobic digestion: effects of leachate recirculation and pH adjustment. Biotechnology and Bioengineering 2003a, 81(1), 66-73. Vavilin, V.A.; Rytov, S.V.; Pavlostathis, S.G.; Jokela, J.; Rintala, J. A distributed model of solid waste anaerobic digestion: sensitivity analysis. Water Science and Technology 2003b, 48 (4), 147-154.

APPENDIX A Note that equation (3.11) has the same structure of (2.1) with ĭሺ‫ݖ‬, ‫ݐ‬, ܺሻ = ߶ሺ‫ݖ‬, ‫ݐ‬, ܺሻ‫ܰܫ‬1 where ߶ሺ‫ݖ‬, ‫ݐ‬, ܺሻ = ‫ݒ‬1

and Ȍ(‫ݖ‬, ‫ݐ‬, ܺ) = ‫ݓܣ‬ሺ‫ݖ‬, ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݖ‬, tሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݖ‬,‫ݐ‬ሻ ߲‫ݖ‬

+

‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ. Thus, the solution of (3.11) is given by the (2.10) as follows ݀‫)ݐ( ݖ‬ ݀‫ݐ‬ ݀‫)ݐ( ݓ‬ ݀‫ݐ‬

= ‫ݒ‬1 = ‫ݓܣ‬ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫)ݐ(ݖ‬, ‫ݐ‬ሻ + ‫ܧ‬

(A.1) ߲ܺ 2 ሺ‫)ݐ(ݖ‬,‫ݐ‬ሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

The solution of (A.1) can be easily obtained and is given by: Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(A.2)

136

Efrén Aguilar-Garnica and Juan P. García-Sandoval ‫ݖ = )ݐ(ݖ‬0 + ‫ݒ‬1 (‫ ݐ‬െ ‫ݐ‬0 )

The above equation described a time-space relationship of a fluid particle within the plug flow reactor which is located at a certain point (‫ݖ‬0 , ‫ݐ‬0 ). In addition, this equation allows that equation (A.2) can be represented as: ݀‫)ݐ( ݓ‬ ݀‫ݐ‬

= ‫ݓܣ‬ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݖ‬0 + ‫ݒ‬1 (‫ ݐ‬െ ‫ݐ‬0 ) , ‫ݐ‬ሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݖ‬0 +‫ݒ‬1 (‫ݐ‬െ‫ ݐ‬0 ) ,‫ݐ‬ሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

whose general solution is: ‫ݐ‬

߲ܺ 2 ሺ‫ݖ‬0 +‫ݒ‬1 (ߣെ‫ ݐ‬0 ) ,ߣሻ

0

߲‫ݖ‬

‫ݓ‬ሺ‫ݐ‬ሻ = ݁ ‫ܣ‬ሺ‫ݐ‬െ‫ ݐ‬0 ሻ ‫ݓ‬ሺ0ሻ + ‫ܣ ݁ ݐ׬‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݖ‬0 + ‫ݒ‬1 (ߣ െ ‫ݐ‬0 ), ߣሻ + ‫ܧ‬

‫ݑ‬1ߣ+‫ݑܪ‬2ߣ ݀ߣ

+ (A.3)

where ‫ݓ‬ሺ0ሻ is the ³GHSDUWLQJSRLQW´RI‫ݓ‬ሺ‫ݐ‬ሻ7KLV³GHSDUWLQJSRLQW´FDQEHLQIOXHQFHGHLWKHU by the initial condition or by the boundary condition of equation (3.11) ‫ݓ‬ሺ‫ݖ‬0 , 0ሻ = ߙ1 ሺ‫ݖ‬0 ሻ + ‫ߙܪ‬2 ሺ‫ݖ‬0 ሻ if ‫ݐ‬0 = 0 and ‫ݖ‬0 > 0 ሺi. e. ‫ݒ > ݖ‬1 ‫ ݐ‬ሻ ‫ݓ‬ሺ0ሻ = ቐ ‫ݓ‬ሺ0, ‫ݐ‬0 ሻ = ߚ1 ሺ‫ݐ‬0 ሻ + ‫ߚܪ‬2 ሺ‫ݐ‬0 ሻ if ‫ݖ‬0 = 0 and ‫ݐ‬0 > 0 ሺi. e. ‫ݒ < ݖ‬1 ‫ ݐ‬ሻ
‫ݓ‬ሺ0,0ሻ = ߙ1 ሺ0ሻ + ‫ߙܪ‬2 ሺ0ሻ if ‫ݐ‬0 = 0 and ‫ݖ‬0 = 0 ሺi. e. ‫ݒ = ݖ‬1 ‫ ݐ‬ሻ Since ‫ݓ‬ሺ0ሻ can take different values then, equation (A.3) can be split in the following parts:

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If ‫ݒ > ݖ‬1 ‫ݐ‬ ‫ݐ‬

‫ݓ‬ሺ‫ݐ‬ሻ = ݁ ‫ݓ ݐܣ‬ሺ‫ݖ‬0 , 0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݖ‬0 + ‫ݒ‬1 ߣ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݖ‬0 +‫ݒ‬1 ߣ ,ߣሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺߣሻ +

‫ݑܪ‬2ߣ ݀ߣ

(A.4)

If ‫ݒ < ݖ‬1 ‫ݐ‬ ‫ݐ‬

߲ܺ 2 ሺ‫ݒ‬1 (ߣെ‫ ݐ‬0 ),ߣሻ

0

߲‫ݖ‬

‫ݓ‬ሺ‫ݐ‬ሻ = ݁ ‫ܣ‬ሺ‫ݐ‬െ‫ ݐ‬0 ሻ ‫ݓ‬ሺ0, ‫ݐ‬0 ሻ + ‫ܣ ݁ ݐ׬‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݒ‬1 (ߣ െ ‫ݐ‬0 ), ߣሻ + ‫ܧ‬

+ ‫ݑ‬1 ሺߣሻ +

‫ݑܪ‬2ߣ ݀ߣ

(A.5)

If ‫ݒ = ݖ‬1 ‫ݐ‬ ‫ݐ‬

‫ݓ‬ሺ‫ݐ‬ሻ = ݁ ‫ݓ ݐܣ‬ሺ0,0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݒ‬1 ߣ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ߣ ,ߣሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ ቃ ݀ߣ(A.6)

Besides, ‫ݖ‬0 = ‫ ݖ‬െ ‫ݒ‬1 ‫ ݐ‬if ‫ݒ > ݖ‬1 ‫ ݐ‬and ‫ݐ‬0 = ‫ ݐ‬െ ‫ݖ‬/‫ݒ‬1 if ‫ݒ < ݖ‬1 ‫ ݐ‬. As a consequence, (A.4) and (A.5) can be expressed as follows: If ‫ݒ > ݖ‬1 ‫ݐ‬

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The Role of the Method of Characteristics in the Solution of Estimation«

137

‫ݓ‬ሺ‫ݖ‬, ‫ݐ‬ሻ = ߲ܺ ሺ‫ݖ‬+‫ݒ‬1 ሺߣെ‫ݐ‬ሻ,ߣሻ

‫ ܺܦ‬ሺ‫ ݖ‬+ ‫ݒ‬1 ሺߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ ܧ‬2 ‫ݐ‬ ݁ ‫ݓ ݐܣ‬ሺ‫ ݖ‬െ ‫ݒ‬1 ‫ݐ‬, 0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ 2 +‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ If ‫ݒ < ݖ‬1 ‫ݐ‬

቉ ݀ߣ

߲‫ݖ‬

(A.7)

‫ݓ‬ሺ‫ݖ‬, ‫ݐ‬ሻ = ݁

‫ ݖܣ‬/‫ݒ‬1

‫ݓ‬ሺ0, ‫ ݐ‬െ ‫ݖ‬/‫ݒ‬1 ሻ +

‫ݐ‬ ‫ݐ׬‬െ‫ݖ‬/‫ݒ‬1

݁

‫ܣ‬ሺ‫ݐ‬െߣሻ

‫ݖ‬

቎‫ܺܦ‬2 ቀ‫ݒ‬1 ቀߣ െ ‫ ݐ‬+ ቁ , ߣቁ + ‫ܧ‬

‫ݖ‬ ቁ,ߣቁ ‫ݒ‬1

߲ܺ 2 ቀ‫ݒ‬1 ቀߣെ‫ݐ‬+ ߲‫ݖ‬

‫ݒ‬1

(A.8)

቏ ݀ߣ

+‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

For a certain specific fixed axial point ‫ ܮ = ݖ‬and introducing ߬ = ‫ܮ‬/‫ݒ‬1 equations (A.7) and (A.8) acquire the following structure If ‫߬ < ݐ‬ ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݐ‬

݁ ‫ݓ ݐܣ‬ሺ‫ݒ‬1 (߬ െ ‫)ݐ‬,0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ

‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ

቉ ݀ߣ

߲‫ݖ‬

+‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

(3.14)

If ‫߬ > ݐ‬ ߲ܺ ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ

‫ ܺܦ‬ሺ‫ ݒ‬ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ ܧ‬2 ‫ݐ‬ ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ݁ ‫ݓ ߬ ܣ‬ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ݐ׬‬െ ߬ ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ 2 1 +‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

቉ ݀ߣ

߲‫ݖ‬

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(3.15) besides, the behavior of ‫ ݓ‬traveling along the characteristic line ‫ݒ = ݖ‬1 ‫ ݐ׊ ݐ‬൑ ߬ is given by equation (A.6) that can also be written in the following form: ߲ܺ ሺ‫ߣ ݒ‬,ߣሻ

‫ ܺܦ‬ሺ‫ߣ ݒ‬, ߣሻ + ‫ ܧ‬2 1 ‫ݐ‬ ߲‫ݖ‬ ‫ݓ‬ሺ‫ݒ‬1 ‫ ݐ‬, ‫ݐ‬ሻ = ݁ ‫ݓ ݐܣ‬ሺ0,0ሻ + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ 2 1 +‫ݑ‬1 ሺߣሻ + ‫ݑܪ‬2 ሺߣሻ

቉ ݀ߣ

(3.16)

thus, it can be said that the application of the MC generates (3.14)-(3.16) when it is applied to (3.11), concluding the demonstration.

APPENDIX B If one computes the time derivative of (3.14) - (3.16) the result is: If ‫ ߬ < ݐ‬at ‫ܮ = ݖ‬ ݀‫ ݓ‬ሺ‫ܮ‬,‫ݐ‬ሻ ݀‫ݐ‬ ‫ݐܣ‬

= ‫ݐ‬ 1

‫ܣ‬ሺ‫ݐ‬െߣሻ

߲ܺ 2 ሺ‫ݒ‬1 ሺ߬+ߣെ‫ݐ‬ሻ,ߣሻ 2

1

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1

138

Efrén Aguilar-Garnica and Juan P. García-Sandoval

‫ݑܪ‬2 ሺߣሻቃ ݀ߣቅ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬ ‫ݐ‬

‫ݒ‬1 ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൜‫ ܦ‬ቂ

߲ܺ2 ሺ‫ݖ‬,ߣሻ ߲‫ݖ‬

ቃ

߲ܺ 2 ሺ‫ܮ‬,‫ݐ‬ሻ ߲‫ݖ‬

‫ݒ=ݖ‬1 ሺ߬+ߣെ‫ݐ‬ሻ

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ െ ‫ݒ‬1 ݁ ‫ ݐܣ‬ቂ

+‫ܧ‬ቂ

߲ 2 ܺ2 ሺ‫ݖ‬,ߣሻ ߲‫ ݖ‬2

ቃ

‫ݒ=ݖ‬1 ሺ߬+ߣെ‫ݐ‬ሻ

݀‫ ݓ‬ሺ‫ݖ‬,0ሻ ݀‫ݖ‬

ቃ

‫ݒ=ݖ‬1 ሺ߬െ‫ݐ‬ሻ

െ

ൠ ݀ߣ

(B.1)

If ‫ ߬ > ݐ‬at ‫ܮ = ݖ‬ ‫ݐ‬ ݀‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ ߲ܺ2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ = ‫ ܣ‬න ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቈ‫ܺܦ‬2 ሺ‫ݒ‬1 ሺ߬ + ߣ െ ‫ݐ‬ሻ, ߣሻ + ‫ܧ‬ + ‫ݑ‬1 ሺߣሻ ݀‫ݐ‬ ߲‫ݖ‬ ‫ݐ‬െ߬

+ ‫ݑܪ‬2 ሺߣሻ൨ ݀ߣ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ ߲ܺ2 ሺ‫ܮ‬, ‫ݐ‬ሻ ݀‫(ݓ‬0, ‫)ݐ‬ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ݁ ‫ ߬ܣ‬൤ ൨ ߲‫ݖ‬ ݀‫ݐ‬ ‫ݐ=ݐ‬െ߬ ሺ0, ߲ܺ ‫ݐ‬ െ ߬ሻ 2 െ ݁ ‫ ߬ܣ‬ቈ‫ܺܦ‬2 ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ܧ‬ + ‫ݑ‬1 ሺ‫ ݐ‬െ ߬ሻ + ‫ݑܪ‬2 ሺ‫ ݐ‬െ ߬ሻ቉ ߲‫ݖ‬

+‫ܧ‬

‫ݐ‬

߲ܺ2 ሺ‫ݖ‬, ߣሻ ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൝‫ ܦ‬ቈ ቉ ߲‫ݖ‬ ‫ݐ‬െ߬ ‫ݒ=ݖ‬

െ ‫ݒ‬1 න

+‫ܧ‬ቈ

1 ሺ߬+ߣെ‫ݐ‬ሻ

߲ 2 ܺ2 ሺ‫ݖ‬, ߣሻ ቉ ߲‫ ݖ‬2 ‫ݒ=ݖ‬

ൡ ݀ߣ

1 ሺ߬+ߣെ‫ݐ‬ሻ

(B.2) If ‫ ݐ‬൑ ߬ along ‫ݒ = ݖ‬1 ‫ݐ‬ ݀‫ ݓ‬ሺ‫ݒ‬1 ‫ݐ‬,‫ݐ‬ሻ

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‫ܧ‬

݀‫ݐ‬ ߲ܺ 2 ሺ‫ݒ‬1 ‫ݐ‬,‫ݐ‬ሻ ߲‫ݖ‬

‫ݐ‬

= ‫ ܣ‬ቄ݁ ‫(ݓ ߬ܣ‬0,0) + ‫׬‬0 ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ቂ‫ܺܦ‬2 ሺ‫ݒ‬1 ߣ, ߣሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ߣ,ߣሻ ߲‫ݖ‬

ቃ ݀ߣቅ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ +

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

(B.3)

These equations can also be written as If ‫ ߬ < ݐ‬at ‫ܮ = ݖ‬ ݀‫ ݓ‬ሺ‫ܮ‬,‫ݐ‬ሻ ݀‫ݐ‬

= ‫ݓܣ‬ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ܮ‬,‫ݐ‬ሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ1 (‫)ݐ‬

(B.4)

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ2 (‫)ݐ‬

(B.5)

If ‫ ߬ > ݐ‬at ‫ܮ = ݖ‬ ݀‫ ݓ‬ሺ‫ܮ‬,‫ݐ‬ሻ ݀‫ݐ‬

= ‫ݓܣ‬ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ܮ‬,‫ݐ‬ሻ ߲‫ݖ‬

If ‫ ݐ‬൑ ߬ along ‫ݒ = ݖ‬1 ‫ݐ‬ ݀‫ ݓ‬ሺ‫ݒ‬1 ‫ݐ‬,‫ݐ‬ሻ ݀‫ݐ‬

= ‫ݓܣ‬ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ + ‫ܧ‬

߲ܺ 2 ሺ‫ݒ‬1 ‫ݐ‬,‫ݐ‬ሻ ߲‫ݖ‬

+ ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ

where

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(B.6)

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139

݀‫ݓ‬ሺ‫ݖ‬, 0ሻ ቉ ߛ1 ሺ‫ݐ‬ሻ = െ‫ݒ‬1 ݁ ‫ ݐܣ‬ቈ ݀‫ݖ‬ ‫ݒ=ݖ‬

1 ሺ߬െ‫ݐ‬ሻ

‫ݐ‬

߲ܺ2 ሺ‫ݖ‬, ߣሻ െ ‫ݒ‬1 න ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൝‫ ܦ‬ቈ ቉ ߲‫ݖ‬ 0 ‫ݒ=ݖ‬

1 ሺ߬+ߣെ‫ݐ‬ሻ

߲ 2 ܺ2 ሺ‫ݖ‬, ߣሻ +‫ܧ‬ቈ ቉ ߲‫ ݖ‬2 ‫ݒ=ݖ‬

ቋ ݀ߣ ߛ2 ሺ‫ݐ‬ሻ

1 ሺ߬+ߣെ‫ݐ‬ሻ

݀‫ݓ‬ሺ0, ‫ݐ‬ሻ = ݁ ‫ ߬ܣ‬ቈ ቉ െ ‫ݓ ߬ܣ ݁ܣ‬ሺ0, ‫ ݐ‬െ ߬ሻ ݀‫ݐ‬ ‫ݐ=ݐ‬െ߬ ߲ܺ2 ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ݑ‬1 ሺ‫ ݐ‬െ ߬ሻ + ‫ݑܪ‬2 ሺ‫ ݐ‬െ ߬ሻ቉ െ ݁ ‫ ߬ܣ‬ቈ‫ܺܦ‬2 ሺ0, ‫ ݐ‬െ ߬ሻ + ‫ܧ‬ ߲‫ݖ‬ ‫ݐ‬

߲ܺ2 ሺ‫ݖ‬, ߣሻ ݁ ‫ܣ‬ሺ‫ݐ‬െߣሻ ൝‫ ܦ‬ቈ ቉ ߲‫ݖ‬ ‫ݐ‬െ߬ ‫ݒ=ݖ‬

െ ‫ݒ‬1 න

1 ሺ߬+ߣെ‫ݐ‬ሻ

+‫ܧ‬ቈ

߲ 2 ܺ2 ሺ‫ݖ‬, ߣሻ ቉ ߲‫ ݖ‬2 ‫ݒ=ݖ‬

ൡ ݀ߣ

1 ሺ߬+ߣെ‫ݐ‬ሻ

Finally note that equations (B.4)-(B.6) can be conveniently represented as the following dynamical system If ‫ ݐ‬൑ ߬ ߲ܺ2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ1 ሺ‫ݐ‬ሻ ߲‫ݖ‬ ߲ܺ2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ ‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ + ‫ܧ‬ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ ߲‫ݖ‬ ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬1 ሺ‫ݐ‬ሻ ‫ݓ‬ሺ‫ݒ‬1 ‫ݐ‬, ‫ݐ‬ሻ = ‫ݔ‬2 ሺ‫ݐ‬ሻ ܺ1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ

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‫ݔ‬ሶ 1 ሺ‫ݐ‬ሻ = ‫ݔܣ‬1 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬

If ‫߬ > ݐ‬ ‫ݔ‬ሶ1 ሺ‫ݐ‬ሻ = 0 ߲ܺ2 ሺ‫ܮ‬, ‫ݐ‬ሻ ‫ݔ‬ሶ 2 ሺ‫ݐ‬ሻ = ‫ݔܣ‬2 ሺ‫ݐ‬ሻ + ‫ܺܦ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ + ‫ܧ‬ + ‫ݑ‬1 ሺ‫ݐ‬ሻ + ‫ݑܪ‬2 ሺ‫ݐ‬ሻ + ߛ2 (‫)ݐ‬ ߲‫ݖ‬ ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݔ‬2 ሺ‫ݐ‬ሻ ܺ1 ሺ‫ܮ‬, ‫ݐ‬ሻ = ‫ݓ‬ሺ‫ܮ‬, ‫ݐ‬ሻ െ ‫ܺܪ‬2 ሺ‫ܮ‬, ‫ݐ‬ሻ which is the set of equations described in (3.17) and (3.18).

APPENDIX C Let us define ߳ሺ‫ܮ‬, ‫ݐ‬ሻ = ܺ෠1 ሺ‫ܮ‬, ‫ݐ‬ሻ െ ܺ1 (‫ܮ‬, ‫ )ݐ‬be the estimation error of the proposed estimation scheme. The estimation error dynamics is given by If

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140

Efrén Aguilar-Garnica and Juan P. García-Sandoval ߳ሶሺ‫ܮ‬, ‫ݐ‬ሻ = ‫߳ܣ‬ሺ‫ܮ‬, ‫ݐ‬ሻ + ܾ1 (‫)ݐ‬

(C.1)

If ‫߬ > ݐ‬ ߳ሶሺ‫ܮ‬, ‫ݐ‬ሻ = ‫߳ܣ‬ሺ‫ܮ‬, ‫ݐ‬ሻ + ܾ2 (‫)ݐ‬

(C.2)

where ܾ1 ሺ‫ݐ‬ሻ = ߛො1 ሺ‫ݐ‬ሻ െ ߛ1 ሺ‫ݐ‬ሻ and ܾ2 ሺ‫ݐ‬ሻ = ߛො2 ሺ‫ݐ‬ሻ െ ߛ2 (‫ )ݐ‬are disturbance terms in (C.1) and (C.2), respectively. According to the well-known concept of input-output stability, the dependent variable of the previous equations (i.e. ߳ሺ‫ܮ‬, ‫ݐ‬ሻ) remains bounded if the disturbance element is bounded and if the unperturbed system is asymptotically stable. Assumption 3.5 states that the disturbance elements ܾ1 ሺ‫ݐ‬ሻ and ܾ2 ሺ‫ݐ‬ሻ remains bounded and besides, the first sentence of Proposition 3.1 establishes that matrix ‫ ܣ‬is Hurwitz. As a consequence, the unperturbed systems are asymptotically stables. Thus, the estimation error remains bounded and the proposed estimation scheme is stable in the region dominated by the initial condition (i.e. when ‫ ݐ‬൑ ߬) and in the region governed by the boundary condition (i.e. when ‫) ߬ > ݐ‬. On the other hand, the general solution of the (C.2) can be written as follows ‫ݐ‬

߳ሺ‫ܮ‬, ‫ݐ‬ሻ = ߰ܿ‫ ݐܣ ݁ ݋‬+ ݁ ‫ ݁ ߬׬ ݐܣ‬െ‫ܾ ݏܣ‬2 ሺ‫ݏ‬ሻ݀‫ݏ‬

(C.3)

where ߰ܿ‫ ݋‬is a constant. Since ܾ2 ሺ‫ݐ‬ሻ is assumed to be bounded then it is possible to set lim‫ݐ‬՜’ ܾ2 ሺ‫ݐ‬ሻ = ܾ2‫ ݏ‬and regarding (C.3) it is easy to verify that lim ߳ሺ‫ܮ‬, ‫ݐ‬ሻ = ߜܿ‫ ݋‬ሺܾ2‫ ݏ‬ሻ

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‫ݐ‬՜’

in this equation, ߜܿ‫ ݋‬ሺܾ2‫ ݏ‬ሻ is a vector of constant elements whose magnitude will depend on ܾ2‫ ݏ‬. Then, even when the observer is stable, it generates a steady-state offset between the real state and its corresponding estimate. In other words, the estimation error of the proposed observer does not converge towards zero but towards a bounded value.

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 ©2011 Nova Science Publishers, Inc.

Chapter 5

SOLITON SOLUTIONS OF ONE KDV EQUATION Zhao-ling Tao* and Yang Yang College of Mathematics & Physics, Nanjing University of Information Science and Technology, Nanjing, P. R

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ABSTRACT We know KdV equations are important, so we join to study the soliton solutions of one KdV equation. Three simple and effective methods are employed, the exp-function method, the tanh method and solitary wave ansatz method for the soliton solutions in varied form. To our knowledge, not only same solutions as those in the open literature are obtained, but also some new are arrived at by the help of the symbolic computation software.

Keywords: soliton solution; KdV equation; the exp-function method; the solitary wave ansatz method; the tanh method PACS numbers: 02.60_cb, 02.30.Hq, 05.45._a. Mathematics Subject Classification (2000) 35F25, 34C30, 34A30

1. INTRODUCTION Soliton theory has a wide range of applications in nonlinear optical, magnetic flux quantum devices, plasma physics, atmospheric physics, and neural networks. Soliton solutions play an important role in soliton theory. With the development of soliton theory, people found many ways to get soliton solutions. Some of them are: Exp-function method [14], Bäcklund transformation method [5-6], Darboux transformation [7], the tanh method [810], the homogeneous balance method [11], the solitary wave ansatz method [12-13, 17]. The exp-function method was proposed by He and Wu in 2006 [1], it should be pointed out that Zhu [3-4] first applied the method to differential-difference equations. Applications * College of Mathematics & Physics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China [email protected] or [email protected] (Tao) Tel: 86-25-58731160

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142

Zhao-ling Tao and Yang Yang

of the method can be found in Ref. [2]. The tanh method in its systematized form is presented by Malfliet [14] and Fan et al. [15]; the technique is based on a priori assumption that the traveling wave solutions can be expressed in terms of the tanh function, and main steps is summarized in [8-10] for convenient use. The solitary wave ansatz method [12-13, 17] is to carry out the integration of a newly formulated nonlinear evolution equation. In this paper, we use the three methods to obtain soliton solutions of one KdV equation.

2. THE KDV EQUATION Many Scholars attach importance to KdV equation in various forms, we will not list here. We consider the KdV equation in [16]

ut  u x  uux  u xxx

0

(1)

The first term represents the evolution term, the second term the linear term, the third term the nonlinear term, while the fourth term the dispersion term. Solitons are the result of a delicate balance between dispersion and nonlinearity.

2.1. Solution in Form of Exp-Function To investigate the travelling wave solution of Eq. (1), use the transformation

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[

Dx  vt

(2)

This in turn transforms Eq. (1) to

vuc  Duc  Duuc  D 3uccc

0.

(3)

By the exp-function method [1-4], the solution expression is assumed to be the form d

u ([ )

¦a

n

exp( n[ )

n c q

(4)

¦b

m

exp( m[ )

m p

with unknown positive integers c , d , p , q and unknown constants a n and b m to be determined. By properly handling the coefficients of uu c and u ccc , the relationship between p and c can be obtained easily as follows. Since

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Soliton Solutions of One KdV Equation

143

uu c

c1 exp[  ( p  2c )[ ]  ... c1 exp[  ( 6 p  2c )[ ]  ... = c 2 exp[ 3 p[ ]  ... c 2 exp[  8 p[ ]  ...

(5)

u ccc

c3 exp[  (7 p  c )[ ]  ... , c 4 exp[ 8 p[ ]  ...

(6)

and

we have

 (6 p  2c) which leads to p Similarly

(7 p  c) ,

(7)

c.

...  d 1 exp[( q  2 d )[ ] ...  d 1 exp[( 6 q  2 d )[ ] = ...  d 2 exp[ 3q [ ] ...  d 2 exp[ 8 q [ ]

uu c

(8)

and

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u ccc

...  d 3 exp[( 7 q  d )[ ] , ...  d 4 exp[ 8q [ ]

(9)

which requires that

6q  2d thus q

7q  d ,

(10)

d.

Considering the simplest case, we choose c

p

1and q

d

1 , the solution reduces

to

u ( x, t )

a 1 exp( [ )  a 0  a1 exp( [ ) exp( [ )  b0  b1 exp( [ )

(11)

Substituting Eq. (11) to Eq. (3), we have

1 [C 3 exp( 3[ )  C 2 exp( 2[ )  C1 exp( [ )  C 0  A C1 exp( [ )  C2 exp( 2[ )  C3 exp( 3[ )] 0

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(12)

144

Zhao-ling Tao and Yang Yang While A= [exp( [ )  b0  b1 exp( [ )] 4 , Ci (i=-3,-2, . . .,2 , 3,) are constants obtained

by symbolic computation. Equating the coefficients of exp ( n[ ) to be zero, we obtain

­C3 0, C 2 0, C1 0, C 0 0, ® ¯ C 1 0, C 2 0, C 3 0

(13)

Solving the system, simultaneously, we obtain the following solution

v

D a 1  D 3  D ,

a 1b0  6D 2 b0 ,

a0

a1

1 a 1b02 , 4

b1

1 2 b0 4

(14)

Substituting (14) to (11), we get the solution of the Eq. (1),

u ( x, t )

D3 D  v

D



exp(  (Dx  vt ))  b0 

Choose c p 1 , q d solutions with the same steps.

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6D 2 b0

2 and c

p

2,q

1 2 b0 exp( Dx  vt ) 4

d

(15)

2 , we can obtain some soliton

2.2. 1-Soliton Solution by the Solitary Wave Ansatz Reconsider the KdV equation in [16]

ut  u x  uux  u xxx

0

(16)

In this subsection, Eq. (1) or Eq. (16) will be handled by the aid of solitary wave ansatz. Let solitary wave ansatz for the 1-soliton solution in [17] be

u ( x, t )

A , cosh [ B ( x  vt )] p

(17)

where A represents the amplitude of the soliton, while B the inverse width and v the velocity. The unknown exponent p will be determined during the soliton solution derivation. Use the notation

W

B( x  vt ) .

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(18)

Soliton Solutions of One KdV Equation

145

It is not difficult to get

ut

BvAp sinh W cosh p 1 W

ux



(19)

BAp sinh W cosh p 1 W

(20)

B 3 Ap 3 sinh W  2 B 3 Ap sinh W  3 B 3 Ap 2 sinh W B 3 Ap 3 sinh W  cosh p  3 W cosh p 1 W

u xxx

(21)

Substituting (19)-(21) into Eq. (16) yields

BvAp sinh W BAp sinh W BA 2 p sinh W   cosh p 1 W cosh p 1 W cosh 2 p 1 W 

B 3 Ap 3 sinh W  2 B 3 Ap sinh W  3 B 3 Ap 2 sinh W B 3 Ap 3 sinh W  cosh p  3 W cosh p 1 W

0

(22)

Now, from (22),

2p 1 ᧹ p  3

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That is

p

2

(23)

Again, note that the functions cosh p 1 W and cosh p 3 W are linearly independent, with simple operation, set the corresponding coefficients in Eq. (22) to zero, respectively. There is 2 ­ °v  1 4 B ® 2 ° A ¯12 B

(24)

From (24), we get

A 3v  3 , B

v 1 4

And the solution of Eq. (16) is

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(25)

146

Zhao-ling Tao and Yang Yang

u ( x, t )

3v  3 ª v 1 º cosh « ( x  vt ) » ¬ 4 ¼

(26)

2

2.3. Solutions in Form of the Tanh Function The standard tanh method was introduced in [8-10] where the tanh is used as a new variable with all derivatives of a tanh are represented by tanh itself. Introducing a new independent variable

tanh(P[ ), [

Y

Dx  vt ,

(27)

leads to the derivatives change:

d d[

P (1  Y 2 )

d2 d[ 2

d , dY

2 P 2Y (1  Y 2 )

d d2 .  P 2 (1  Y 2 ) 2 dY dY 2

(28)

The tanh±coth method admits the use of the finite expansion

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M

u ( P[ )

S (Y )

M

¦ a k Y k  ¦ bk Y  k , k 0

(29)

k 1

where M is a positive integer, in most cases, that will be determined. a k and bk are unknown constants to be determined. Reconsider Eq. (3)

vuc  Duc  Duuc  D 3uccc

0.

(30)

Balance the linear terms of highest order u ccc with the highest order nonlinear terms uu c , we have

M  M 1 M  3 , which implies that M

u ( x, t )

S (Y )

(31)

2 . So a0  a1Y  a2Y 2  b1Y 1  b2Y 2 .

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(32)

Soliton Solutions of One KdV Equation

147

Now substitute (32) into (30), collect the coefficients of the powers of Y i ,  5 d i d 5 . Let all the coefficients be zero, and solve the resulting system of algebraic equations; we obtain the following three cases. Case 1.

c 8D 3 P 2  D  Da0 ,

a1

0,

a2

0, b1

0, b2

12D 2 P 2 ,

(33)

the solution for Case 1 is

u1 ( x , t )

8D 3 P 2  D  c

 12D 2 P 2 tanh  2 ( P (D x  vt )) .

D

(34)

Case 2.

c

8D 3 P 2  D  Da0 ,

a1

0,

a2

12D 2 P 2 , b1

0, b2

0,

(35)

the solution for Case 2 is

u 2 ( x, t )

8D 3 P 2  D  c

D

 12D 2 P 2 tanh 2 ( P (D x  vt )) .

(36)

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Case 3.

c 8D 3 P 2  D  Da0 ,

a1

0,

12D 2 P 2 , b1

a2

0, b2

12D 2 P 2 ,(37)

the solution for Case 3 is

u1 ( x , t ) Set P Ref [16]).

8D 3 P 2  D  c

D

>

 12D 2 P 2 tanh 2 ( P (D x  vt ))  tanh  2 ( P (D x  vt ))

@

(38)

1 , the solutions (34) and (36) are the same as that in Ref [16] (See Eq.(28) in

3. REMARKS It is well known that KdV equation attracts much attention recently. Here we join to study soliton solution for one KdV equation. Several methods including the exp-function method, the tanh method and the solitary wave ansatz method are involved. To our knowledge, not only the same solutions as those in the open literature, but also some new are arrived at. The methods¶ calculation process is simple and convenient, and the methods used can be used to many other nonlinear equations.

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Zhao-ling Tao and Yang Yang

4. ACKNOWLEDGMENTS This research is jointly sponsored by National natural science foundation in China (No:71071073),Meteorology Commonweal Special Project, Ministry of Science and Technology of China (No: GYHY200806029) and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (NO: 708051).

REFERENCES [1] [2]

[3] [4] [5]

[6]

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[7] [8] [9] [10] [11]

[12]

[13] [14] [15]

Ji-Huan He, Xu-Hong Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30(3), 2006, 700-708. Zhou XW, Wen YX, He JH. Exp-function method to solve the nonlinear dispersive K(m,n) equations, International Journal of Nonlinear Sciences and Numerical Simulation, 9(3),2008, 301-306. S.D. Zhu, Exp-function method for the Hybrid-Lattice system, International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), 2007, 461-464. S.D. Zhu, Exp-function method for the discrete mKdV lattice, International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), 2007, 465-468. V.O. Vakhnenko , E.J. Parkes , A.J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalized Vakhnenko equation, Chaos, Solitons and Fractals, 17(4), 2003, 683±692. Shu-Fang Deng, Zhen-Yun Qin, Darboux and Bäcklund transformations for the nonisospectral KP equation, Physics Letters A, 357(6), 2006, 467±474. Xue mei Li, Ai hua Chen, Darboux transformation and multi-soliton solutions of Boussinesq±Burgers equation, Physics Letters A, 342(5-6), 2005, 413±420. Abdul-Majid Wazwaz, The tanh±coth method for solitons, Applied Mathematics and Computation, 188(2), 2007, 1467-1475. Abdul-Majid Wazwaz, The tanh±coth and the sech methods for exact solutions of the Jaulent-Miodek equation, Physics Letters A, 366(1-2), 2007, 85-90. Abdul-Majid Wazwaz, The tanh-coth and the sine±cosine methods for kinks, Applied Mathematics and Computation, 195(1), 2008, 24-33. Mingliang Wang, Yubin Zhou, Zhibin Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216(1-5), 1996, 67-75. Anjan Biswas. 1-Soliton solution of the generalized Zakharov±Kuznetsov equation with nonlinear dispersion and time-dependent coefficients. Physics Letters A, 373(33), 2009, 2931-2934. Anjan Biswas. 1-Soliton solution of the generalized Camassa±Holm Kadomtsev± Petviashvili equation. Commun Nonlinear Sci Numer Simulat, 14(6), 2009, 2524-2527. W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (7) (1992) 650±654. E. Fan, Y.C. Hon, Generalized tanh method extended to special types of nonlinear equations,Z. Naturforsch 57a (2002) 692±700.

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[16] Ibrahim E. Inan, Exact solutions for coupled KdV equation and KdV equations, Physics Letters A, 371(1-2), 2007, 90±95. [17] Anjan Biswas. 1-soliton solution. Phys Lett A , 372(25), 2008, 4601±4602.

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 ©2011 Nova Science Publishers, Inc.

Chapter 6

NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS Jiunn-Lin Wu Dept. of Computer Science and Engineering, National Chung Hsing University

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ABSTRACT Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body. Fractional calculus is an extension of derivatives and integrals to noninteger orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this chapter, the numerical methods for fractional partial differential equations will be reviewed, especially the approach based on the operational matrices of the orthogonal functions. It transforms the problem to a simple Lyapunov matrix equation solving. Advantages of the operational method include (1) the computation is simple and computer oriented, (2) it can solve the partial differential equations numerically, even the ones with fractional order, (3) the scope of application is wide and (4) the step size used could be large and the result obtained is still satisfactory. The numerically unstable problem does not occur in the operational method.

1. INTRODUCTION The numerical treatment of partial differential equations is, by itself, a vast subject. Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body, and so on [1]. In most mathematics books, partial differential equations (PDEs) are classified into the three categories, hyperbolic, parabolic, and elliptic, on the basis of their characteristics, or

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152

Jiunn-Lin Wu

curves of information propagation. The prototypical example of a hyperbolic equation is the one-dimensional wave equation

w 2u wt 2

v2

w 2u wx 2

where v =constant is the velocity of wave propagation. The prototypical parabolic equation is the diffusion equation:

wu wt

w § wu · ¨D ¸ wx © wx ¹

where D is the diffusion coefficient. The prototypical elliptical equation is the Poisson equation

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w 2u w 2u  wx 2 wy 2

U  x, y

where the source term U is given. If the source term is equal to zero, the equation is /DSODFH¶VHTXDWLRQ Analytical solutions to partial differential equations exist only for a limited number of cases, whereas so called numerical methods can be used to obtain an approximate solution. The two most common numerical methods are finite difference and finite element methods. The main idea of the above two numerical methods is to calculate the values of the unknown head only at limited (finite) number of points which are usually called nodes. Analytical solution ±if it exists- could be used to calculate values of the unknown at any point in the problem domain. The use of fractional calculus of modelling physical systems has been widely considered in the last decades [2]. The partial differential equations involving derivatives with noninteger orders have shown to be adequate models for various physical phenomena in areas such as damping laws, diffusion processes, etc. Other applications include electromagnetics, electrochemistry, arterial science, the theory of ultra-slow processes and finance. However, a small number of algorithms for the numerical solution of fractional differential equations have been suggested. Podlubny [2] used the Laplace Transform method to solve the fractional differential equations numerically with the Riemann-Liouville (RL) derivatives definition, as well as the fractional partial differential equations with constant coefficients. Podlubny [2] VXJJHVWHGDJHQHUDOL]DWLRQRIWKHGHILQLWLRQRI*UHHQ¶VIXQFWLRQWRVROYHWKH problems of the fractional-order systems and controllers. If the Laplace transform methods or the fractional *UHHQ¶VIXQFWLRQDUHLPSRVVLEOHMeerschaert and Tadjeran [3] proposed the finite difference method to solve the numerical solution of two-sided space-fractional partial differential equations. By discretizing the fractional derivatives by the shifted Grunwald estimates, it provides accurate approximations and the numerical solutions are also stable. To deal with nondifferentiable functions, Jumarie [4, 5] derived new expressions for fractional Taylor¶s

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Numerical Solution of Fractional Partial Differential Equations

153

series and modified Riemann-Liouville definition. Baeumer et al. [6] used the sequential operator splitting scheme to solve the fractional reaction-diffusion equations numerically. They defined the fractional derivatives by the generalized Levy representation. However most of the above methods are based on the approximation of the two definitions of fractional derivatives, Grunwald-Letnikov (GL) definition and the Riemann-Liouville (RL) definition [2, 7]. They are complicated and time consuming. In recent studies, many reports have been devoted to the development of algebraic methods for the analysis, identification, and optimization of systems. The aim of these studies has been to obtain effective algorithms that are suitable for the digital computer. Their major effort has been concentrated on the methods of the orthogonal polynomial and functions. Typical examples are the applications of Walsh functions [8], block pulse functions [9], Laguerre polynomials [10], Legendre polynomials [11], Chebyshev polynomials [12], Taylor series [13], Fourier series [14] and Haar wavelets functions [15]. The main characteristic of the operational method is to convert a differential equation into an algebraic one. It not only simplifies the problem but also speeds up the computation. Let us start with the integral property of the basic orthonormal matrix, ) (t ) . We write the following approximation, t

t

t

 ³ ) W dW # Q)k ) t ³ 0 ³0

0  k

times

where ) (t )

&

k

&

>I (t ) 0

&

&

I1 (t ) ... I m 1 (t )

@

T

&

&

in which the elements I 0 (t ) , I1 (t ) , ... ,

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I m1 (t ) are the discrete representation of the basis functions which are orthogonal on the interval [0,1) and Q ) is the operational matrix for integration of )t . In view of the simple structure of the operational matrix of integration Q ) , the computation of the powers of Q ) is very easy. This elegant operational property is useful for the simplification of problems. Using the operational matrix of an orthogonal function to perform integration for solving, identifying and optimizing a linear dynamic system has several advantages: (1) the method is computer oriented, thus solving higher order differential equation becomes a matter of dimension increasing; (2) the solution is a multi-resolution type; (3) the answer is convergent, even the size of increment is very large. In this chapter, we present a new method based on the operational matrices of orthogonal functions to solve the fractional partial differential equations numerically. It transforms the problem into a simple Lyapunov matrix equation. Two examples solved by the new method are demonstrated. Advantages of the presented operational method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; (3) the step size used could be large and the result obtained is still satisfactory; (4) the numerically unstable problem never occurs in our method; and (5) it can solve the partial differential equations with not only integer but also fractional (non-integer) orders.

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Jiunn-Lin Wu

2. THE UNIFIED DERIVATION METHOD OF OPERATIONAL MATRICES The conventional method of deriving the operational matrix is not only complicated but also time-consuming. And the main problem is that they are not unified, in other words, the method for deriving the operational matrix of one orthogonal functLRQ FDQ¶W EH DSSOLHG WR derive the operational matrix of any other orthogonal functions. In this section, we present a new unified method to derive the operational matrix of orthogonal functions for integration and differentiation. It can derive the operational matrix not only with the integer order but also with the non-integer order. This method is simple, efficient, computer-oriented and unified in which many orthogonal functions can be used.

B0 (t )

1 t

1/4 B1 (t )

1

1/4

t

1/2

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B2 (t )

1

1/2

t

3/4

B3 (t )

1

3/4 Figure 1. Block pulse functions for

1

t

m 4.

Although historically the idea of the operational matrix was established via the Walsh function, logically the one via the block pulse function is more basic [17]. Let us start with the block pulse function. In the duration [0,1) , the following function is defined:

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Numerical Solution of Fractional Partial Differential Equations

bi t

­ ª i i  1º °1, t  « , » ¬m m ¼ ® °0, elsewhere ¯

i

0,1,2,, m  1

155

(1)

2D usually, where D is a positive integer. The 0,1,2,, m  1 are called block functions in >0,1@ . If m 4 , for

Here m is the dimension and m functions bi t , i

example, we have a graph shown in Fig.1. For any >a, b @ , a similar block pulse function can be defined. The corresponding matrix representation of Fig.1 can be written as follows:

B4

ª1 «0 « «0 « ¬0

0

0

1

0

0

1

0

0

0º

0» » 0» » 1¼

(2)

³

B0 (t )dt

1/4

1/4

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³

1

t

B1 (t )dt

1/4

1/4

³

1

t

B2 (t )dt

1/4

1/2

³

1

t

B3 (t )dt

1/4

3/4 Figure 2. The integration of the Block pulse functions for

1

t

m 4.

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156

Jiunn-Lin Wu Performing the integration on the functions in Fig.1 obtains Fig.2. That is, t

³ B W dW # Q 4

0

B4

B4

(3)

where

QB 4

ª1 «8 « «0 « « «0 « «0 ¬

1

1

4 1

4 1

8

4 1

0

8

0

0

1º 4» 1» » 4» 1» 4» 1» » 8¼

ª1 «2 « «0 1« 4 «0 « « «0 ¬

1 1 2 0 0

1 1 1 2 0

º 1» » 1» » 1» » 1» » 2¼

(4)

Q B4 is the operational matrix of thh block pulse functions for integration with m Eq. (4) can be easily extended to higher dimensions, m

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QB

ª1 «2 « 1 «0 m « « «0 «¬



1

  1 0 2 

0

4.

2D and D is a positive integer,

º 1» » » 1» » 1» 2 »¼ mu m

(5)

The operational matrix of integration of Eq. (3) is so intuitive that it is obtained by inspection. Now we consider the integration of a general orthogonal function ) (t ) given by t

³ ) (W )dW # Q

)

0

where

Q)

˜ ) (t )

(6)

is the operational matrix for integration of ) (t ) . Since the block pulse matrix

B(t ) is the identity matrix with the appropriate order, Eq. (6) can be expressed as

³

t

0

) (W ) dW

³

t

0

) ˜ B (W ) dW

t

) ˜ ³ B (W ) dW

(7)

0

If )t in Eq. (6) is the block pulse function (i.e. )(t )

B(t ) ), we have

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Numerical Solution of Fractional Partial Differential Equations

157

t

³ B W dW # Q B t B4

0

(8)

where QB is the operational matrix for integration of the block pulse function. From Eqs. (7) and (8), we obtain t

³ ) (W )dW

) ˜ QB ˜ B ( t )

0

(9)

B(t ) in Eq. (9) is an identity matrix and can be omitted, we have t

³ ) (W )dW

) ˜ QB

0

(10)

The left hand sides of equations Eqs. (6) and (10) are identical; so are their right hand sides, we obtain

Q) ˜ )

) ˜ QB

(11)

) ˜ QB ˜ ) 1

(12)

Thus,

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Q)

Formula (12) is new and its form is much simpler than all precious works [17]. And it can be extended to derive the operational matrix of any other orthogonal functions. If the bases

&

&

&

I0 (t ), I1 (t ),, Im1 (t ) are not only orthogonal but also orthonormal and real, ) (t ) is called the unitary matrix, and it has the wonderful property: the inverse of ) (t ) is equal to its 1

transpose, (i.e. ) (t )

Q)

)T (t ) ), thus Eq. (12) becomes

)(t ) ˜ QB ˜ ) T (t )

(13)

This above equation not only is derived in a unified framework but also needs much less computation than those obtained by all previous works [17].

3. THE OPERATIONAL MATRICES OF HAAR WAVELETS As mentioned above, the main benefit of operational matrix is to transfer the differential equations into the algebraic ones or the Lyaponov form. Therefore they can be solved by the computer oriented methods; it is much easier and more time efficient. Wavelets have become an increasingly popular tool in the computational sciences. They have numerous applications in a wide range of areas such as signal analysis, data compression

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Jiunn-Lin Wu

and many others. The Haar wavelets have the following features: (1) highly energy packing; (2) the base functions are consisted of three simple integers, 0 and ±1, and 1 only. The properties are useful in speeding up the computation. So we will use the Haar wavelets and its operational matrix for demonstration throughout this chapter. Let us begin by briefly reviewing the Haar functions [18, 19]. The Haar functions are an orthogonal family of switched rectangular waveforms where amplitudes can differ from one function to another. They are defined in the interval [0,1) by

1

h 0 t

m ­ j k  1 k 1 2 °2 2 , dt  2j 2j ° ° j k  1 1 ° 2 dt  k 2 ® 2 , j 2 2j m ° otherwise in [ 0,1) ° 0, ° ° ¯

hi t

0, 1, 2,  , m  1 , m

where i

(14)

2 D and D is a positive integer. j and k represent the

2 j  k  1 . Theoretically, this set of functions is complete [18]. In the construction, h0 (t ) is called the scaling function and h1 (t ) the Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

integer decomposition of the index i , i.e. i

mother wavelet. There are two basic operations involved in this set of Haar functions: (1) translation and (2) dilation [19]. Starting from the mother wavelet, h1 (t ) , compression and translation are performed to obtain h2 (t ) and h3 (t ) as shown in Fig. 3. Any function y (t ) which is square integrable in the interval 0 d t  1 , that is 1

³y

2

0

(t ) dt  f

(15)

, can be expanded into Haar series by

y (t ) where ci

c0h0 (t )  c1h1 (t )  c2h2 (t )  c3h3 (t )  .....

(16)

1

³ y (t )h (t )dt . In usual, the series expansion of Eq. (16) contains infinite terms 0

i

for a general smooth function y (t ) . However, if y (t ) is approximated as piecewise constant during each subinterval, Eq. (16) will be terminated at finite terms, i.e.

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Numerical Solution of Fractional Partial Differential Equations

159

h0 t

1 12

0

14

1/ 2

3/ 4

1

14

1/ 2

3/ 4

1

1/ 2

3/ 4

1

1/ 2

3/ 4

1

t

1 2

-1 h1 t

1 12

0 1 2

t

-1 h2 t

1 1 2 12 0

14

1 2 1 2 -1

t

h3 t

1 1 2 12 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0

14

1 2 1 2 -1

m

Figure 3. Haar wavelet functions with

t

4.

m 1

¦ c h (t )

y (t )

(17)

i i

i 0

The continuous curve of Eq. (17) can be written into the discrete form by

& y

where

& yT

& & & c0 h0  c1h1  ... cm1hm1

m

>y 0

is the dimension and usually

y1

...

(18)

m

2D , D

is a positive integer.

y m 1 @ is the discrete form of the continuous function y(t ) , the

discrete values yi are obtained by sampling the continuous curve y (t ) at a space 1 Similarly,

& h0T

>h

0,0

h0,1

... h0, m 1 @ ,

& h1T

>h

1, 0

h1,1

... h1, m 1 @

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«

m

. 

160

Jiunn-Lin Wu

& hmT1

>h

... hm 1, m 1 @ are the discrete form of the Haar wavelet bases; the

hm 1,1

m 1, 0

discrete values are taken from the continuous curves h0 (t ) , h1 (t ) « h m1 (t ) respectively. The Haar wavelet matrix H of dimension m is defined by:

H

& ª h 0T º « &T » « h1 » « ... » « &T » ¬« h m 1 ¼»

ª h0,0 « h « 1, 0 « ... « ¬ h m 1, 0

h 0 ,1

...

h1,1

...

...

...

h m 1,1

...

&T

Eq. (18) is then expressed as y

h 0 , m 1 º h1, m 1 » » ... » » h m 1, m 1 ¼

(19)

& & c T ˜ H , where c &T

&

the coefficient vector of y and it can be calculated from c

>c 0

c1

... c m 1 @ is called T

& y T ˜ H 1 . Similarly, a two-

dimensional function y( x, t ) which is square integrable in the interval 0 d x d 1 and

0 d t d 1 can be expanded into Haar series by m 1 m 1

¦¦c

y ( x, t )

ij

hi ( x ) h j (t )

(20)

i 0 j 0

1

³

where c ij

0

1

y ( x, t ) hi ( x ) dx ˜ ³ y ( x, t ) h j (t ) dt . Eq. (20) can be written into the discrete 0

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form by

Y  x, t

where C

H T x ˜ C ˜ H t ª c 0,0 « c « 1, 0 «  « ¬c m 1, 0

c 0 ,1 c1,1  c m 1,1

(21)



c 0 , m 1 º  c1, m 1 » » is the coefficient matrix of Y , and it can be   » »  c m 1, m 1 ¼

calculated by

C

H ˜Y ˜ H T

For deriving the operational matrix of Haar wavelets, we let ) obtain

QH

(22) H in Eq.(13), and

H ˜ QB ˜ H T

where QH is the operational matrix for integration of H . For example, the operational matrix of the Haar wavelet in the case of m

4 is given by

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Numerical Solution of Fractional Partial Differential Equations

H 4 ˜ QB4 ˜ H 4

QH 4 ª « « « « « « « « ¬«

1 2 1 2 1 2



0

1 2 1 2 1

1 2 1  2 0

2 1

0

2

ª 0 .5 « 0.25 « «0.0884 « ¬0.0884

161

T

1 2 1  2

º ª1 » « » «2 » «0 »˜1 « 0 » 4 «0 » « 1 » « » «0  2 ¼» ¬

 0.25

 0.0884

0

 0.0884

0.0884

0

 0.0884

0

1

1

1 2

1

0

1 2

0

0

º ª 1» « » « 1» « »˜« « 1» « » 1» « » « 2 ¼ ¬«

1 2 1 2 1



2 0

1 2 1 2 1

1 2 1  2 0

2 0

1 2

1 2 1  2

º » » » » 0 » » 1 » »  2 ¼»

T

 0.0884 º 0.0884 » » » 0 » 0 ¼

For any m 2D where D is a positive integer, we can establish the corresponding operational matrix accordingly.

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4. NUMERICAL EVALUATION OF FRACTIONAL CALCULUS Fractional calculus is a generalization of integration and differentiation to non-integer order In this section, we use the operational matrix of orthogonal functions to express the fractional derivatives. It consists of the following three steps [20]:

& f

f t into the discrete vector form & > f 0 f1  f m1 @ , where f is a column vector expression. & Step 2: Transfer the vector f into the Haar wavelet domain by using the Haar wavelets

Step 1: Sample the continuous function T

transform

& fT

& cT ˜ H

& where c is the coefficient vector and H is the Haar wavelet matrix. Step 3: For finding the numerical solution of

& dD f dt D

dD & (c ˜ H ) dt D

dD f dt D

(23)

, where D is a real number, we use,

& dDH c˜ D dt

From the definition of the operation matrix, it yields

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(24)

162

Jiunn-Lin Wu

dDH dt D

Q HD ˜ H

(25)

where Q H is the operational matrix for integration of Haar wavelets. Substituting Eq. (25) into Eq. (24) yields:

& dD f dt

D

dD & c ˜ H dt D

& c ˜ Q HD ˜ H

(26)

D

where Q H is the operational matrix with fractional order. And it can be derived from the following equation:

QHD

H ˜ QBD ˜ H t

(27)

D

where Q B is the operational matrix of the block pulse function for integration with the order

D. Let us take f1 t

t , and f 2 t

sin 2St as examples. We calculate the numerical 2S

solution of fractional derivatives, D f1 x and D f 2 x , where 0 d D d 1, of f1 t and D

D

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f 2 t respectively by using Eq. (26). The results are shown in Figs. 4 and 5 respectively.

Figure 4. The numerical solution of the fractional derivative, and in case of Haar wavelets basis,

m

D D f (t ) , where y t

t , 0 d D d 1,

64 . The exact solution in terms of Riemann-Liouville

* 2 1D t [2]. definition is * 2  D Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Numerical Solution of Fractional Partial Differential Equations

Figure 5. The numerical solution of the fractional derivative,

0 d D d 1, and in case of Haar wavelets basis, m

D D f (t ) , where f t

163

sin 2St , 2S

64 .

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5. NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS The numerical method for the linear fractional partial differential equation is illustrated in this section, it is based on the operational matrices of Haar wavelets. Obviously, Eq. (26) can be extended to the case of a function with two variables, the integration of

Y ( x, t )

H T x ˜ C ˜ H t with respect to variable t yields D

§w· ¨ ¸ Y © wt ¹

D

§w· T ¨ ¸ H  x ˜ C ˜ H t © wt ¹

ª§ w · D º H  x ˜ C ˜ «¨ ¸ H t » ¬«© wt ¹ ¼» T

(28)

H T  x ˜ C ˜ Q HD ˜ H t or D

§w· ¨ ¸ Y © wt ¹

H T C ˜ Q HD ˜ H

(29)

Similarly, the fractional integration order E of Y ( x, t ) with respect to variable x can be expressed as: Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

164

Jiunn-Lin Wu E

§ w · ¨ ¸ © wx ¹

§ w · ¨ ¸ Y © wx ¹

>Q

E H

E

T

ª§ w ·  E º «¨ ¸ H  x » ˜ C ˜ H t «¬© wx ¹ »¼

H x ˜ C ˜ H t T

@



T

˜ H  x ˜ C ˜ H t

H T  x ˜ QHE

T

˜ C ˜ H t

(30)



H T ˜ QHE

T

˜C ˜ H

In general, performing the double integration to the function Y ( x, t ) with fractional order D to variable t and fractional order E to variable x , we obtain D

E

§w· § w · ¨ ¸ ¨ ¸ Y © wt ¹ © wx ¹



H T ˜ QHE

T

˜ C ˜ QHD ˜ H

(31)

Equations (29), (30) and (31) are the main formulas for solving a fractional partial differential equation numerically via the Haar wavelet operational method [21]. For illustrating the above procedure, we consider the following first-order PDE

wy wy  wx wt

k,

with initial condition y( x,0) a and boundary condition y(0, t ) b , where a , b and k are constants. First we integrate the above equation with respect to t and yield t

wy

t

³ wx dt   y ( x, t )  y ( x,0) ³ kdt

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0

(32)

0

then integrate Eq. (32) with respect to x , we obtain t x

³³

0 0

x x wy dxdt  ³ ydx a ³ 1dx 0 0 wx

t x

³³

0 0

k dxdt

(33)

or t

x

0

0

³ > y ( x, t )  y (0, t )@dt  ³

x

t x

ydx a ³ 1dx

³³

0

0 0

k dxdt

(34)

Rearranging gives: t

x

0

0

³ ydt  ³

ydx

x

t

x

0

0

0

a ³ 1dx  b ³ 1dt  k ³

t

³ 1dtdx 0

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(35)

Numerical Solution of Fractional Partial Differential Equations

165

For solving the partial differential equation in Eq. (31) by the presented operational method, we let Y  x, t (35), it gives:

H T x ˜ C ˜ H t and substitute Eqs. (25), (28) and (30) into Eq.

H T CQH H  H T QHT CH

a ˜ H T QHT JH  b ˜ H T JQH H  k ˜ H T QHT JQH H

(36)

where

ª1 «1 J is the coefficient matrix of « « « ¬1

1  1º 1  1» » , or J   » » 1  1¼ mum

ª1 «1 H ˜« « « ¬1

1  1º 1  1» » ˜ H T . By   » » 1  1¼ mum

multiplying H T to the right side and H to the left side of each term in Eq. (30), it yields:

QHT C  CQH

k ˜ QHT JQH  a ˜ QHT J  b ˜ JQH

(37)

Eq. (37) is the Lyapunov equation which can be solved by the packages [22, 23].

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6. EXAMPLES AND RESULTS The presented operational method can be used to evaluate the numerical solution of the linear fractional partial differential equation with constant coefficients, including the homogenous linear fractional partial difference equations and nonhomogeneous fractional partial difference equations [5].

Example 1. Solve the Following Linear Homogeneous Partial Differential Equation wy wy  wx wt

0 , x, t t 0 ,

(38)

2

with the boundary condition y (0, t ) t and y ( x,0) x . By integrating Eq. (38) with respect to variables x and t with the given conditions, we obtain: t

x

0

0

³ y ( x, t )dt  ³

y ( x, t ) dx

t t

x

x

x

0 0

0

0

0

³ ³ 1dtdt  2 ˜ ³ ³ ³ 1dxdx

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(39)

166

Jiunn-Lin Wu

If the Haar wavelets with order m 16 is used, the matrix form of Eq. (39) can be simplified as:

QHT 16 C16u16  C16u16QH16

J16QH2 16  2QH3 16 J16

(40)

Eq. (40) is a Lyapunov-type matrix equation, which can be solved and obtain the coefficient matrix C16u16 . The discrete form of y( x, t ) is then given by

Y16u16 ( x, t )

T H16 ˜ C16u16 ˜ H 16

The plot of Y16u16 ( x, t ) is shown in Fig. 6. The exact solution obtained from an analytical method is given by

y ( x, t )

[t  x  (t  x) 2 ] ˜ u (t  x)  (t  x) 2

(41)

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where u ˜ is the Heaviside step function. It is shown in Fig. 7 for comparison.

Figure 6. The approximate solution of the partial differential equation

y ( x,0)

wy wy  wx wt

0 , y (0, t )

x 2 , x, t t 0 in the case of m 16 .

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t,

Numerical Solution of Fractional Partial Differential Equations

Figure 7. The exact solution of the partial differential equation

y ( x,0)

wy wy  wx wt

0 , y(0, t )

167

t,

x 2 , x, t t 0 .

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Example 2. Solve the Following Nonhomogeneous Partial Differential Equation wy wy  w x wt

1 , x,t t 0 ,

with the initial conditions y(0, t ) If m

4, k

1, a

QHT 4 C 4u4  C 4u4 QH 4

0 and b

(42)

y( x,0)

0.

0 are used, Eq. (37) becomes

QHT 4 J 4 QH 4

Solving the above equation yields:

C 4u4

ª 1.3125 «  0 .5 « «  0.2652 « ¬  0.0884

 0 .5

 0.2652

0.3125

0.0884

0.0884

0.0625

0.0884

0

 0.0884 º 0.0884 » » » 0 » 0.0625 ¼

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168

Jiunn-Lin Wu The matrix form of y( x, t ) is then given by

H 4T ˜ C 4u4 ˜ H 4

Y4u4 ( x, t ) ª0.0625 « 0.125 « « 0.125 « ¬ 0.125

0.125

0.125

0.3125

0.375

0.375

0.5625

0.375

0.625

0.125 º 0.375 » » 0.625 » » 0.8125 ¼

The plot of Y4u4 ( x, t ) is shown in Fig. 8. The exact solution of the partial differential equation is given by

y ( x, t )

­t , x t t , ® ¯ x, x  t

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which is shown in Fig. 9.

Figure 8. The approximate solution of the partial differential equation

y (0, t )

y ( x,0)

0 , x, t t 0 in the case of m

wy wy  w x wt

1,

4.

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Numerical Solution of Fractional Partial Differential Equations

169

Figure 9. The exact solution of the partial differential equation.

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Example 3. Finally, We Use the Operational Method to Solve the Below Fractional Partial Differential Equation w 1/ 2 y w 1/ 2 y  w 1 / 2 x w 1 / 2t

1 , with the zero initial conditions, x, t t 0

(43)

In the case of m 16 , we get the numerical solution by solving the Lyapunov equation as below: T

1 §¨ Q 1 2 ·¸ C 2 16u16  C16u16 Q H16 H16 © ¹

T

§¨ Q 1 2 ·¸ ˜ J ˜ Q 1 2 4 H 16 © H16 ¹

(44)

, then yield

Y16u16 ( x, t )

T H16 ˜ C16u16 ˜ H 16

The plot of the numerical solution Y16u16 ( x, t ) is shown in Fig. 10. They demonstrate the simplicity, and powerfulness of the operational method.

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170

Jiunn-Lin Wu

w 1/ 2 y w 1/ 2 y Figure 10. The approximate solution of the partial differential equation 1 / 2  1 / 2 w x w t zero initial conditions, x, t t 0 in the case of m 16 .

1 , with the

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7. CONCLUSIONS In this chapter, a new numerical method for partial differential equation with fractional orders is presented. It is based on the operational matrices of orthogonal functions. Two examples of the fractional-order partial differential equations are used to demonstrate the powerfulness of the presented operational method. Advantages of the presented method include (1) it is much simpler than the conventional numerical method for fractional differential equations; (2) the computation is computer oriented; and (3) the step size used could be large and the result obtained is still satisfactory.

REFERENCES [1] [2] [3]

W. H. Press, S. A. Teukolsky, W. T. Vettering and B. P. Flannery: Numerical Recipes in C, the art of Scientific computing, second edition, Cambridge University Press, 1988. I. Podlubny, Fractional Differential Equations, Academic press, 1999. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56 (2006) 80-90.

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Numerical Solution of Fractional Partial Differential Equations [4]

[5]

[6]

[7]

[8] [9]

[10] [11]

[12]

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[13] [14]

[15] [16]

[17]

[18] [19] [20]

171

G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51 (2006) 1367-1376. G. Jumarie, Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, Journal of Applied Mathematics and Computing, 24 (2007) 31-48. B. Baeumera, M. Kov´acsa, and M. M. Meerschaertb, Numerical solutions for fractional reaction±diffusion equations, Computers and Mathematics with Applications, 55 (2008) 2212-2226. I. B. Petris, M. Vinagre, L. Dorcak and V. Fwliu, , Fractional Digital Control of a Heat Solid: Experimental Results, The Proceedings of International Carpathian Control Conference, Malenovice, Czech Republic, (2002) 365-370. C. F. Chen and C. H. Hsiao, Design of piecewise constant gains for optimal control via Walsh functions, IEEE Transactions on Automatic Control, AC-20 (1975) 596-603. C. F. Chen, Y. T. Tsay and T. T. Wu, Walsh operational matrices for fractional calculus and their application to distributed parameter systems, Journal of Franklin Institute, 503 (1977), 267-284. D. S. Shih, F. C. Kung and C. M. Chao, Laguerre series approach to the analysis of a linear control system incorporation observers, Int. J. Control, 43 (1986) 123-128. R. Y. Ckang and M. L. Wang, Legendre polynomials approximation to dynamic linear state equations with initial or boundary value condition, International Journal of Control, 40 (1984) 215-232. P. N. Paraskevopoulos, Chebyshev series approach to system identification analysis and optimal control, Journal of Franklin Institute, 316 (1983) 135-157. S. G. Mouroutsos and P. D. Sparis, Taylor series approach to system identification, analysis and optimal control, Journal of Franklin Institute, 319 (1985) 359-371. P. N. Paraskevopoulos, P. D. Sparis, and S. G. Mouroutsos, The Fourier series operational matrix of integration, International Journal of System Science, 16 (1985) 171-176. C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc. Control Theory and Applications, 144 (1997) 87-94. J. L. Wu, C. H. Chen, and C. F. Chen, Numerical inversion of Laplace transform using Haar wavelet operational matrices, IEEE Trans. on Circuits and Systems-Part I: Fundamental Theory and Applications, 48 (2001) 120-122. J. L. Wu, C. H. Chen, and C. F. Chen, Numerical inversion of Laplace transform using Haar wavelet operational matrices, IEEE Trans. on Circuits and Systems-Part I: Fundamental Theory and Applications, 48 (2001) 120-122. A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transform, Subbands and Wavelets, Academic Press, Inc., 1981, pp. 60-61. M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, Englewood Cliffs, New Jersey, 1995, pp. 32-34. J. L. Wu and C. H. Chen, A New Operational Approach for Solving Fractional Calculus and Fractional Differential Equations Numerically, IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences, Special Section on Discrete Mathematics and Its Applications, Vol. E87-A, No. 5, pp.1077-1082.

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

172

Jiunn-Lin Wu

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[21] - / :X ³$ :DYHOHW 2SHUDWLRQDO 0HWKRG IRU 6ROYLQJ )UDFWLRQDO 3DUWLal Differential (TXDWLRQV 1XPHULFDOO\´ Applied Mathematics and Computation, Vol. 214, Issue 1, August 2009, pp. 31-40. [22] R. H. Bartels and G. W. Stewart, Solution of the matrix equation AX+XB=C, Communications of The ACM, 15 (1972) 669-713. [23] S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, Second edition, W.H. Press, 1992.

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In: Partial Differential Equations Editor: Christopher L. Jang,

ISBN: 978-1-61122-858-8 c 2011 Nova Science Publishers, Inc. 

Chapter 7

B OUNDARY C ONTROL OF S YSTEMS D ESCRIBED BY PARTIAL D IFFERENTIAL E QUATIONS BY I NPUT-O UTPUT L INEARIZATION

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Ahmed Maidia, b , Moussa Diafa, Jean-Pierre Corrioub,∗ a Universit´e Mouloud MAMMERI, Facult´e de G´enie Electrique et d’Informatique, D´epartement Automatique, 15 000 Tizi-Ouzou, Alg´erie. b Nancy Universit´e, Laboratoire R´eactions et G´enie des Proc´ed´es, CNRS-ENSIC-INPL, 1, rue Grandville BP 20451, 54001 Nancy Cedex, France

1.

Introduction

The characteristic variables and parameters of many systems depend on space, so that their dynamic behavior leads to models by partial differential equations (PDE). PDE are adequate models for problems arising in many modern applications, for instance chemical and biological processes, thermochemical flow phenomena, fluid dynamics and vibrations. This kind of systems are termed distributed parameter systems or infinite dimensional systems and include the transport-reaction processes (Christofides, 2001a,b), particulate processes (Christofides, 2004; Christofides et al., 2007), processes involving fluid flows (Aamo et al., 2002; el Hak, 2000) and wave equation problems (Liu, 2010). Distributed parameter systems (DPS) occupy an important place in control theory and constitute an active research area with sustained interest (Padhi and Faruque Ali, 2009). The mathematical model of DPS is constituted by a set of PDEs accompanied by initial and boundary conditions that can be either homogeneous or not. From the point of view of control, when a boundary condition is not homogeneous, it is called boundary control. The examination of the DPS control literature shows that most proposed control methodologies deal with distributed control rather than boundary control of PDE systems, which remains an open problem. In the case of distributed control of PDE system, a series of actuators are attached to the body of the system, whereas in the case of boundary control and

∗ E-mail address: [email protected] (Author to whom the correspondence should be Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

addressed)

174

Ahmed Maidi, Moussa Diaf, Jean-Pierre Corriou et al.

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observation, one actuator and one sensor are needed. In practice, boundary control is actually implementable on a real system whereas distributed control is rarely implementable as such due to the infinite continuous variation of control which is theoretically required. Thus boundary control is easy and convenient for real implementation. In addition, it is an economic approach since it does not need a series of actuators attached to the inner spatial domain of the DPS but requires only one actuator and one sensor placed at the boundaries. Thus, for boundary control, two situations are possible. The first one occurs when the sensor and actuator are placed at the same boundary (collocated sensor and actuator) and the second situation when the sensor and actuator are placed at opposite boundaries (anti-collocated sensor and actuator). Design methodologies of boundary control of a system described by PDE can be split into two approaches (Christofides, 2001a; Ray, 1989). The first one called early lumping represents the conventional approach. It consists in performing a spatial discretization of the PDE to derive a set of ordinary differential equations (ODE) that constitute an approximation of the original PDE model, and the controller design is performed in the framework of the classical control theory of lumped parameter systems (LPS). To obtain the ODE model, different approximations of the PDE are possible such as finite difference methods (method of lines) or the method of weighted residuals (Galerkin’s method, collocation methods). It must be noted that through early lumping, the fundamental control theoretical properties (controllability, observability and stability) are lost (Christofides, 2001a; Ray, 1989). This generally leads to high dimension controllers which are difficult to implement (Christofides, 2001a). The second approach, termed as late lumping, uses the PDE model without approximation for the controller design. The approximation is performed only for implementation purposes of the controller. Late lumping allows the control designer to avoid losing the distributed nature of the PDE system and to take full advantage of its natural properties. However, direct handling of PDEs is difficult and generally leads to a state feedback control law, which requires the design of an observer for practical implementation. In recent years, several control methods that directly take into account the distributed nature of the processes have been developed and deal with distributed control rather than boundary control. In this chapter, the late lumping approach is considered to design a boundary control by means of an input-output linearization for systems described by PDEs. Both collocated and anti-collocated cases are addressed. Thus, two design approaches and two control strategies are presented. These design approaches use directly the PDE model and are based on the notion of characteristic index (Christofides, 2001a,b) which is a generalization with respect to distributed parameter systems of the concept of relative degree used in lumped parameter systems (Corriou, 2004; Isidori, 1995; Kravaris and Kantor, 1990a). The characteristic index characterizes the spatio-temporal interactions between the controlled and manipulated variables. Control approaches are illustrated by three applications concerning temperature control at the outlet of counter-current (case of collocated sensor and actuator) and co-current (case of anti-collocated sensor and actuator with boundary output) heat exchangers by manipulating the temperature of the entering external stream. The third example concerns the boundary control of the temperature of a heated rod at a given position (case of anti-collocated sensor and actuator with punctual output). The stability and practical implementation of the proposed control strategies are also discussed and their control Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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performances are evaluated through numerical simulation by considering the tracking and perturbation rejection problems. The chapter is structured as follows. In Section 2., the general boundary control problem of PDE systems is formulated. Then the design approaches of boundary controller in the case of collocated and anti-collocated sensor and actuator are developed in Section 3.. For the anti-collocated case, both boundary and punctual outputs are considered. The developed approaches are illustrated by applications for practical systems. The stability of the closedloop system is addressed in Section 4.. The practical implementation of the control laws is discussed in Section 5. whereas Section 6. is devoted to the conclusion.

2.

Boundary Control Problem of PDE System

Consider the PDE dynamical system with respect to one spatial dimension described by the following state-space representation ∂x(z, t) = Ax(z, t), ∂t

z ∈ Ω = [0, L]  L y(t) = Cx(z, t) = δ(z − zy ) x(z, t) dz = x(zy , t),

(1)

0

zy ∈ Ω = [0, L]

(2)

subject to the inhomogeneous boundary condition x(zu , t) = u(t), Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

and the initial condition

zu ∈ ∂Ω = {0, L}

x(z, 0) = x∗ (z)

(3) (4)

where x(z, t) is the vector of state variables, u(t) is the manipulated variable, y(t) is both the controlled and measured variable. A and C are bounded linear operators. The variables z and t denote position and time respectively. ∂Ω = {0, L} are the boundaries of the spatial domain Ω = [0, L]. zu and zy are the positions of the actuator and sensor, respectively. δ(z) is the Dirac function. As the manipulated variable appears only in the boundary condition, the boundary condition is inhomogeneous and the control problem is termed boundary control with Dirichlet actuation. In addition, the operator C defines a punctual output if 0 < zy < L or a boundary output if zy ∈ ∂Ω. Thus, according to the given positions zu and zy , the sensor and actuator can be either collocated when zy = zu or anti-collocated when zy = zu . Another type of controlled variable that will be considered in this chapter is defined as the spatial weighted average given as  L y¯(t) = Lx(z, t) = c(z) x(z, t) dz (5) 0

where L is a linear operator and c(z) is a smooth shaping function. The different considered outputs are illustrated by Figure 1 in the case of the problem of controlling the temperature at a given position zy in a rod by manipulating the heating source u(t). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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y(t) = x(zy , t) 0 u(t)

L zy

  y¯(t) = L x(z, t)

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Figure 1. Type of outputs in case of heated rod. u(t) = x(zu , t) is the manipulated variable (source of heating) applied at zu = 0, y(t) = x(zy , t) punctual output (temperature at position zy ) and y¯(t) = L x(z, t) is the spatial weighted average temperature. The control problem consists to design a boundary control law u(t) that achieves a desired performance of the output y(t) using directly the PDE model to enhance the performances (Christofides, 2001a). Recently geometric control has proved to be very successful as a control approach of PDE system and successful applications are reported in literature (Christofides, 2001b; Christofides and Daoutidis, 1996; Maidi et al., 2008, 2009a; Shang et al., 2005). This is explained by the fact that control based on the geometric theory presents the advantage that the PDE model can be used in control design without any approximation, which allows to preserve the fundamental control theoretical properties as the distributed nature of the system is taken into account (Christofides, 2001a; Ray, 1989). The design approach adopted to solve this control problem uses some concepts of geometric control (Corriou, 2004; Isidori, 1995; Kravaris and Kantor, 1990a,b), and the control law is derived on the basis of the notion of the characteristic index (Christofides, 2001a; Christofides and Daoutidis, 1996). The characteristic index is the smallest order of the time derivative of a given controlled variable which explicitly depends on the manipulated variable. Its is a generalization of the concept of relative degree used in lumped parameter systems (ODE systems) for PDE systems. In the following, all used functions and operators are assumed to be defined in appropriate functional spaces.

3.

Control Law Design and Strategies

This section addresses the design methodology of boundary control for PDE systems, in the framework of geometric control, for both collocated and anti-collocated sensor and actuator cases. The design approaches are presented and illustrated by applications to physical systems (heat exchangers and heated rod). For the anti-collocated configuration, both posPartial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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sibles cases for the controlled output are considered. The former concerns the control of a boundary output (zy ∈ ∂Ω) and the later concerns the control of a punctual output (zy ∈ Ω). The design approaches are based on the characteristic index, denoted in the following by σ. If such an integer does not exist, then the characteristic index σ is infinite, which leads to loss of controllability of the controlled output y(t) by means of the manipulated variable u(t).

3.1.

Collocated Sensor and Actuator

The design of a boundary control law u(t) in the case of collocated sensor and actuator (zy = zu ) can be summarized as follows: Step 1. Determine the characteristic index of the controlled output y(t) by calculating its successive derivatives with respect to time: dy(t) d2 y(t) dσ y(t) , , . . . , dt dt2 dtσ

(6)

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Step 2. The characteristic index being equal to σ suggests requesting dynamics between an external input v(t) and the controlled output y(t) characterized by the following equation: dσ y(t) dσ−1 y(t) τσ + τ + · · · + y(t) = v(t) (7) σ−1 dtσ dtσ−1 Tune the adjustable parameters τσ , τσ−1 . . . , τ1 in order to ensure the stability and to enforce the desired performance of the closed-loop system described by (7). dσ y(t) dσ−1 y(t) dy(t) , , . . ., by their expressions caldtσ dtσ−1 dt culated in Step 1, and deduce the control law u(t) under the following state-feedback

Step 3. Substitute the derivatives

u(t) = K x(z, t) + F v(t)

(8)

where K and F are bounded operators. Step 4. The control robustness dealing with model and parameter uncertainty and unmodeled dynamics is provided in (8) by application of the linear control theory to the resulting linear system [external input v(t)–output y(t)] given by (7). Consequently, define the external input v(t) by a robust controller:  v(t) =

t 0

  G(t − ξ) y d (t) − y(t) dξ

(9)

where y d (t) is the desired set point of the controlled variable y(t) and the function G(t), for instance, can be chosen as the inverse of an appropriate transfer function. 2

The global control strategy for PDE system in case of collocated sensor and actuator is given by Figure 2. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Ahmed Maidi, Moussa Diaf, Jean-Pierre Corriou et al. y d (t) −

u(t)

v(t)

Robust controller

m

x(z, t)

C( . )

y(t)

Figure 2. Control strategy for PDE system in case of collocated sensor and actuator. 3.1.1.

Control of Counter-Current Heat Exchanger

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A fluid of constant density ρi and heat capacity Cpi flows through the internal tube of a counter-current heat exchanger (Fig. 3), of length L with a constant velocity vi . This fluid enters at temperature Ti (0, t) and exchanges heat with the external fluid, a non condensating vapor fluid, of constant density ρe and heat capacity Cpe , which flows in the jacket in the opposite direction with a velocity ve . This fluid enters at temperature Te (0, t) and leaves at temperature Te (L, t). At the outlet of the exchanger, the internal fluid leaves at temperature Ti (L, t). In the present study, the internal and external cross sections Si and Se of the heat exchanger are supposed to be uniform. Both temperatures Te (z, t) of the internal fluid and Ti (z, t) of the external fluid depend on time and spatial position along the tube. The objective is to control the internal fluid temperature Ti (L, t) at the outlet of the heat exchanger zy = L (boundary output), by manipulating the inlet external fluid temperature Te (L, t) at zu = L (boundary control). Note that zu = zL so that the sensor and the actuator are collocated. Thus, by denoting the control variable as u(t) = Te (L, t) and the controlled variable as y(t) = Ti (L, t), the dynamic model of the counter-current heat exchanger is described by following energy balances ∂Ti(z, t) ∂t ∂Te (z, t) ∂t

  ∂Ti(z, t) + hi Te (z, t) − Ti (z, t) ∂z   ∂Te (z, t) = ve + he Ti (z, t) − Te (z, t) ∂z = −vi

(10) (11)

with boundary conditions Ti (0, t) = Ti0 (t)

(12)

Te (L, t) = TeL (t) = u(t)

(13)

and as initial conditions the given temperature profiles

 y(t) =

0

Ti (z, 0) = Ti∗ (z)

(14)

Te (z, 0) =

(15)

L

Te∗ (z)

δ(z − L) Ti(z, t) dz = Ti(L, t)

(16)

Ui S Ue S and he = are heat transfer coefficients, vi and ve velocities, Ui and ρiSi Cpi ρe Se Cpe Ue overall heat transfer coefficients, S is the surface area per unit length devoted to heat transfer. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. hi =

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Te (L, t) Te (zi , t) Ti (0, t)

Ti (zi , t)

Ti (L, t)

Te (zi , t) Te (0, t) L

0

z

Figure 3. Counter-current heat exchanger. 3.1.2.

Control Law

Following the steps given in Section 3.1., it results Step 1. The calculation of the first derivative of the outlet temperature of the internal fluid gives   dy(t) ∂Ti (z, t)  dTi(z, t)  = −vi (t) + hi [Te (L, t) − Ti(L, t)] (17) = dt dt  ∂z 

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z=L

z=L

The manipulated variable u(t) = Te (L, t) appears linearly in the first time derivative (17) of the output y(t) = Ti (L, t), as for the heat exchanger hi = 0, thus the characteristic index is σ = 1. Step 2. As σ = 1, a first order linear behavior can be imposed for the pair [external input v(t)- controlled output y(t)] in closed-loop dy(t) + y(t) = v(t) (18) dt where τ is the desired time constant for the closed loop linear behavior between the external input v(t) and controlled output y(t). τ1

dy(t) Step 3. By substituting the first derivative by its expression (17), the following condt trol law writes   1  vi ∂Ti(z, t)  1 u(t) = 1 − Ti (L, t) + + v(t) (19)  τ1 hi hi ∂z τ1 hi z=L Step 4. For this application, the external input v(t) is defined by means of a PI controller as follows   1 t d d v(t) = Kc (y (t) − y(t)) + (y (ξ) − y(ξ)) dξ (20) τI 0 where Kc and τI are the proportional gain and the integral time constant, respectively.

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3.1.3.

Ahmed Maidi, Moussa Diaf, Jean-Pierre Corriou et al. Simulation Results

The nominal parameters of the studied heat exchanger are hi = 2.92 s−1 , ve = 1 m . s−1 , he = 5 s−1 , ve = 2 m . s−1 and L = 1 m. The initial conditions Ti (z, 0) and Te (z, 0) are taken as the spatial profiles of both fluid temperatures obtained at steady-state corresponding to Ti(0, t) = 25 ◦ C and Te (L, t) = 50 ◦ C. The parameters used for the control law are τ = 1 s, Kc = 7.8 and  τI = 1.03 s and are tuned in order for the denominator of the d closed loop y (t) − y(t) to approach a polynomial minimizing an ITAE criterion (Maidi et al., 2008). Note that, for simulation purpose, the control is held constant over the sampling period equal to 0.02 s. The simulation of the closed-loop system is performed using the method of lines (Van de Wouwer et al., 2004) based on the backward finite difference scheme by assuming a number of discretization points N = 100. In the simulation run, the capabilities of the controller for tracking the reference output and rejecting disturbances are studied. A change of the fluid temperature at the inlet of the heat exchanger is considered as a disturbance that affects the operation of the heat exchanger. Thus, a step set point at time t = 1 s corresponding to y d (t) = 60 ◦ C is imposed followed by a step disturbance of +20% of the entering temperature Ti(0, t) at time t = 5 s. To avoid the consequences due to brutal set point steps, the set point has been filtered by a first order filter with a time constant equal to 0.2 s. It appears that the controlled output (Fig. 4a) is not influenced by the disturbance, owing to a compensation by the variation of the manipulated variable (Fig. 4b). Nevertheless, if the temperature at any position zi with 0 < zi < L is examined, it can be noticed that the step disturbance visible at z = 0 is attenuated as the position zi moves from the inlet (zi = 0) towards the outlet of the heat exchanger (zi = L) where it becomes nearly undetectable (Fig. 4a). From the obtained results, it is clear that the output y(t) follows perfectly the imposed set point and the disturbance is rejected whereas the control moves of u(t) are physically acceptable. Again, the spatial profiles of temperature at time t = 10 s (Fig. 5) are typical of the behavior of a counter-current heat exchanger. The robustness of the control strategy is investigated by considering a perturbed model corresponding for a change of +20% of the heat transfer coefficient he and a change of +40% of the velocity ve (t) of the external fluid assumed as parametric uncertainties or sudden fluctuations since the heat exchangers are commonly connected to other thermal equipments which can affect the physical parameters and the operating flow rates. The parameter change occurs at t = 5 s after having imposed a step set point y d (t) = 50 ◦C at time t = 1 s. Fig. 6 shows the performance of the control system. It should be noted that in spite of these large parametric variations, the controlled output (Fig. 6a) is not influenced by the parameter changes and still tracks very correctly its reference set point, the parameter changes being compensated by admissible variations of the manipulated variable (Fig. 6b).

3.2.

Anti-collocated Sensor and Actuator

By considering a boundary or punctual output in the case of collocated sensor and actuator, the characteristic index always exists. For the case of anti-collocated sensor and actuator zu = zy , when a boundary or punctual output is considered, the problem of system controllability occurs (Maidi et al., 2009a,b, 2010). Thus, the characteristic index does not exits (σ → ∞) and this can be justified by the fact that the system is infinite dimensional. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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62

Controlled output y(t)=Ti(L, t) (°C)

60 58 56 54 52 50 48 Set point yd(t) Output y(t)

46 44 0

2

4

6

8

10

8

10

Time (s) (a) Controlled output y(t).

Manipulated input u(t)= Te(L, t) (°C)

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80 75 70 65 60 55 50 45 0

2

4

6 Time (s)

(b) Manipulated variable u(t).

Figure 4. Counter-current heat exchanger: set point tracking at t = 1s and disturbance rejection at t = 5s.

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T (z, t) i

65

Te(z, t)

Temperature (°C)

60 55 50 45 40 35 30 0

0.2

0.4 0.6 Length (m)

0.8

1

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Figure 5. Counter-current heat exchanger: spatial profiles of temperatures at t = 10 s. Thus to overcome this difficulty and design a control law, the control law must be inserted in the state equation (1) by means of Dirac function δ, to make the boundary condition homogeneous. Furthermore, the spatial weighted average output y¯(t) must be taken into account instead of the output y(t) in the design approach. This allows us to obtain a finite characteristic index (Maidi et al., 2009a,b, 2010). Then, to control the boundary or punctual output, a control strategy will be proposed. The different steps of the design approach are summarized as follows : Step 1. Make the boundary condition (3) homogeneous by inserting the manipulated variable u(t) in the state equation (1) using the Dirac function δ, Step 2. Choose the shaping function c(z) in (5) so that the condition of controllability is ensured (existence of a finite characteristic index) (Maidi et al., 2009a,b, 2010), Step 3. Define the spatial weighted average output y¯(t) as in (5), Step 4. Determine the characteristic index of the output y¯(t) by calculating its successive derivatives with respect to time t: d¯ y (t) d2 y¯(t) dσ y¯(t) , . . . , , dt dt2 dtσ

(21)

Step 5. The characteristic index σ suggests requesting a dynamics between an external input v(t) and the output y¯(t) characterized by the following equation: dσ y¯(t) dσ−1 y¯(t) + τ + · · · + y¯(t) = v(t) σ−1 σ Partial Differential Equations: Theory, Analysis and Applicationsdt : Theory, Analysis and Applications, dtσ−1 Nova Science Publishers, Incorporated, 2011. τσ

(22)

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Controlled output y(t)=Ti(L, t) (°C)

51 50 49 48 47 46 45 44 0

Set point yd(t) Output y(t) 5

10

15

10

15

Time (s) (a) Controlled output y(t).

Manipulated input u(t)= Te(L, t) (°C)

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62 60 58 56 54 52 50 48 0

5 Time (s) (b) Manipulated variable u(t).

Figure 6. Counter-current heat exchanger: robustness study, control performance for both a change of +20% of the heat transfer coefficient and a change of +40% of the external fluid velocity at t = 5s.

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Ahmed Maidi, Moussa Diaf, Jean-Pierre Corriou et al.

y d (t) −

External robust controller

y¯d (t)

Internal Robust controller

−

v(t)

u(t)

m

x(z, t)

L( . )

y¯(t)

C( . )

y(t)

Figure 7. Control strategy for PDE system in case of anti-collocated sensor and actuator. Choose the adjustable parameters τσ , τσ−1 . . . , τ1 in order to ensure the stability and to enforce the desired performance of the closed-loop system described by (22). dσ y¯(t) dσ−1 y¯(t) d¯ y (t) , , . . ., by their expressions calσ σ−1 dt dt dt culated in Step 4, and deduce the control law u(t) under the state-feedback form (8).

Step 6. Substitute the derivatives

Step 7. Define the external input v(t) by an internal robust controller:  t

 v(t) = G1 (t − ξ) y¯d (t) − y¯(t) dξ 0

(23)

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where y¯d (t) is the desired set point of the output y¯(t) and the function G1 (t), for example, can be chosen as the inverse of an appropriate transfer function. Step 8. Define the desired set point y¯d (t) of the output y¯(t) by an external robust controller:  t

 d y¯ (t) = G2 (t − ξ) y d (t) − y(t) dξ (24) 0

with respect to y d (t) desired set point of the boundary or the punctual controlled output y(t). 2 The global control strategy of an anti-collocated sensor and actuator PDE system is summarized in Fig. 7 3.2.1.

Boundary Output Case: Co-current Heat Exchanger

Contrary to the counter-current heat exchanger, in the case of a co-current heat exchanger, both fluids move in the same direction. Thus, the PDE model takes the form   ∂Ti (z, t) ∂Ti (z, t) (25) = −vi + hi Te (z, t) − Ti(z, t) ∂t ∂z   ∂Te (z, t) ∂Te(z, t) (26) = −ve + he Ti (z, t) − Te (z, t) ∂t ∂z with boundary conditions Ti (0, t) = Ti0 (t)

(27)

T (0, t) = T (t) = u(t)

(28)

e e0 Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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185

and initial conditions

 y(t) =

0

Ti (z, 0) = Ti∗ (z)

(29)

Te (z, 0) =

(30)

L

Te∗ (z)

δ(z − L) Ti(z, t) dz = Ti(L, t)

(31)

The objective is to control the internal fluid temperature y(t) = Ti(L, t), at the outlet of the heat exchanger zy = L, by manipulating the inlet external fluid temperature u(t) = Te (0, t) at zu = 0. Note that zu = zL , so the sensor and the actuator are anti-collocated. For cocurrent heat exchanger, the controlled variable y(t) is termed boundary output. Control Law The design approach presented in Section 3.2. gives: Step 1. Using the Dirac δ function, the manipulated variable is inserted in equation (26) as follows (Maidi et al., 2009b, 2010)   ∂Te (z, t) ∂Te (z, t) = −ve + he Ti(z, t) − Te (z, t) + ve δ(z) u(t) ∂t ∂z

(32)

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To obtain (32), the initial values of the variables must be set as Te (0− , t) = 0 and Ti (0− , t) = 0

(33)

so that equation (32) appears as the final form of a punctual control. Step 2. The shaping function that ensures the controllability condition is c(z) = L − z (Maidi et al., 2009a). Step 3. The spatial weighted average output is then  y¯(t) =

L 0

c(z) Ti(z, t) dz

(34)

Step 4. Calculation of the characteristic index d¯ y(t) dt



∂Ti(z, t) dz ∂t 0

 L ∂Ti(z, t) = + hi [Te (z, t) − Ti (z, t)] dz c(z) −vi ∂z Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and Applications 0 : Theory, Analysis and Applications, =

L

c(z)

(35)

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 L 2 ∂Te(z, t) ∂Ti(z, t) dym (t) ∂ ∂Ti(z, t) = c(z) −vi + hi − dz dt2 ∂t ∂z ∂t ∂t 

0 L ∂ ∂Ti(z, t) = c(z) −vi ∂z ∂t 0 ∂Te (z, t) +hi −ve + he [Ti(z, t) − Te (z, t)] + ve b(z) u(t) ∂z

∂Ti(z, t) +vi − hi [Te (z, t) − Ti(z, t)] dz  L ∂z  ∂ 2 Ti(z, t) ∂Ti (z, t) ∂Te (z, t) = c(z) vi2 − hi (vi + ve ) + 2hi vi 2 ∂z ∂z ∂z 0  L +hi (he + hi )(Ti(z, t) − Te (z, t) dz + hi ve c(z) b(z) dz u(t) 0

To simplify the notations, the second derivative of the output is set as  L 2 (t) dym = Ji Ti (z, t) + Je Te (z, t) + hi ve c(z) b(z) dz u(t) dt2 0

(36)

(37)

Note that the manipulated variable u(t) appears in the second derivative and since L 0 c(z) δ(z) dz = L = 0 with c(z) = L − z, thus the characteristic index is σ = 2. Step 5. As σ = 2, a second order linear behavior can be imposed for the pair [external input v(t)-output y¯(t)] in closed-loop, i.e. d¯ y 2 (t) d¯ y (t) + τ1 + y¯(t) = v(t) (38) dt2 dt where τ2 and τ2 are the desired time constants for the closed-loop linear behavior between the external input v(t) and output y¯.

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τ2

d¯ y (t) d¯ y 2 (t) Step 6. By substituting by their expressions (35) and (36), the control and dt dt2 law results   1 d¯ y(t) u(t) = v(t) − y¯(t) − τ1 − τ2 Ji Ti (z, t) − τ2 Je Te (z, t) (39) hi ve τ2 L dt where Ji and Je are bounded linear operators corresponding to (36) and (37). Step 7. The external input v(t) is defined by a PI controller   1 t d i d v(t) = Kc (¯ y (t) − y¯(t)) + i (¯ y (ξ) − y¯(ξ)) dξ τI 0

(40)

where Kci and τIi are the proportional gain and integral time constant, respectively. Step 8. The desired set point of the spatial weighted average output y¯d (t) is defined also by a PI controller with respect to the set point y d (t) of the output y(t)   t 1 d e d d y¯ (t) = Kc (y (t) − y(t)) + e (y (ξ) − y(ξ)) dξ (41) τI 0 where Kce and τIe are the proportional gain and integral time constant, respectively.

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Simulation Results The parameters of the studied heat exchanger are ve = 2 m . s−1 , vi = 1 m . s−1 , he = 1 s−1 , hi = 1 s−1 and L = 1 m (Friedly, 1972). The initial conditions Ti (z, 0) and Te (z, 0) are the steady state profiles resulting from the inlet temperatures Ti(0, t) = 25◦ C and Te (0, t) = 50◦ C. For the internal PI controller and the state feedback, the parameters are tuned following the same approach as for the counter-current heat exchanger. The obtained parameters are Kci = 0.0240, τIi = 0.0469 s, τ1 = 0.22 s, and τ2 = 0.05 s2 . The tuning of the external PI controller that provides the set point y¯d (t) is performed by a pole placement approach, using the approximation of the closed-loop desired performances between y¯d (t) and y¯(t) according to a second order system. The obtained parameters are Kce = 0.0268 and τIe = 0.3253 s. The other simulation conditions are the same as for the counter-current heat exchanger and similar tests of tracking and perturbation rejection are studied. For that reason, a step of +10% of the temperature of the entering internal fluid Ti(0, t) is imposed as a disturbance at time t = 30 s, after having imposed a step set point at time t = 1 s corresponding to y d (t) = 70 ◦C (Fig. 8a). Clearly, the control objective is achieved as the output y(t) follows perfectly the imposed set point and the controller behaves adequately to reject the disturbance effect with small well damped oscillations after t = 30s (Fig. 8a). The control moves of u(t) are physically admissible (Fig. 8b). Again, the spatial profiles at time t = 60 s given by Fig. 9 are typical of the behavior of a co-current heat exchanger. To evaluate the robustness of the controller, a change of +10% and −10% of the external and internal heat transfer coefficients he , hi , respectively is considered. The parameter variations occur at t = 30 s after having imposed a step set point y d (t) = 70 ◦C at time t = 1 s. Fig. 10 shows that in spite of these significant fluctuations, the control objective is achieved since the controlled output still tracks very correctly its reference trajectory with admissible moves of the manipulated variable. 3.2.2.

Punctual Output Case: Rod with Heat Exchange

The second example considered for the case of anti-collocated sensor and actuator is the rod exchanging heat with the environment (Fig. 11). The evolution of the temperature of the rod is described by the following one dimensional parabolic partial differential equation ∂T (z, t) ∂ 2 T (z, t) − β T (z, t) (42) = ∂t ∂z 2 where T (z, t) is the rod temperature. In (42), it is assumed that all variables T (z, t), z, t, L are dimensionless and in general, L = 1. β can be considered grossly as a heat transfer coefficient and is positive. The left-hand boundary condition is of Dirichlet type and the source of heating u(t) (manipulated control) is applied at the same left-hand end (zu = 0), leading to the following inhomogeneous boundary condition T (0, t) = u(t)

(43)

At the right-hand end (z = L), the boundary condition is of Dirichlet type T (L, t) = TL

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Controlled output y(t)=Ti(L, t) (°C)

80

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30 Set point yd(t) Output y(t) 20 0

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e

Manipulated input u(t)=T (0, t) (°C)

120

90 80 70 60 50 0

10

20

30 Time (s)

(b) Manipulated variable u(t).

Figure 8. Co-current heat exchanger: set point tracking and disturbance rejection at t = 30s.

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110 100

Temperature T(z,t) [°C]

90 80 70 60 50 40

Te(z,t)

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Ti(z,t) 20 0

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0.4 0.6 Length [m]

0.8

1

Figure 9. Co-current heat exchanger: spatial profiles of temperatures at t = 60 s. and the initial condition is given as the spatial profile

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T (z, 0) = T ∗ (z)

(45)

The objective is to design a control law u(t) that achieves a given set point y d (t) for the L temperature at location zy = . Thus, the controlled variable is a punctual output defined 2 as



 L L L y(t) = δ z− T (z, t) dz = T ,t (46) 2 2 0 Control law The design procedure described in Section 3.2. results in: Step 1. Using the Dirac δ function, the manipulated variable is inserted in equation (42) as follows (Maidi et al., 2009c) ∂T (z, t) ∂ 2 T (z, t) ˙ − β T (z, t) − δ(z) u(t) = ∂t ∂z 2

(47)

In a similar way to the case of the co-current heat exchanger, equation (47) is derived from (42) by applying a Laplace transform with respect to space which takes into account boundary conditions, followed by an inverse Laplace transform.

Step 2. The shaping function that ensures the controllability condition is c(z) = z (Maidi et al., 2009c). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Controlled output y(t)=Ti(L, t) (°C)

70 65 60 55 50 45 40 35 0

Set point yd(t) Output y(t) 10

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30 Time (s)

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(a) Controlled output y(t).

120

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e

Manipulated input u(t)=T (0, t) (°C)

110 100 90 80 70 60 50 40 0

10

20

30 Time (s)

(b) Manipulated variable u(t).

Figure 10. Co-current heat exchanger: robustness study, control performance for a change of +10% and −10% of the heat transfer coefficients he , hi , respectively at t = 30s.

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zy

Water bath

u(t) L 2

0

191

L

z

Figure 11. Rod exchanging heat with the environment with Dirichlet actuation and a fixed temperature at the right-hand end z = L. Step 3. The spatial weighted average output is  L c(z) T (z, t) dz y¯(t) = 0

(48)

Step 4. Calculation of the characteristic index  L  L  L  2 d¯ y(t) dT (z, t) ∂ T (z, t) ˙ z z − β T (z, t) dz− z δ(z) dz u(t) = dz = dt dt ∂z 2 0 0 0 (49) Note that the manipulated variable u(t) appears in this first derivative and since L ˙ 0 z δ(z) dz = −1 = 0, thus the characteristic index is σ = 1. Step 5. As σ = 1, a first order linear behavior can be imposed for the pair [external input v(t)-output y¯(t)] in closed-loop d¯ y (t) + y¯(t) = v(t) (50) dt where τ1 is the desired time constant for the closed-loop linear behavior between the input v(t) and output y¯.

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τ1

d¯ y(t) Step 6. Substituting by its expression (49) and after some calculations, the control dt law follows 

 L 1 ∂T (z, t)  1 u(t) = β − z T (z, t) dz − L + TL − T0 + v(t) (51)  τ1 ∂z τ1 0 z=L Step 7. The external input v(t) is defined by a PI controller   1 t d i d v(t) = Kc (¯ y (t) − y¯(t)) + i (¯ y (ξ) − y¯(ξ)) dξ τI 0

(52)

where Kci and τIi are the proportional gain and integral time constant, respectively. Step 8. The desired set point of the spatial weighted average output y¯d (t) is defined by a PI controller with respect to the set point of the output   t 1 d e d d y¯ (t) = Kc (y (t) − y(t)) + e (y (ξ) − y(ξ)) dξ (53) τI 0 where Kce and τIe are the proportional gain and the integral time constant, respectively.

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Simulation Results For numerical simulation, the heat transfer coefficient is β = 3, and the controller parameters are τ1 = 1, Kci = 159, τii = 0.0636, Kce = 0.2 and τie = 0.02. Note that the similar simulation conditions as for the heat exchangers are considered for the heated rod. The reference input tracking capabilities of the controller are studied. Thus, two step set points have been specified at times t = 0.1 s and t = 0.5 s corresponding respectively to y d (t) = 0.5 and y d (t) = 0.2. The obtained results show that the set points are perfectly reached by the controlled output y(t) (Fig. 12a). The manipulated input first reacts rapidly, then goes back to its steady value in the same way as the outputs reach the set points (Fig. 12b). Thus, the control strategy shows its effectiveness.

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4.

Stability Analysis

As the controllers that provide v(t) or y d (t) (PI controllers in the studied applications) are designed so that the closed-loop system y d (t) − y(t) is stable, in order to ensure the stability of the whole control strategy, it is important to examine the stability of the PDE system with the state feedback (8). The exponential stability is the most desirable type of stability, among many stability concepts, because of its robustness to bounded disturbances and small modeling errors (Christofides, 2001a; Christofides and Daoutidis, 1996). Thus, it is easy to show that the property that the zero dynamics of the PDE system is exponentially stable, or equivalently that the open-loop PDE system is a minimum phase system, ensures the stability of the closed-loop system, i.e. the PDE system with the state feedback (8). In this section the stability of the closed-loop system in case of co-current heat exchanger is demonstrated, however a similar approach applies for the other studied systems. The co-current heat exchanger model can be written under the following form ∂T (z, t) = A T (z, t) + B u(t) ∂t  Ti (L, t) Z(t) = DT (z, t) = Te (L, t) where A, B, D are the following operators ⎡

⎤ ∂   hi 0 C 0 ⎢ −vi ∂z − hi ⎥ A=⎣ ,D= ⎦, B = ∂ ve δ(z) 0 C he −ve − he ∂z and Z(t) is the vector of measurements.

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(54) (55)

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Time (s) (a) Controlled output y(t).

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2.5 2 1.5 1 0.5 0 −0.5 0

0.2

0.4

0.6 Time (s)

(b) Manipulated variable u(t).

Figure 12. Heated rod: set point tracking and disturbance rejection.

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194

Ahmed Maidi, Moussa Diaf, Jean-Pierre Corriou et al. Using the previous operators, the control law (8) can be transformed as   1 ∂Ti(z, t) u(t) = v(t) − Cm Ti (z, t) − τ1 Cm − τ2 J T (z, t) hi ve τ2 α ∂t   1 ∂T (z, t) = v(t) − [Cm 0] T (z, t) − τ1 [Cm 0] − τ2 J T (z, t) hi ve τ2 α ∂t   1 = v(t) − [Cm 0] T (z, t) − τ1 [Cm 0] A T (z, t) − τ2 J T (z, t) hi ve τ2 α   1 = v(t) − F T (z, t) (56) hi ve τ2 α

with the operator F = [Cm 0] + τ1 [Cm 0] A + τ2 J

(57)

Now, by using the control law (56) in the open-loop system (54), the closed-loop system is expressed as ∂T (z, t) 1 = A T (z, t) + B [v(t) − F T (z, t)] ∂t hi ve τ2 α  1 dym (t) = A T (z, t) + B v(t) − ym (t) − τ1 − τ2 J T (z, t) (58) hi ve τ2 α dt

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By constraining the measured output ym (t) to zero (Christofides, 2001a; Christofides and Daoutidis, 1996), the zero dynamics associated with the open-loop system results ∂T (z, t) 1 = A T (z, t) − B J T (z, t) = X T (z, t) ∂t hi ve α ym (t) = Cm Ti(z, t) ≡ 0 with: T (0, t) = [Ti0 (t) 0] and T (z, 0) = T

[Ti∗ (z)

(59) (60)

Te∗ (z)]T

1 BJ. hi ve α If the open-loop system is minimum phase, the zero dynamics is exponentially stable, or equivalently the operator X generates a stable semigroup (Curtain and Zwart, 1995). This property can be checked using spectral theory for operators in infinite dimensions (Pohjolainen, 1981) but requires difficult calculations. Thus an alternative approach consists to verify the stability of zero dynamics through simulations (Christofides, 2001a; Christofides and Daoutidis, 1996) of the open-loop system or of the zero dynamics (59). In the present case, it has been observed that no inverse open-loop responses were obtained, so that the zero dynamics is assumed to be stable in the following. dym (t) Thus, by defining the state variables w1 (t) = ym (t) and w2 (t) = , the closeddt loop system (58) can be written in the form of the following interconnection subsystems with the operator X = A −

w˙ 1 (t) = w2 (t) 1 τ1 1 w˙ 2 (t) = − w1 (t) − w2 (t) + v(t) τ2 τ2 τ2 ∂T (z, t) 1 = A T (z, t) + B [W (t) − J T (z, t)] ∂tApplications : Theory, Analysis and Applications, hi ve αNova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and

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v(t) − w1 (t) τ1 − w2 (t). τ2 τ2 Since τ1 and τ2 are chosen so that the stability of (38) is ensured, the w-subsystem of the interconnection is exponentially stable (|w(t)| ≤ Kw |w(0)| e−awt ; Kw ≥ 1, aw > 0). As the external input v(t) is defined by a PI controller, thus lim v(t) = 0 when t → ∞ and with W (t) =

|W (t)| ≤ KW |W (0)| e−aW t ,

KW ≥ 1 and aW > 0

(62)

Now, as the zero dynamics (59) is exponentially stable, its spatial differential operator X generates an exponentially stable semigroup U (t) so that U (t) 2 ≤ KU e−aU t with KU ≥ 1, aU > 0 and . 2 the norm in H2 [(0, L), ] (Curtain and Zwart, 1995). Consequently, the state T (z, t) of the closed-loop system (58) verifies    t  1   −aU t −aU (t−ξ) 

T (z, t) 2 ≤ KU T (z, 0) 2 e + KU e B   |W (ξ)| dξ (63)   h v α i e 0 2

Substituting W (ξ) by its expression in (62) gives    t  1   

T (z, t) 2 ≤ KU T (z, 0) 2 e−aU t +KU KW  e(aU −aW ) ξ dξ B |W (0)| e−aU t  hi ve α  0 2

If aU = aW , −aU t

T (z, t) 2 ≤ KU T (z, 0) 2 e

+ KU KW

   1    B |W (0)| t e−aU t (64)   hi ve α 

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2

where lim t e−aU t = 0 when t → ∞. Thus, the closed-loop system is exponentially stable. If aU = aW ,    1  e−aW t − e−aU t  

T (z, t) 2 ≤ KU T (z, 0) 2 e−aU t + KU KW  B |W (0)|  hi ve α  aU − aW 2

as the exponential difference fraction is always positive, the closed-loop system is exponentially stable. Consequently, the stability of ym (t) implies the stability of y(t). In conclusion, as the zero dynamics (59) of the heat exchanger is exponentially stable, the closed-loop system (58) is exponentially stable.

5.

Practical Implementation

For practical implementation, the control law (8) requires that the complete state x(z, t) must be available. From a practical point of view, this is impossible since the state x(z, t) is infinite. State estimation is widely used for on-line implementation of state-space control laws that require the knowledge of many of the state variables, if not all of them. In this way, several techniques have been introduced to estimate variables from the available measurements (Meurer et al., 2005). Estimation theory has occupied a prominent place in distributed parameter systems theory, in particular linear systems (Banks and Kunisch, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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1989; Krstic and Smyshlyaev, 2008; Omatu and Seinfeld, 1989; Ray, 1989). For DPS, it is not possible to measure the whole state. Thus to estimate the entire state, partial measurements are used by a filter or estimator. To show the practical implementation of the control strategy, let us consider the example of the counter-current heat exchanger. To perform an estimation of the required state temperatures to be used in the controller law (19), a Kalman filter is used, and the estimation is performed using the available measurements: the outlet internal temperature y(t) = Ti (L, t) and the inlet internal temperature Ti0 (t) considered as a measured disturbance. Before designing the Kalman filter, it is necessary to study the observability of the system. Concerning the observability study of the heat exchanger, as the dynamic model (10)(11) is a set of two first-order hyperbolic equations, the checkout of the general conditions for observability of first-order PDE systems discussed by Ray (1989, page 293) confirms that the process is observable. As the process is in continuous form and measurements are in discrete form, the Kalman filter is used in its continuous-discrete formulation (Corriou, 2004). The design of the Kalman filter is based on a finite-dimensional approximate model and the matrix gain of the estimator is evaluated using the Kalman filtering algorithm. The equations for the Kalman filter are given by a set of prediction and correction equations. For more details about the Kalman filter design, the reader can refer to the original works by Maidi et al. (2009a, 2010) For simulation purpose, the available measurements are corrupted by Gaussian white noise signals of standard deviation equal to 0.2◦ C, and two set point steps have been specified at times t = 1s and t = 5s corresponding respectively to y d (t) = 60◦ C and y d (t) = 50◦ C. From Figure 13, it is clear that the designed Kalman filter provides an acceptable estimation of the unmeasured variable that leads to an acceptable tracking performance (Fig. 13a) and admissible variations of the manipulated variable (Fig. 13b).

6.

Conclusion

In this chapter, two design approaches of a boundary control, by input-output linearization, for collocated and anti-collocated sensor and actuator on a PDE system are presented and summarized by two control strategies. The general approach to obtain the control law is clearly described. To illustrate the validity of the developed control strategies, three applications are studied. Thus, the collocated case is illustrated by a counter-current heat exchanger, whereas the anti-collocated case is displayed by two physical systems, a parallelcurrent heat exchanger in the case of a boundary output, and a heated rod in the case of a punctual output. The control performances are evaluated through numerical simulation by considering both tracking and disturbance rejection problems. The simulation results show that the developed control strategies ensure both satisfactory tracking and disturbance rejection. The simulation results concerning the sudden fluctuations of system parameters reveal also the robustness of the control strategies. The stability is studied and the closed-loop system is shown to be exponentially stable if the open-loop PDE system is a minimum phase system. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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2

4

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(b) Manipulated variable u(t).

Figure 13. Counter-current heat exchanger: set point tracking and disturbance rejection in case of noisy measurements and Kalman filtering.

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Finally, as the control law requires the whole vector of state variables, the practical implementation of the proposed control strategy using a Kalman filter, that provides the state vector, is presented. The obtained results show that the Kalman filter allows us to achieve an excellent performance. This study demonstrates that the design of the boundary control of PDE system, based on input-output linearization, is a very successful control approach since it leads to a control law that enhances the control performance by preserving the fundamental control properties, consequently the distributed nature of the PDE system.

References O. M. Aamo, S. Simani, and M. Krstic. Flow Control by Feedback: Stabilization and Mixing. Springer, 2002. H. T. Banks and K. Kunisch. Estimation Techniques for Distributed Parameter Systems. Birkh¨auser, Boston, 1989. P. D. Christofides. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Birkh¨auser, Boston, 2001a. P. D. Christofides. Control of nonlinear distributed parameter process systems: Recent developments and challenges. AIChE Journal, 47(3):514–518, 2001b.

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P. D. Christofides. Model-based control of particulate processes. Kluwer Academic Publishers, Netherlands, 2004. P. D. Christofides and P. Daoutidis. Feedback control of hyperbolic PDE systems. AIChE Journal, 42:3063 – 3086, 1996. P. D. Christofides, M. Li, and L. M¨adler. Control of particulate processes: Recent results and future challenges. Powder Technology, 175:1–7, 2007. J. P. Corriou. Process control - Theory and applications. Springer, London, 2004. R. F. Curtain and H. Zwart. An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, 1995. M. Gad el Hak. Flow Control: Passive, Active, and Reactive Flow Management. Cambridge University Press, 2000. J. C. Friedly. Dynamic Behaviour of Processes. Prentice-Hall, New Jersey, 1972. A. Isidori. Nonlinear Control Systems. Springer-Verlag, New York, 1995. C. Kravaris and J.C. Kantor. Geometric methods for nonlinear process control. 1. Background. Industrial & Engineering Chemistry Research, 29:2295–2310, 1990a.

C. Kravaris and J.C. Kantor. Geometric methods for nonlinear process control. 2. Controller synthesis. Industrial & Engineering Chemistry Research, 29:2310–2323, 1990b. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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M. Krstic and A. Smyshlyaev. Boundary Control of PDEs: A Course on Backstepping Designs. SIAM, 2008. W. Liu. Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation. Springer, 2010. A. Maidi, M. Diaf, and J.-P. Corriou. Distributed geometric control of wave equation. In 17th IFAC World Congress, Seoul, Korea, July 6-11 2008. A. Maidi, M. Diaf, and J.-P. Corriou. Boundary geometric control of a counter-current heat exchanger. Journal of Process Control, 19:297–313, 2009a. A. Maidi, M. Diaf, and J.-P. Corriou. Boundary geometric control of co-current heat exchanger. In 7th IFAC International Symposium on Advanced Control of Chemical Processes (ADCHEM 2009), Istanbul, Turkey, July 12-15 2009b. A. Maidi, M. Diaf, and J.-P. Corriou. Boundary geometric control of a heat equation. In European Control Conference (ECC’09), pages 4677–4682, Budapest, Hungary, August 23-26 2009c. A. Maidi, M. Diaf, and J.-P. Corriou. Boundary control of a parallel-flow heat exchanger by input-output linearization. Journal of Process Control (under press), 2010.

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T. Meurer, K. Graichen, and E. D. Gilles (Eds.). Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, volume 322. Springer-Verlag, Berlin Heidelberg, 2005. S. Omatu and J. H. Seinfeld. Distributed Parameter Systems. Theory and Applications. Clarendon Press, Oxford, 1989. R. Padhi and Sk. Faruque Ali. An account of chronological developments in control of distributed parameter systems. Annual Reviews in Control, 33:59–68, 2009. S. Pohjolainen. Computation of transmission zeros for distributed parameter systems. Journal of Process Control, 33:199–212, 1981. W. H. Ray. Advanced Process Control. Butterworths, Boston, 1989. H. Shang, J. F. Forbes, and M. Guay. Feedback control of hyperbolic distributed parameter systems. Chemical Engineering Science, 60:969 – 980, 2005. A. Van de Wouwer, P. Saucez, and W. E. Schiesser. Simulation of distributed parameter systems using a Matlab-based method of lines toolbox: Chemical engineering applications. Industrial and Engineering Chemical Research, 43(14):3469 – 3477, 2004.

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN 978-1-61122-858-8 c 2010 Nova Science Publishers, Inc. 

Chapter 8

R OBUST N O PARAMETRIC I DENTIFIER FOR A C LASS OF C OMPLEX PARTIAL D IFFERENTIAL E QUATIONS R. Fuentes1 , I. Chairez2,∗, A. Poznyak1 and T. Poznyak3 1 CINVESTAV 2 UPIBI 3 ESIQIE

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Abstract In this chapter a strategy based on differential neural networks (DNN) for the identification of a class of models described by partial differential equation with a complex-valued state is proposed. The identification problem is reduced to finding an exact expression for the weights dynamics using the DNNs properties. In this case, the DNN can be viewed as two coupled networks where one of them reproduces the real part of the complex valued equation and the other provides the identification of the imaginary part, where each stimated state is a complex valued state. The adaptive laws for complex weights ensure the convergence of the DNN trajectories to the PDE complex-valued states. To investigate the qualitative behavior of the suggested methodology, here the non parametric modeling problem for two distributed parameter plants is analyzed: the Sch¨odinger and Ginzburg-Landau equations.

PACS 05.45-a, 52.35.Mw, 96.50.Fm. Keywords: Complex Valued Differential Neural Networks Key Words: Complex Partial Differential Equations, Differential Neural Networks, Non Parametric Identificaction.

1.

Introduction

The complex Schr¨odinger and Ginzburg-Landau equations are two of the most-studied nonlinear equations in the physics and mathematics community. Landau equation describes var∗ E-mail address: isaac [email protected] (Corresponding Author) Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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ious phenomena from nonlinear waves to superconductivity and Schr¨odinger’s is an equation that describes how the quantum state of a physical system changes in time. These are examples of complex partial differential equations (C-PDE) [5]. There exists some methods to achive analytic solutions for complex PDE, and some others to approximate the numerical solution. There are some general techniques available such as the Perturbation theory, the Variational and Quantum Monte Carlo methods, Density functional theory, etc. which let us find an approximation of the exact solution of C-PDEs [30]. In general, the solution of linear or non-linear partial differential equations (PDE) has not been practicable when the PDE dynamics are affected by perturbations or if there is some degree of uncertain in the mathematical structure. The goal of the chapter is to give an overview of the strategy based on differential neural networks (DNN) for the non parametric identification of a mathematical model described by a partial differential equation with a complex-valued state. √ In what follows, j denotes the imaginary unit, −1.

2.

Complex Valued Neural Networks

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This chapter describes a specific approximative solution for a class of complex-valued partial differential equations (C-PDE). Here a strategy based on Differential Neural Networks (DNN) to avoid the non parametric modeling problem for a complex distributed parameter plant is presented. The aplications of artificial neural networks has been increasing in a lot of research interests. A lot of neural models has been made in order to apply this tool to various fields. Particullary, several models of complex-valued neural networks has been proposed, studied and investigated with the idea of afford complex numbers problem in physics and pure mathematics [19]. The Radial Basis Function Neural Networks (RBFNN) and Multi-Layer Perceptrons (MLP) are known as universal approximators of continuous functions defined on a compact set. One can show that they are efficient numerical approximators and can be implemented as learning machines. With the success of this real-valued neural networks, an extension to complex valued neural networks has taken special attention. Complex valued neural networks are the prototype of networks that extend real valued architectures. In this way, as the solutions to the complex partial differential equation are known to be uniformily continuous, neural networks seem like ideal candidates for approximating viability problems. There are, however, relatively few works that exploit complex valued neural networks to solve C-PDE. The reference [21] introduces a complex-valued network inversion method to solve inverse problems with complex numbers. The description of a complex-variable version of the Hopfield neural network (CHNN) made by [22] shows that adaptive conections can exist in both fixed point and oscillatory modes. A complex back propagation algorithm for complex back propagation neural networks is introduced by [23]. It consists of suitable node activation functions having multi-saturated output regions. Nevertheless, within the NN framework, if a mathematical continuous model is incomplete or partially known, as in the mathematical representation of complex valued equation, which presents many sources of uncertainties, DNN methodology can provide an effective instrument for Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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avoids this lack of information. Even more if the complex valued equation is modelling a distributed parameter system. In this chapter, it is presented a generalship based on DNN properties for the non parametric identification of a mathematical model described by complex-valued partial differential equations, the so called Complex Valued DNN (CVDNN). The solution is presented as the finite composition of N coupled uncertain ordinary differential equations. This representation was prepared by using the finite differences methodology. The adaptive laws for CVDNN weights ensure the ”practical stability” of the DNN trajectories to the ”complex states”, according to the DNN features. Convergence of the identification method is obtained by a modified Lyapunov function in infinite dimensional spaces. The identification problem is reduced to finding an exact expression for the weights dynamics. To verify the qualitative behavior of the suggested methodology, a non parametric modeling problem for two distributed parameter plants described in a complex way is analyzed.

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3.

Complex Partial Differential Equations

Complex Valued Partial Differential equations (CVPDE) provide excellent examples of infinite dimensional dynamical systems which possess diverse physics phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatiotemporal chaos, quantum state changes in time, superconductivity and some others [5]. Nonlinear dispersive waves occur throughout physical and natural systems wherever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical fibers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial differential equations include the Korteweg-de Vries equation, nonlinear Klein-Gordon equations, nonlinear Schr¨odinger equations, Ginzburg-Landau equation, and many others [30]. In this chapter, we choose a class of nonlinear Schr¨odinger and the classical GinzburgLandau equations as prototypal examples, to show the qualitative efficiency of the approximation method based on DNN presented in a complex way.

3.1.

Schr¨odinger Equation (Hyperbolic Type)

One of the most important equations in mathematical physis is the Schr¨odinger’s equation, one of the fundamental equations in non-relativistic quantum mechanics. In pure mathematics, the Schr¨odinger equation and its variants is one of the basic equations studied in the field of partial differential equations, and has applications to geometry, to spectral and scattering theory, and to integrable systems [18]. There are actually two (closely related) variants of Schr¨odinger’s equation, the time dependent Schr¨odinger equation and the time independent Schr¨odinger equation. The following PDE describes the time-dependent Schr¨odinger equation

∂Ψ = HΨ Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and ∂tApplications, Nova Science Publishers, Incorporated, 2011. j

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where H is now a quantum observable rather than a classical one, and more precisely H=

|P |2 + V (Q) 2m

In other words, we have ∂Ψ 2 (1) (t, q) = HΨ(t, q) = − Δq Ψ(t, q) + V (q)Ψ(t, q) ∂t 2m n ∂ 2 Ψ where Δq Ψ = i=1 ∂q 2 Ψ is the Laplacian of Ψ. Equation 1 is the parabolic type i Schr¨odinger Equation. It is possible that the quantum state Ψ oscillates in time according to the formula j

E

Ψ(t, q) = e j t Ψ(0, q) for some real number E in which case it is true that

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EΨ(t) = HΨ(t), ∀t which is known as the time-independent Schr¨odinger equation, and E is referred as the energy level of the state Ψ. In particular for this work, we employ a special case of Schr¨odinger equation. The coupling between real object particles and light quantums can be discussed by Minkowski space’s directional strangeness, through the Galilei transfomation [31]. The classical quantum mechanics can be introduced into non-Euclidean geometry by the correspondence between hyperbolic Minkowski space and Galilei transformation [31]. This gives us a way to find the logic correlation between the classical quantum mechanics and the restricted theory of relativity. All these theories make possible to discuss microscopic low speed foreign body’s law of motion of Schr¨odinger equation. Taking from 1 the differential quotient for time, we have ∂Ψ p2 = −j Ψ ∂t 2m which is the so called hyperbolic Schr¨odinger equation.

3.2.

Ginzburg-Landau Equation (Parabolic Type)

In physics,complex Ginzburg-Landau equation (CGLE) is used to model a vast variety of phenomena, from nonlinear waves to superconductivity, liquid crystals and second order phase transitions to others. The equation describes the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation, for which a continuous spectrum of unstable wavenumbers is taken into account [30]. It can be considered as a general normal form for a large class of bifurcations and nonlinear wave phenomena in spatially extended systems. The equation is given by

∂A ∂ 2A = A + (1 + jα) 2 − (1 + jβ) |A|2 A Partial Differential Equations: Theory, Analysis and Applications Nova Science Publishers, Incorporated, 2011. ∂t : Theory, Analysis and Applications, ∂x

(2)

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where A is a complex function of (scaled) time t and space x (often in reduced dimension D = 1 or 2) and the real parameters b and c characterize linear and nonlinear dispersion terms. The equation can be reduced to the ”real” Ginzburg-Landau equation, ∂A ∂ 2A − |A|2 A =A+ ∂t ∂x2 which one might also call the ”complex nonlinear diffusion equation”. In this chapter, both models are treated with the propose of show the alternative identification method presented in the following sections.

4.

Approximation of Complex-Valued Partial Differential Equation

Let us consider the partial differential equation with a complex-valued state ut (x, t) = f (u (x, t) , ux (x, t) , uxx (x, t)) +g (u (x, t) , ux (x, t) , uxx (x, t)) j

(3) (4)

with boundary conditions of Dirichlet and Neumann type:

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u(0, t) = u0 , u(1, t) = 1 ux (0, t) = 0

(5)

The PDE domain is x ∈ [0, 1], t ≥ 0. Let f (x, t) and g (x, t) be piecewise continuous in t. Suppose that the uncertain nonlinear functions f (x, t) and g (x, t) satisfie the Lipschitz condition f (t, x) − f (t, y) ≤ L1 x − y, and g (t, x) − g (t, y) ≤ L2 x − y, ∀ x, y ∈ B := {x ∈ n | x − x0  ≤ r}, ∀ t ∈ [t0 , t1 ], where L1 and L2 are constants and f 2 = (f, f ), g2 = (g, g) [10]. The norm defined above in (14) is given in a Sobolev space. Definition. Sobolev space [1], H m,p (Ω): Let Ω be an open set in Cn and let u ∈ m C (Ω). Define a norm on u by

um,p :=

 0≤|α|≤m

⎛ ⎝



⎞1/p |D α u (x)|p dx⎠

,1≤p 0, Sri > 0, Tri > 0, Pci > 0, Sci > 0 and Tci > 0) of the algebraic Riccati equations given by  −1   i  i i i i i i i i PD AD + AD PD +PD ΛD,α PD +λmax ΛP,D F3,D In×n +QiP,D = 0   −1  i  i i i i i i i i SD AD + AD SD + SD ΛS,D SD + λmax ΛS,D F8,D In×n + QiS,D = 0  (12) −1   i  i i i i i i i i i TD AD + AD TD + TD ΛT ,DTD + QT ,D + λmax ΛP,D F0,D In×n +      −1 −1  i i λmax ΛiS,D +λmax ΛiT ,D f0.D In×n =0 F5,D

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where D indicates the corresponding real part (denoted by the subindex r) or the complex part (denoted by the subindex c). Employing the Schur Complement, equation 12 can be rewriting as a Linear Matrix Inequality (LMI) following the nonlinear matrix inequiality equivalence technique [14]. Special class of Riccati equation P A + AP + P RP + Q = 0 has positive solution if and only if the LMI 13 has feasible solution. It is ∀P > 0 

−P A − AP − Q P P R

 >0

(13)

where A, Q = QT , R = RT > 0 are given matrices of appropiate sizes. This LMI could be solved by several efficient methods in order to determine whether or not the LMI is feasible, if it is, comupute the solution. Some existing numerical methods for LMI resolution are Ellipsoid algorithm or Interior-point method, implemented in a computational way employing the SeDuMi or Yalmip toolboxes. State estimation problem for uncertain nonlinear systems analyzed in this study, could be now stated as follows: Problem Statement. Under the complex nonlinear system with an adequate selection of matrices Air and Aic , and with the complex value neural network identifier structure supplied with the adjustment law (11) (including the selection of Wi∗ , i = 1, 2, 3), the upper bound for the estimation error β defined as β := lim ˆ u (t, x) − u (t, x)2P

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t→∞

(14)

P > 0, P = P  ∈ Rn×n must be obtained, and if it is possible, to reduce to its less achievable value, using any of the free parameters participating within the NN structure. Following the descomposition described in 8, one can reorganize the problem as follows: 5.0.1.

Practical Stability

The following definition and proposition are needed for the main results of the paper. Consider the following ODE nonlinear system z˙t = g(zt, vt) + t

(15)

with zt ∈ n , vt ∈ m and t an external perturbation or uncertainty such that  t 2 ≤

+ Definition 1. Definition (Practical Stability): Assume that a time interval T and a fixed function vt∗ ∈ m over T are given. Given ε > 0, the nonlinear system (15) is said to be ε-practically stable over T under the presence of t if there exists a δ > 0 (δ depends on and the interval T ) such that zt ∈ B[0, ε], ∀ 0 ≤ t ≤ T, whenever zt0 ∈ B[0, δ].

Definition 2. Definition (Practical Stabilizability): The system (15) is said to be ε−stabilizable if for some fixed ε > 0 there exists a function v (·) and the number δ > 0 Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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such that for any solution zt of the system (15) with any initial condition z0  ≤ δ is bounded as zt  ≤ ε (16) In other words, system (15) is ε−stable if the designed adaptive function v (·) mantains any system trajectory inside the given ε−neighborhood B[0, ε] of the origin. Actually, there exists the uniform practical stability that appears when for all δ > 0, the upper bound (16) holds. Similarly to the Lyapunov stability theory for nonlinear systems, it was applied the aforementioned direct method for the ε-practical stability of nonlinear systems usingpractical Lyapunov-like functions under the presence of external perturbations and model uncertainties. Note that these functions have properties differing significantly from the usual Lyapunov functions in classic stability theory. The subsequent proposition requires the following Lemma. Lemma 1. Lemma : Let a nonnegative function V (t) sastisfying the following differential inequality V˙ (t) ≤ −αV (t) + β

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where α > 0 and β ≥ 0. Then



μ 1−  V (t)



 with μ = β/α and the function [·]+ defined as  z if [z]+ := 0 if

→0 +

z≥0 z 0 such that   V˙ (z, t) ≤ −αV (z, t) + H + with H a bounded non-negative nonlinear function with upper bound H + . Moreover, the H+ trajectories of zt belongs to the zone ε := when t → ∞. In this proposition V˙ (zt , t) α denotes the derivative of V (z, t) along zt , i.e., V˙ (z, t) = Vz (z, t) · (g(zt, vt) + t ) + Vt(x, t). Proof. The proof follows directly from Lemma 1.

Definition 3. Definition: Given a time interval T and a function v (·) over T , nonlinear system (15) is ε-practically stable, T under v if there exists an ε−practical Lyapunov-like function V (x, t) over T under v. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Theorem. Let be the non linear system described on C-PDE’s unknown and perturbed on the state and the output (3) with the conditions at the border of Dirichlet and Neumman type defined on (5). Moreover, suppose the structure of non-parametric adaptive identifier (8) whose parameters are adjusted as the adaptable law given in (11). If there exists matrices QiP R , QiSR , QiT R , QiP C , QiSC and QiT C positive defined such that the LMI’s (13) have positive definit solutions Pri , Sri , Tri, Pci , Sci and Tci (i = 3, N), then the following upper bound lim ˆ ui (t, x) − ui (t, x)P ≤ ρ (17) t→∞

is ensured for the state nonparametric identification process where ρ has the following form       −1 −1 i i ρ := min (αm ) N max λmax [Λ ] Fi i

i

Besides the following upper bound is achieved for the gradient of the estimation error   ∂   lim  [ˆ ui (t, x) − ui (t, x)]  ≤ ρx t→∞ ∂x with ρx with the following form  −1

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ρx := S

min (αim )−1 N i

     −1 i Fi max λmax [Λ ] i

Moreover, the weights W1,t , W2,t and W3,t are bounded with the following bounds  R     W  ≤ K R ρ , W R  ≤ K R ρ , W R  ≤ K R ρ 1,t 1 r 2,t 2 r 3,t 3 r  C     W  ≤ K C ρc , W C  ≤ K C ρc , W C  ≤ K C ρc 4,t

4

5,t

5

6,t

6

Proof. The detailed proof is given in the appendix.

6. 6.1.

Numerical Results Schr¨odinger Simulation

As a case study, for the purpose of illustrating the main theoretical results derived in previous Sections, here it is considered the basic equation of quantum mechanics: Schr¨odinger’s complex partial differential equation (1). In particular the simplest complex-valued PDE is the linearized Schr¨odinger equation (hyperbolic type), for this we take p,  and m equal to one, ψ t = −jψ xx ψ x (0) = 0

(18) (19)

where ψ(1) is actuated and ψ(x, t) is a complex-valued function. This model will be used just to generate the data to test the identifier based on complex valued DNN. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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(20)

The initial conditions and boundary conditions used in this numerical simulations are u(0, t) = rand(1), u(1, t) = 1, u(x, 0) = 2, ux(0, t) = 0

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The Schr¨odinger equation (18) trajectories are showed here in their corresponding magnitude and phase (Fig. 1, 2).

Figure 1. Magnitude of Schr¨odinger equation trajectories. The CVDNN identifier for C-PDE produces a trajectory very close to the Schr¨odinger equation magnitude and phase as can be seen in Fig. (2) and Fig. (3). There is an important zone where there exists a big difference between original and estimated trajectories. This dissimilarity is dependent on the learning period required to adjust the CVDNN identifier. The difference between the C-PDE trajectories and the estimated state produced by the CVDNN identifier is shown in Figures (4) and (5). This error is close to zero almost during all x and all t.

6.2.

Landau Simulation

Here, there is presented the numerical simulation of the complex GLE: ∂A ∂ 2A = A + (1 + iα) 2 − (1 + iβ) |A|2 A ∂t ∂x

which one might also call the ”complex nonlinear diffusion equation”. The simulation is made with the propose of show, in a qualitative way, the effectiveness of the proposed Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 2. Phases of Schr¨odinger equation (continuos line) and CVDNN trajectories (dash line), comparition at 3 seconds of simulation for the whole space domain.

Figure 3. Identifier trajectories for the Schr¨odinger equation magnitude.

algorithm. In this section we compare the behavior of the CGLE with the trajectories of the CVDNN. The magnitude and phase of the CGLE are presented in comparission with the CVDNN as well as the identification error. The difference between the C-PDE magnitude trajectories (see Fig. (6)) and the estiPartial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 4. Error trajectories between Magnitude Schr¨odinger Equation and Magnitude CVDNN.

Figure 5. Error dynamics between the Schr¨odinger Equation Phase and CVDNN at 3 seconds of simulation for the whole spatial domain.

mated state produced by the CVDNN identifier (Fig. (7)) shows that the error (figure 8) is close to zero almost during all x and all t, except for a zone where the difference between CVDNN identifier and the C-PDE model can be seen as a consequence of the training process necesary in the adjustment of the identifier. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 6. Magnitude of CGLE trajectories.

Figure 7. CVDNN trajectories to approximate the CGLE magnitude.

Trajectories of CGLE and the CVDNN phases (Figures 9 and 10) show similar behaviour. There exists a big difference between original C-PDE model and estimated trajectories. This dissimilarity is dependent on the learning period required to adjust the CVDNN identifier. In figure 11 it can be seen the convercence of the error, which mantains close to zero Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 8. Difference between CGLE magnitude and the CVDNN.

Figure 9. Pashe of CGLE trajectories.

almost during all x and t. In this way, the proposed technique can be useful, when we have an unknown system, and this system, is modeled by complex PDE. Here, we show some numerical examples employing two different mathematical complex valued models, but in fact, for the implementation of this technique there is no necessary having a mathematical model to apply the Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 10. CVDNN trajectories to approximate the phase of CGLE.

Figure 11. Error dynamics of the Phase approximation made by the CVDNN.

CVDNN. This approach can be applied with any complex valued system, even if it is uncertain or if the system is under the effect of some perturbations. Besides the Schr¨odinger and Ginzburg-Landau equations, there are in physics many other examples of this type. Equations like Korteweg-de Vries, nonlinear Klein-Gordon, and some others, provide excellent examples of infinite dimensional dynamical systems which possess diverse physics Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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phenomena, and which show complex uncertain behavior.

7.

Conclusions

The suggested approach solves the problem of non parametric identification of uncertain nonlinear systems described by complex partial differential equations. Asymptotic convergence for the identification error has been demonstrated applying a Lyapunov-like analysis using a special class of Lyapunov functional. Besides, the same analysis leads to the generation of the corresponding conditions for the upper bound of the weights involved in the CVDNN identifier structure. Numerical examples showing the magnitude and phase of the Schr¨odinger and Ginzburg-Landau complex partial differential equations demonstrating the workability of this new methodology based on continuous complex valued differential neural networks, what is more it was proof, in a qualitative way, the convergence of the error. The technique is useful for any system which behaviour can be seen like a complex partial differential equation, even if the system is uncertain or partially known.

8.

Appendix

Consider the Lyapunov-like functional

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V (t) :=

1  N

V¯i (t, x) dx+

6 

 tr

˜ i (t) W k



˜ i (t) Kk W k

!

k=1 x=0 i=1 2  2 V¯i (t, x) := Δir (t, x)P i + Δix,r (t, x)S i + ˆ ui,r (t, x)2T i r r r  2  2 + Δic (t, x)P i + Δix,c (t, x)S i + ˆ ui,c (t, x)2T i c c c

(21)

Following the procedure for the second Lyapunov method, the time derivative of Vt is V˙ (t) = 2

Z1 X N “

x=0 i=1 Z1 N

X

+2

Δir (t, x)

(t) Pri

Z1 X N h

d i Δr (t, x) dx + 2 dt

Δix,r (t, x)

i

x=0 i=1

u ˆi (t, x) Tri

x=0 i=1 Z1 X N h

+2

”

Δix,c

d u ˆi (t, x) dx + 2 dt

Z1 X N “

x=0 i=1

(t, x)

x=0 i=1

+2

i

Sci

d i Δx,c (t, x) dx + 2 dt

Z1 X 6 N X

Δic (t, x)

Z1 X N

”

(t) Pci

d i Δx,r (t, x) dx dt

d i Δ (t, x) dx dt c

u ˆi (t, x) Tci

x=0 i=1

Sri

d u ˆi (t, x) dx dt

nh i o ˙ ki (t) dx ˜ ki (t) Kk W tr W

x=0 i=1 k=1

(22)

    d d Using the results for the terms 2 Δir (t, x) Pri Δir (t, x) , 2 Δic (t, x) Pci Δic (t, x) , dt dt     d d 2 Δix,r (t, x) Sri Δix,r (t, x) , 2 Δix,c (t, x) Sci Δix,c (t, x) , dt dt   i d i d 2ˆ ui,r (t, x) Tr u ui,c (t, x) Tc u ˆi,r (t, x) , 2ˆ ˆi,c (t, x) , and the following matrix indt dt equality XY  + Y X  ≤ XΛX  + Y Λ−1 Y  valid for any X, Y ∈ Rr×s and any 0 < Λ Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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= Λ ∈ Rs×s the next inequality is achieved V˙ (t) ≤

Z1 X N h h i h i i i i i i AiD + AiD PD + PD Λiα,D PD ΔiD PD ΔiD dx x=0 i=1

+

Z1 X N “

Δix,D

x=0 i=1 Z1 X N

+

Z1 X N

+

x=0 i=1 Z1 X N

+

” „

« h iT i i i i SD AiD + AiD SD + SD ΛiS,D SD Δix,D dx

h “ i ” i i i i ui,D TD AiD + AiD TD + TD ΛiT,D TD u ˆi,D dx

x=0 i=1

“

λmax ΛiP,D

« ‚ ‚2 ”−1 „ ‚ i‚ i i i i i F0,D ui 2 + F1,D ui−1 2 + F2,D ui−2 2 + F3,D ‚Δ ‚ + F4,D dx

‚ ‚2 « h i−1 „ ‚ i‚ i i i i i F5,D λmax ΛiS,D ˆ ui 2 + F6,D ui−1 2 + F7,D ˆ ui−2 2 + F9,D + F8,D ‚Δx ‚ dx

x=0 i=1

Z1 X N “ ” “ ” i ˜ hi (t) ϕ(ˆ ˜ li (t) γ(ˆ ˜ gi (t) σ(ˆ 2 ΔiD PD xi)ˆ ui + W xi)ˆ ui−1 + W xi )ˆ ui−2 dx + W x=0 i=1

Z1 X N “ ” “ ” i ˜ hi (t) ∇x ϕ(ˆ ˜ li (t) ∇x γ(ˆ ˜ gi (x) ∇x σ(ˆ 2 Δix,D SD xi )ˆ ui + W xi )ˆ ui−1 + W xi )ui−2 dx W + x=0 i=1 Z1

+

N X

“ ” i Wgi (t) σ(ˆ 2ui,D TD xi )ˆ ui (x, t) + Whi (t) ϕ(ˆ xi )ui−1 + Wli (x) γ(ˆ xi)ui−2 dx

x=0 i=1 Z1 X N

+

“ ”−1 “ ” i i i i λmax ΛiT,D ui 2 + f1,D ui−1 2 + f2,D ui−2 2 + f4,D f0,D dx

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x=0 i=1

where D indicates the corresponding real part (denoted by the subindex r) or the complex part (denoted by the subindex c), and h = 1, 4, g = 2, 5 and l = 3, 6. By the Riccati equations defined in (12) and in view of the adjust equations of the weights (11), the previous inequality is simplified to N “ N “ ` ´−1 i ” P ` ´−1 i ” P F9 + F9,c λmax ΛiS,R λmax ΛiS,C i=1 i=1 “ ” “ N N ` ´−1 i ` ´−1 i ` ´−1 i ` ´−1 i ” P P f3 + λmax ΛiP,R F4 + f3,c + λmax ΛiP,C F4,c λmax ΛiT,R λmax ΛiT,C +

V˙ (t) ≤ −αim V (t) +

i=1

i=1

Taking the maximum value over i, we obtain “ “ˆ “ “ˆ ` ´ ˜−1 ” i ” ˜−1 ” i ” V˙ (t) ≤ −min αim V (t) + N max λmax ΛiP,R F4 + N max λmax ΛiP,C F4,c i i i “ “ˆ “ˆ ˜−1 ” i ˜−1 ” i ” i i +N max λmax ΛS,R F9 + λmax ΛT,R f3 i “ “ˆ “ˆ ˜−1 ” i ˜−1 ” i ” i i +N max λmax ΛS,C F9,c + λmax ΛT,C f3,c i

Applying the Lemma 1, one has: 

 μ 1− →0 V (t) + Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications,

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with

r

r “ “ˆ “ “ˆ ˜−1 ” i ” ˜−1 ” i ” N min (αim )−1 max λmax ΛiP,R N min (αim )−1 max λmax ΛiS,R F4,c F9,c i i i i r “ “ˆ ” ” “ “ ” ” ˜−1 i ˜−1 ˆ i N min (αim )−1 max λmax ΛiT,R N min (αim )−1 max λmax ΛiP,C f3,c F4,c i i i r r i “ “ˆ “ “ˆ ˜−1 ” i ” ˜−1 ” i ” −1 −1 i i i N min (αm ) max λmax ΛS,C N min (αm ) max λmax ΛiT,C F9,c f3,c

μ := r

i

i

i

i

References [1] R. Adams and J. Fournier, Sobolev spaces, 2nd ed. New York: Academic Press, 2003. [2] Akira Hirose, Complex Valued Neural Networks, Theories and Applications, Series on Innovative Intelligence. Vol. 5, Tokyo Jap´on, 978-981-238-464-5, 2003. [3] N. E. Cotter, The stone-weierstrass theorem and its application to neural networks IEEE Transactions on Neural Networks, vol. 1, pp. 290–295, December 1990. [4] Delyon, B., Juditsky, A., Benveniste, A. Accuracy analysis for wavelet approximations. IEEE Trans. Neural Network. Vol. 6 pp. 290-295, 1995. [5] Fick A., Ann. Physik, Leipzig, 170, pp59, 1855.

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[6] R. Fuentes, A. Poznyak, T. Poznyak and I. Chairez, Neural Numerical Modeling for Uncertain Distributed Parameter System. IJCNN 2009. [7] S. Haykin, Neural Networks. A comprehensive Foundation. New York: IEEE Press, Prentice Hall U.S., 2nd Ed. 1999. [8] S. He, K. Reif, and R. Unbehauen, Multilayer neural networks for solving a class of partial differential equations Neural Networks, vol. 13, pp. 385–396, 2000. [9] Kalman, R.E. and Bertran, J.E. Control Systems Analysis and Design via the Second Method of Lyapunov, I: Continuos-time Systems, J. Basic Engineer Trans. ASME 82,Vol. 2, pps. 371-393, 1960. [10] H. K. Khalil, Nonlinear systems. Upper Saddle River: Prentice-Hall, 2002. [11] I. Daubechies, Ten lectures on Wavelets. Philadelphia: SIAM, 1992. [12] I. E. Lagaris, A. Likas, and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations IEEE Trans- actions on Neural Networks, vol. 9, no. 5, pp. 987–1000, September 1998. [13] K.S. Narendra and S. Mukhopadhyak, Adaptive Control Using NN and Approximate Models, IEEE Trans. NN, Vol. 8, No. 3, 475-485, 1997.

[14] A. Poznyak,Advanced Mathetical Tools for Automatic Control Engineers, Vol. 1, Deterministic Technique, Elservier, 2008. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[15] A. Poznyak, E. Sanchez, and W. Yu, Differential Neural Networks for Robust Nonlinear Control (Identification, Estate Estimation an trajectory Tracking). Worl Scientific, 2001. [16] G.A. Rovithakis and M.A. Christodoulou, Adaptive Control of Unknown Plants Using Dinamical NN, IEEE Transactions and Systems, Vol. 24, 400-412, 1994. [17] G. D. Smith, Numerical solution of partial differential equations: Infinite difference methods. Oxford: Clarendon Press, 1978. [18] Griffiths, David J. Introduction to Quantum Mechanics (2nd Ed.). Prentice Hall. ISBN 0-13-111892-7, 2004. [19] Tohuru Nitta, et al, Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, Information Science Reference, United Kingdom, 978-160566-215-2, 2009. [20] Richard Bellman and George Adomian. Partial differential equations : new methods for their treatment and solution. Dordrecht : D. Reidel, 1985. [21] Takehiko Owaga, Takushoku Complex-Valued Neural Network and Inverse Problems, Chapter II, Complex Valued Neural Networks, pp. 27-34, 2009.

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[22] A Complex-Valued Hopfield Neural Network: Dynamics and Applications ComplexValued Neural Network and Inverse Problems, Chapter IV, Complex Valued Neural Networks, pp. 79-104, 2009. [23] Cheolwoo You, Myongit and Daesk Hong, Yansei Learning Algorithms for ComplexValued Neural Networks in Communication Signal, Chapter IX, Complex Valued Neural Networks, pp. 194-236, 2009. [24] G.Cybenko, Approximation by Superposition of Sigmoidal Activation Function, Math.Control, Sig Syst, Vol.2, 303-314, 1989. [25] M. W. M. G. Dissanayake and N. Phan-Thien, Neural-network based approximations for solving partial differential equations Communications in Numerical Methods in Engineering, vol. 10, pp. 195–201, 2000. [26] F.L. Lewis, A. Yesildirek and K. Liu, Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Trans. Neural Network, vol. 7, No. 2, pp. 1-11, 1996. [27] N. Mai-Duy and T. Tran-Cong, Numerical solution of differential equations unsing multiquadric radial basis function networks Neural Networks, vol. 14, pp. 185–199, 2001. [28] K.Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks, Vol.4 No.5,251-257,1991.1 [29] T. J. R. Hughes, The finite element method. New Jersey: Prentice Hall, 1987.

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[30] I.S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation (CGLE), Rev. Mod. Phys. 74: 99-143, 2002.

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[31] Zhao Zheng and Yu Xuegang, Hyperbolic Schr¨odinger Equation, Advances in Applied Clifford Algebras 14 No. 2, 207-213 (2004).

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 c 2011 Nova Science Publishers, Inc. 

Chapter 9

T HE G ENERALIZED W EIERSTRASS S YSTEM I NDUCING S URFACES IN E UCLIDEAN T HREE S PACE AND H IGHER D IMENSIONAL S PACES Paul Bracken∗ Department of Mathematics University of Texas Edinburg, TX 78540 USA

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Abstract An inducing mechanism for describing minimal surfaces imbedded in threedimensional Euclidean space was first formulated by Enneper and Weierstrass one and a half centuries ago. More recently, this idea has been substantially generalized by B. Konopelchenko, who established the connection between certain classes of constant mean curvature surfaces and the trajectories of an infinite dimensional Hamiltonian system. Here we begin by reviewing his formulation of the generalized Weierstrass system which consists of a system of nonlinear Dirac-like partial differential equations. It is the solutions of this system which can be used to generate surfaces in Euclidean three space which have constant mean curvature. The correspondence between this system and the two-dimensional nonlinear sigma model is presented. A linear spectral problem is established. The integrability of the system is discussed and symmetry reduction is systematically applied to derive several classes of invariant solutions for both the two-dimensional nonlinear sigma model and the generalized Weierstrass system. Extensions of this system to higher dimensional spaces are considered. Finally, a physical application in the area of string theory will be presented, including an application in Minkowski space, and the path integral quantization in three space using the Weierstrass representation. An introduction to the case of inducing surfaces of nonconstant mean curvature is briefly treated.

1. Introdution The dynamics of surfaces is an important part of many interesting phenomena in mathematics and especially physics. The theory of immersion and deformations of surfaces has been

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an important area of study of classical differential geometry, and many methods have been used to describe immersions and types of deformations as well. The generalized Weierstrass representation put forward first by Konopelchenko and Taimanov [1-2] is particularly useful in considering these particular kinds of problems. Surfaces and their dynamics are very important ingredients in a great number of phenomena in physics and applied mathematics [3]. They appear in the study of surface waves, shock waves, deformations of membranes, and many problems in hydrodynamics connected with the motion of boundaries between regions of differing densities and viscosities [3-4]. Of special interest is the case of surfaces which have zero mean curvature and such surfaces are referred to as minimal surfaces. The most general method for constructing minimal surfaces in three-dimensional Euclidean space was introduced by Enneper and Weierstrass in the ninteenth century. Such subjects as that of waves, deformation of membranes, dynamics of vortex sheets, two-dimensional gravity and string theory have areas of mutual interest with the theory of surfaces [5]. Such dynamics can be modeled by nonlinear partial differential equations that describe the evolution of surfaces in time. The theory of constant mean curvature surfaces has had a great impact on many problems with physical applications. It is worth discussing some of these applications at greater length by way of introduction. Consider the propagation of a string through space-time. It describes a surface called its world sheet. When one quantizes a string, the result is an ordinary two-dimensional point particle quantum field theory on a surface. It will be seen here that generalized Weierstrass representations provide a setting for formulating string theory and even for quantizing strings. Another area of recent interest with regard to applications is to the area of liquid crystals and the theory of membranes. Fluid membranes may be idealized as two-dimensional surfaces with each membrane being made up of a double layer of long molecules. Various physical properties of interest such as elastic free energy per unit area can be calculated in terms of quantities which are directly related to the geometry of the surface. In fact, the curvature elastic free energy per unit area of the membrane can also be formulated rigorously in terms of two-dimensional differential invariants of the surface. It is of considerable interest with regard to these types of physical applications to obtain shape equations for the membrane. These interrelate the basic parameters and functions which determine the form of a given membrane surface, as in a liquid crystal. In an equilibrium state, the energy of any physical system must be minimized. One usually writes down a shape energy function F in terms of the basic parameters and then minimizes it, and the result is a shape equation. An example of such a shape function is given by    1 F = kc (2H + c0 )2 dA + Δp dV + λ dA, (1.1) 2

where kc is the bending rigidity of the membrane, H the mean curvature and the spontaneous curvature c0 takes account of the assymmetry effect of the membrane or the surrounding environment [6]. The pressure difference between the outside and the inside of the membrane is called Δp, λ the tensile stress acting on the membrane. Mathematically, Δp and λ may be considered as Lagrange multipliers. The shape equation is obtained from the first variation of F . Specific Delauney’s surfaces of constant mean curvature can be written Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Generalized Weierstrass System Inducing Surfaces...

225

as sin ψ(ρ) = aρ + d/ρ. This equation is then substituted into a given shape equation and results in constraint equations between the parameters a and b, which give a characterization of the surface. There are many other applications of the shape equation as well [7]. Also, one of the classical problems in differential geometry is the study of the connection between the geometry of submanifolds and partial differential equations [8]. The Liouville and sine-Gordon equations, which describe minimal and pseudospherical surfaces are well-known examples. These two equations were also the first nonlinear partial differential equations to reveal a deep connection between differential geometry and soliton theory. Let us now review some basic facts concerning the classical theory of surfaces which can be approached in a number of ways [9]. Let us consider a surface in three-dimensional Euclidean space R3 and denote local coordinates for the surface by X i = xi (u1 , u2 ),

i = 1, 2, 3,

where X i are the coordinates of the variable point of the surface and xi are scalar functions. The basic characteristics of the surface are given by the first and second fundamental forms I = gαβ duαduβ ,

II = hαβ duα duβ .

(1.2)

where α, β = 1, 2, 3 and gαβ and hαβ are symmetric tensors such that gαβ =

∂X i ∂X i , ∂uα ∂uβ

hαβ =

∂ 2X i N i. ∂uα∂uβ

(1.3)

Here N i are the components of the normal vector

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N i = (det g)−1/2 ikl

∂X k ∂X i . ∂u1 ∂u2

(1.4)

The matrix gαβ completely defines the intrinsic properties of the surface. The Gaussian curvature K of the surface is given by K = R1212(det g)−1 , where Rαβγδ is the Riemann tensor. Extrinsic properties of surfaces are described by the Gaussian curvature K and the mean curvature H = g αβ hαβ . Embedding the surface into R3 is described by both gαβ and hαβ as governed by the Gauss-Codazzi equations i ∂2X i γ ∂X − Γ − hαβ N i = 0, αβ ∂uα ∂uβ ∂uγ

i ∂N i γβ ∂X + h g = 0. αγ ∂uα ∂uβ

(1.5)

γ

Here, Γαβ are the Christoffel symbols which can be determined from the first fundamental form. Among the global characteristics of surfaces, there is the Euler characteristic  1 χ= K(det g)1/2 du1 du2 , (1.6) 2π S

where integration is performed over the entire surface. For compact, oriented surfaces χ = 2(1 − n) where n is the genus of the surface. It is perhaps useful at this point to expand on some of these classical results concerning surfaces from the classical point of view which arise out of classical differential geometry. This will give a basic review of surface theory and some preparation for what is to follow. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Paul Bracken

Let (u, v) represent the parameters now and let r = r(u, v) denote the position vector of a generic point P on a surface Σ in R3 . Then the vectors ru and rv are tangential to Σ at P and at such points at which they are linearly independent, the quantity N=

ru × rv , |ru × rv |

(1.7)

determines the unit normal to Σ. The first and second fundamental forms of Σ are defined by I = ds2I = dr · dr = E du2 + 2F dudv + G dv 2, (1.8) 2 2 2 II = dsII = −dr · dN = e du + 2f dudv + g dv . In (1.8), the coefficients are defined by E = ru · ru ,

F = ru · rv ,

G = rv · rv , (1.9)

e = −ru · Nu , g = −rv · Nv , f = −ru · Nv = −rv · Nu . There is a classical result of Bonnet which states that {E, F, G, e, f, g} determines the surface Σ up to its position in space. The Gauss equations associated with Σ are given as ruu = Γ111 ru + Γ211 rv + eN,

ruv = Γ112 ru + Γ212 rv + f N,

rvv = Γ122 ru + Γ222 rv + gN, (1.10)

while the Weingarten equations are given by

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Nu =

f F − eG eF − f E ru + rv , 2 H H2

Nv =

gF − f G f F − gE ru + rv , 2 H H2

(1.11)

where H 2 = |ru × rv |2 = EG − F 2 . The Γijk are the Christoffel symbols and since the derivatives of all the {E, F, G, e, f, g} with respect to u and v can be calculated from (1.9) and (1.10), the derivatives of all the Γijk can be calculated as well. Thus, using these derivatives, the compatibility conditions (ruu )v = (ruv )u and (ruv )v = (rvv )u applied to the linear Gauss system (1.10) produces the nonlinear MainardiCodazzi system ev −fu = eΓ112 +f (Γ212 −Γ111 )−gΓ211 ,

fv −gu = eΓ122 +f (Γ222 −Γ112 )−gΓ212 . (1.12)

The Theorema egregium of Gauss provides an expression for the Gaussian or total curvature K=

eg − f 2 , EG − F 2

(1.13)

or in terms of E, F , and G alone in Liouville’s representation K=

1 H 2 H [( Γ11 )v − ( Γ212 )u ]. H E E

(1.14)

If the total curvature of Σ is negative, that is, if Σ is a hyperbolic surface, then the asymptotic lines on Σ may be taken as parametric curves. Then e = g = 0 and the Mainardi-Codazzi equations reduce to

f f f f )u + 2Γ212 = 0, ( )v + 2Γ112 = 0. Partial Differential Equations: Theory, Analysis and Applications : Theory,H Analysis and Applications, H H Nova Science Publishers, H Incorporated, 2011. (

(1.15)

The Generalized Weierstrass System Inducing Surfaces... Moreover, we have K=−

227

f2 1 = − 2, H2 ρ

GEv − F Gu EGu − F Ev , Γ212 = . 2 2H 2H 2 The angle between the parametric lines is such that Γ112 =

F , cos ω = √ EG

H sin ω = √ , EG

(1.16)

(1.17)

and since E, G > 0, we may take without loss of generality E = ρ2 a2 ,

G = ρ2 a2 ,

f = abρ sin ω.

(1.18)

Then the Christoffel symbols are given by Γ112 =

ρv a + ρav − cos ω(ρu b + ρbu) , ρa sin2 ω

Γ212 =

ρu b + ρbu − cos ω(ρv a + ρav ) . ρb sin2 ω (1.19)

Substituting (1.19) into the pair (1.15), there results 2ρav + 2ρv a − 2 cos ω(ρub + ρbu ) − ρv a sin2 ω = 0, 2ρbu + 2ρu b − 2 cos ω(ρv a + ρav ) − ρu b sin2 ω = 0.

(1.20)

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Solving the linear system in (1.20) for av and bu , we obtain av +

a ρv 1 ρu − b cos ω = 0, 2 ρ 2 ρ

bu +

b ρu 1 ρv − a cos ω = 0. 2 ρ 2 ρ

(1.21)

The representation for the total curvature is 1 ρu b 1 ρv a ωuv + ( sin ω)u + ( sin ω)v − ab sin ω = 0. 2 ρ a 2 ρ b

(1.22)

For the particular case in which K = −1/ρ2 < 0 is a constant, Σ is referred to as a pseudospherical surface. Then (1.21) implies that a = a(u), b = b(v), and if Σ is now parametrized by arc length along asymptotic lines, then ds2I = du2 + 2 cos ω dudv + dv 2 ,

ds2II =

2 sin ω dudv. ρ

(1.23)

Equation (1.22) then reduces to the sine-Gordon equation ωuv =

1 sin ω. ρ2

(1.24)

Thus, there is a clear indication of a relationship between surfaces and an integrable equation, which has soliton solutions [10]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Weierstrass and Enneper [11] started off by introducing two holomorphic functions ψ(z) and φ(z) and three complex-valued functions ω1 , ω2 and ω3 which satisfy the following system of equations ∂ω1 = i(ψ 2 + φ2 ),

∂ω2 = ψ 2 − φ2 ,

¯ = 0, ∂ψ

∂ω3 = −2ψφ,

¯ = 0. ∂φ

(1.25)

where z = x + iy and the derivatives are abbreviated as ∂ = ∂/∂z and ∂¯ = ∂/∂ z¯, such that the bar denotes complex conjugation. They show that if the system of three real-valued functions X i (z, z¯) are considered to be a coordinate system for a surface immersed in R3 defined by  X 1 =  ω1 = 

X 2 =  ω2 = 

i(ψ 2 + φ2 ) dz,

C



C

X 3 =  ω3 = −

(ψ 2 − φ2 ) dz,

(1.26)

 2ψφ dz, C

where C is any contour in the domain of holomorphicity of ψ and φ, then the resulting functions X i(z, z¯) determine a minimal surface. To begin to generalize this idea, suppose the functions ψ1 and ψ2 satisfy the more general system of equations ∂ψ1 =

1 p(z, z¯)Hψ2, 2

¯ 2 = − 1 p(z, z¯)Hψ1 , ∂ψ 2 2

(1.27)

2

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p(z, z¯) = |ψ1 (z, z¯)| + |ψ2 (z, z¯)| . and their complex conjugates, with real potential p(z, z¯). Solutions of system (1.27) can then be used to define the coordinates of a surface in three-dimensional Euclidean space by means of integrals similar to those in (1.26). This program was first put forward by Konopelchenko and Taimanov [1], [12]. The mean curvature function is H(z, z¯) in (1.27) and need not be constant in general. It will be seen momentarily how (1.27) arises. The coordinates (z, z¯) are conformal and in terms of these, the metric and Gaussian curvature are given by ∂ ∂¯ log p I = p(z, z¯)2 dzd¯ z, K=− . (1.28) p2 We can now ask how wide the class of surfaces represented by the Weierstrass formulas (1.28) is. Let F : Σ → R3 be a regular mapping of the domain Σ of the complex plane with coordinates (z, z¯) into three-dimensional Euclidean space, and metric tensor given by (1.28). In this case, the vector G(z) = (∂F1 , ∂F2 , ∂F3 ), (1.29) satisfies the equation

(∂F1 )2 + (∂F2 )2 + (∂F3 )2 = 0.

Therefore, (Fx − iFy , Fx − iFy ) = (Fx , Fx) − (Fy , Fy ) = 0.

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(1.30)

The Generalized Weierstrass System Inducing Surfaces...

229

This immediately follows from the formula G(z) = ∂F = 21 (Fx − iFy ) and the condition that the metric is conformally Euclidean (Fx , Fx) = (Fy , Fy ), (Fx, Fy ) = 0. The subvariety Q ⊂ CP 1 is defined in terms of the homogeneous coordinates (φ1 , φ2 , φ3 ) by φ21 + φ22 + φ23 = 0. It is diffeomorphic to the Grassmann manifold G3,2 formed by two-dimensional subspaces of R3 . This diffeomorphism is given by the mapping G3,2 → Q, which assigns the point (a1 +ib1 , a2 +ib2 , a3 +ib3 ) ∈ Q to the plane generated by the pair of unit vectors (a1 , a2 , a3 ) and (b1 , b2, b3 ). Thus, G can be regarded as the Gauss map. The Gauss map defined in this way for the surface takes the form i 1 G(z) = ( (ψ¯12 + ψ22 ), (ψ¯12 − ψ22 ), −ψ¯1ψ2 ). 2 2

(1.31)

Solving (1.30) and (1.31) for ψ12 and ψ22 , we obtain ¯ 2 + i∂F ¯ 1, ψ12 = ∂F

ψ22 = −∂F2 − i∂F1 .

These results give rise to the following Proposition. Proposition 1.1. Every regular conformally Euclidean immersion of a surface into three-dimensional Euclidean space is locally defined by the generalized Weierstrass formulas (1.27). Proof: Assume that ∂F3 = 0, otherwise change coordinates in R3 . Let us compare G(z) = 12 (Fx − iFy ) with G in the form of the Gauss map and define the functions

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¯ 2 + i∂F ¯ 1, ϕ21 = ∂F

ϕ22 = −∂F2 − i∂F1 ,

(1.32)

and their conjugates. In fact, these imply that −(∂F1 )2 − (∂F2 )2 = ϕ¯21 ϕ22 and therefore, (∂F1 )2 + (∂F2 )2 + (∂F3 )2 = 0. Also (1.32) can be solved for ϕ1 and ϕ2 as square roots. Recall the definition of the second fundamental form hij . Let the metric tensor on the surface F : Σ → R3 be given by (1.28). Take an orthonormal basis in the tangent plane at the point z, e1 =

1 Fx , p

e2 =

1 Fy , p

and extend it to a basis in R3 by including a unit normal vector e3 = e1 × e2 . Components of the curvature tensor are defined by the decomposition formulas Fxx = pxe1 −py e2 +p2 h11 e3 ,

Fxy = py e1 +px e2 +p2 h12 e3 ,

Fyy = −px e1 +py e2 +p2 h22 e3 . (1.33) Given the system (1.33), we derive the associated system satisfied by (ϕ1 , ϕ2 ). We make use of the identity 1 ∂ ∂¯ = (∂x2 + ∂y2 ). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and 4Applications, Nova Science Publishers, Incorporated, 2011.

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Paul Bracken

¯ 2 + i∂F ¯ 1 with respect to ∂, we obtain Differentiating ϕ21 = ∂F ¯ 2 + i∂ ∂F ¯ 1= 2ϕ1 ∂ϕ1 = ∂ ∂F

1 2 i (∂ F2 + ∂y2 F2 ) + (∂x2 F1 + ∂y2 F1 ). 4 x 4

(1.34)

An explicit formula for e3 is required and can be obtained by starting with the representations 1 1 e1 = Fx = (F1,x, F2,x, F3,x), p p

e2 =

1 1 Fy = (F1,y , F2,y , F3,y ). p p

Taking the cross product of e1 and e2 , we have e1 × e2 =

1 (F2,xF3,y − F3,x F2,y , −(F1,xF3,y − F1,y F3,x ), F1,xF2,y − F1,y F2,y ). p2

Thus, 8ϕ1 ∂ϕ1 = +i(

py py px px F2,x− F2,y +h11 (−F1,x F3,y +F1,y F3,x )− F2,x+ F2,y +h22 (−F1,x F3,y +F1,y F3,x ) p p p p

px py px py F1,x − F1,y +h11 (F2,x F3,y −F3,x F2,y )− F1,x + F1,y +h22 (F2,xF3,y −F3,xF2,y )) p p p p = (h11 + h22 )(−F1,xF3,y + F1,y F3,x) + i(h11 + h22 )(F2,xF3,y − F3,x F2,y )

(1.35)

= (h11 + h22 )[iF3,y (iF1,x + F2,x ) − F3,x (iF2,y − F1,y )].

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Finally, it is required to replace the derivatives of Fj in terms of the functions ϕj and their complex conjugates. To do this, we write explicitly, 1 ϕ21 = (F2,x + iF2,y + iF1,x − iF1,y ), 2

1 ϕ22 = (−F2,x + iF2,y − iF1,x − F1,y ), (1.36) 2

and their complex conjugate equations. From (1.36), it follows that ϕ21 − ϕ22 = F2,x + iF1,x,

ϕ21 + ϕ22 = iF2,y − F1,y .

Moreover, ∂F3 =

1 ¯1 ϕ2 , (∂x − i∂y )F3 = −ϕ 2 ∂x F3 = −(ϕ¯1 ϕ2 + ϕ1 ϕ ¯2 ),

¯ 3 = 1 (∂x + i∂y )F3 = −ϕ1 ϕ¯2 , ∂F 2 i∂y F3 = ϕ¯1 ϕ2 − ϕ1 ϕ ¯2 .

Substituting these results into the final expression produced in (1.35), we obtain 8ϕ1 ∂ϕ1 = (h11 + h22 )[iF3,y (ϕ21 − ϕ22 ) − F3,x (ϕ21 + ϕ22 )] = (h11 + h22 )[(ϕ¯1 ϕ2 − ϕ1 ϕ¯2 )(ϕ21 − ϕ22 ) + (ϕ¯1 ϕ2 + ϕ1 ϕ¯2 )(ϕ21 + ϕ22 )] = 2(h11 + h22 )(|ϕ1|2 + |ϕ2 |2 )ϕ1 ϕ2 . Solving for ∂ϕ1 and using the definition of mean curvature in terms of hij , we have the first equation in (1.27). Similarly, we can work out ¯ 2 = −(h11 + h22 )(−F1,xF3,y + F1,y F3,x + iF2,xF3,y − iF3,xF2,y ). 8ϕ2 ∂ϕ

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The Generalized Weierstrass System Inducing Surfaces...

231

The right-hand side of this result is identical except for sign to what was worked out in the previous case, hence, ¯ 2 = −2ϕ1 ϕ2 (h11 + h22 )p. 2ϕ2 ∂ϕ This is the second equation in (1.27). This finishes the proof. It remains to specify, given solutions ψi of (1.27) how the surface inducing is to take place. This was done by Konopelchenko, and makes use of the fact that the solutions of (1.27) satisfy a specific set of conservation laws. Thus, Konopelchenko has established a connection between certain classes of constant mean curvature surfaces and the trajectories of an infinite-dimensional Hamiltonian system of the form (1.27). He considered the nonlinear Dirac-type system of equations in terms of two complex valued functions ψ1 and ψ2 which, after absorbing constants into the derivative variables, satisfy ∂ψ1 = pψ2 ,

¯ 2 = −pψ1 ∂ψ

∂¯ψ¯1 = pψ¯2 ,

∂ ψ¯2 = −pψ¯1 ,

(1.37)

p = |ψ1 |2 + |ψ2|2 . System (1.37) has been referred to as the Generalized Weierstrass (GW) system in the literature recently [13-17]. Using (1.37), it can be verified that the following conservation laws hold

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¯ 2) = 0, ∂(ψ12) + ∂(ψ 2

¯ ψ¯2 ) + ∂(ψ¯2) = 0, ∂( 1 2

¯ ψ¯1 ψ2 ) = 0. ∂(ψ1 ψ¯2 ) + ∂(

(1.38)

Making use of these conserved quantities, there exist three real-valued quantities Xi (z, z¯) which are completely determined by the following integrals   X1 + iX2 = 2i (ψ¯12 dz  − ψ¯22 d¯ z  ), X1 − iX2 = 2i (ψ22 dz  − ψ12 d¯ z  ), Γ

 X3 = −2

Γ

Γ

(ψ¯1 ψ2 dz  + ψ1 ψ¯2 d¯ z  ).

(1.39)

On account of conservation laws (1.38), these integrals are found to be independent of the path Γ chosen. To put it another way, the differentials integrated in equations (1.39) are exact ones. Thus, the integrals (1.39) have the general form  ¯ z¯) d¯ G(z, z¯) dz + G(z, z Γ

where the function G satisfies the condition ¯ = ∂ G, ¯ ∂G by virtue of conservation laws (1.38). Integrals of this form are independent of Γ, hence integrals (1.39) are independent of Γ. The functions Xi(z, z¯) can be treated as the coordinates of a surface immersed in R3 . The Gaussian curvature and the first fundamental form of the surface are given by

¯ ∂ ∂(log p) , Ω = 4p2 dzd¯ z, 2 p Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. K=−

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Paul Bracken

in isothermic coordinates. There is also a current which is conserved and given by J = ψ¯1 ∂ψ2 − ψ2 ∂ ψ¯1 .

(1.40)

¯ = 0 modulo (1.37). Proposition 1.2. The current (1.40) satisfies ∂J Proof: Differentiating (1.40) and substituting (1.37), there results ¯ = ∂¯ψ¯1 ∂ψ2 + ψ¯1 ∂∂ψ ¯ 2 ∂ ψ¯1 − ψ2 ∂∂ ¯ 2 − ∂ψ ¯ ψ¯1 = p(ψ¯1 ∂ ψ¯1 + ψ¯2 ∂ψ2 − ∂p). ∂J However, differentiating p in (1.37), we have ∂p = ψ1 ∂ ψ¯1 + ψ¯2 ∂ψ2 , the result follows. The integrability of system (1.37) has been examined extensively [15-17] by using Cartan’s theorem on systems in involution using a set of differential forms which are equivalent to system (1.37). A B¨acklund transformation has also been determined for GW system (1.37) [18].

2. Two-Dimensional Sigma Model and Correspondence with Generalized Weierstrass System 2.1. Second-Order System Associated with the generalized Weierstrass System

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At this point, a correspondence between system (1.37) and the two-dimensional nonlinear sigma model can be made. Introduce the new variable ρ which is defined in terms of the ψi as ψ1 ρ= ¯ . (2.1) ψ2 Using (1.37), it can be seen that ∂ψ1 ψ1 ∂ρ = ¯ − ¯2 ∂ ψ¯2 = (1 + |ρ|2)ψ22 . ψ2 ψ2

(2.2)

Solving for ψ22 in (2.2), using (2.1) to get ψ1 , the following transformation from ρ to the set of ψi is produced ψ1 = ρ

(∂¯ρ¯)1/2 , 1 + |ρ|2

ψ2 = 

(∂ρ)1/2 , 1 + |ρ|2

 = ±1.

(2.3)

Proposition 2.1. If ψ1 and ψ2 are solutions of GW system (1.37), then the function ρ defined by (2.1) is a solution of the second order sigma model system ¯ − ∂ ∂ρ

2¯ ρ ¯ = 0, ∂ρ∂ρ 1 + |ρ|2

∂ ∂¯ρ¯ −

2ρ ∂ ρ¯∂¯ρ¯ = 0. 1 + |ρ|2

(2.4)

Proposition 2.2. If ρ is a solution to sigma model system (2.4), then the functions ψ1 and ψ2 defined in terms of ρ by the expressions ψ1 = ρ satisfy GW system (1.37).

(∂¯ρ¯)1/2 , 1 + |ρ|2

ψ2 = 

(∂ρ)1/2 , 1 + |ρ|2

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(2.5)

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Equivalently, given a solution to sigma model (2.4), a surface can be obtained by calculating the ψi by means of (2.5) in Proposition 2.2 and then substituting the ψi into (1.39) to obtain the coordinates of the corresponding surface. This may seem involved, but the classical symmetry group can be determined and it yields large classes of nontrivial solutions to (2.4). This procedure is very useful for producing nontrivial solutions of (2.4) and hence surfaces using (2.5) and then (1.39). It will be seen that the symmetry structure of (2.4) is sufficiently complicated enough to be able to generate a great variety of solutions ρ, and by means of (2.3) to GW system (1.37) as well. Once the ψi have been calculated from (2.3), the coordinates of a surface follow from (1.39). The symmetry group of (1.37) has also been found but will not be presented here. The solutions it leads to are rather standard, but the results for it are given in [14].

2.2. Group Invariant Solutions of the Sigma Model Explicit solutions of GW system (1.37) based on transformation (2.3) will be found which uses a variety of invariant solutions of sigma model system (2.4) obtained by means of the symmetry reduction method to ODEs. In this approach, we need to find the symmetry group G for (2.4), and then classify all subgroups Gi of G having generic orbits of codimension one in the space of independent variables. We then find the associated invariants of each of its subgroups Gi , and perform for each of these invariants the symmetry reduction of (2.4) to a system of ODEs which can then be solved [14]. Let us examine the system of real PDEs which are equivalent to the two-dimensional sigma model (2.4). Introduce into (2.4) a polar representation for ρ

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ρ = Reiφ .

(2.6)

The real and imaginary parts which result from doing this are given by φxx + φyy +

2(1 − R2 ) (Rxφx + Ry φy ) = 0, R(1 + R2 ) (2.7)

Rxx + Ryy −

2

R(1 − R ) 2 2R (φx + φ2y ) − (R2 + R2y ) = 0. 1 + R2 1 + R2 x

Note that if we put R = 1, then (2.7) is identically satisfied and the first one reduces to the Laplace equation for the phase φ. This implies that φ has to be a periodic, harmonic function with a period equal to 2π. If the period of φ is not 2π, then the solution will become multivalued. Equation (2.7) is invariant under the discrete transformations generated by the reflections x → 1 x,

y → 2 y,

R → 3 R,

φ → 4 φ,

i = ±1,

i = 1, · · · , 4,

and also the inversion given by 1 , φ → φ. η Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. R→

234

Paul Bracken

The symmetry algebra L can be decomposed into a direct sum of two infinitedimensional simple Lie subalgebras with direct sum of a one-dimensional algebra generated by Φ = ∂φ L = {α+ } ⊕ {α− } ⊕ {Φ}. (2.8) There is only one finite-dimensional subalgebra spanned by the vector fields P1 = ∂x , D = x∂x + y∂y , 2

2

C1 = (x − y )∂x + 2xy∂y ,

P2 = ∂y , L3 = y∂x − x∂y ,

(2.9)

2

2

C2 = 2xy∂x − (x − y )∂y .

The operators P1 and P2 generate translations in the xy directions, D and L3 correspond to dilation and rotation, C1 and C2 generate two different types of conformal transformations. The nonvanishing commutation relations for algebra (2.9) can be calculated showing it closes. The commutators are given by [C1 , L3 ] = C1 ,

[C1 , D] = −C2 ,

[C1 , P1 ] = −2D,

[C1 , P2 ] = 2L3 ,

[C2 , L3 ] = −C2 ,

[C2 , D] = C1 ,

[C2 , P1 ] = −2L3 ,

[C2 , P2 ] = −2D,

[L3 , P1 ] = P2 ,

[L3 , P2 ] = −P1 ,

[D, P1] = −P1 ,

[D, P2 ] = −P2 .

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These relations show that this algebra is isomorphic to the O(3, 1) algebra. In this case, among all nonconjugate subalgebras, the ones that have generic orbits of codimension one in the space of independent variables and three in the space of independent and dependent variables {x, y, R, φ} reduce the original system (2.7) to a system of ODEs by means of the symmetry reduction method. The one-dimensional subalgebras are given by P1 + bΦ,

L3 + bΦ,

D + bΦ,

D + aL3 + bΦ,

(2.10)

where a and b are real parameters. In order to find the relation associated with subalgebras (2.10), we compute for each of them the corresponding invariants by solving the partial differential equations XH(x, y, R, φ) = 0, (2.11) where H is an auxiliary function of four variables (x, y, R, φ), and X is one of the generators listed in (2.10). The solution of (2.11) is found by integrating the associated characteristic system. Three invariants ξ, R and F are found which are given in Table 1. The orbits of the subgroups of G can be expressed in terms of two functions R and φ in the following form R = R(ξ), φ = α(x, y) + F (ξ), ξ = ξ(x, y), (2.12) where α and ξ are given functions of x and y for each subalgebra. The function ξ is the symmetry variable of the invariance subgroup having generic orbits of codimension one. Substituting each specific form (2.12) into system (2.7) leads to the coupled system of ODEs in terms of the symmetry variable ξ

2R ˙ 2 R(1 − R2 ) ˙ 2 g˙ ˙ R(1 − R2 ) R(1 − R2 ) ˙ R φ φ − l R − 2h − − = 0, 2 1 +Analysis R2 and Applications 1 +: Theory, R2 Analysis andg Applications, Nova 1+ R2Publishers, Incorporated, 1 + R2011. Partial Differential Equations: Theory, Science ¨− R

The Generalized Weierstrass System Inducing Surfaces...

235

1 − R2 ˙ ˙ g˙ ˙ 1 − R2 ˙ φ¨ + 2 R φ − φ + 2h R + s = 0, R(1 + R2 ) g R(1 + R2 )

(2.13)

where the functions g, h, l and s are given for each of the subalgebras in Table 1, and dot means differentiation with respect to ξ. The differential equations for R and φ in (2.13) can be decoupled if we perform the transformation φ˙ = V − h

(2.14)

on the second equation of (2.13). Then V must satisfy the equation 1 − R2 g˙ V˙ + 2R˙ V − V − m = 0, 2 R(1 + R ) g

(2.15)

where m = h˙ − (g/g)h ˙ − s. There are two cases to consider. (1) When m = 0, the V equation is a homogeneous ODE for the function V . This corresponds to the three subalgebras P1 + bΦ, L3 + bΦ, D + aL3 + bΦ. The general integral has the form V = Ag

(1 + R2 )2 , R2

A ∈ R,

(1 + R2 )2 φ˙ = Ag − h. R2

(2.16)

Eliminating φ˙ from (2.16) and the first equation of (2.13) gives

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¨− R

2 2 2R ˙ 2 g˙ ˙ R(1 − R2 ) 2 2 (1 − R )(1 + R ) 2 − g + (h − l) = 0. R R − A 1 + R2 g R3 1 + R2

(2.17)

(2) When m = 0 which corresponds to the subalgebra D + bΦ, the general solution is obtained by the method of variation of parameters, and has the following form  (1 + R2 )2 mR2 V = A(ξ)g , A(ξ) = dξ  . (2.18) R2 g(1 + R2 )2 These can be substituted to eliminate φ in favor of R as ¨− R

2 2 3 2R ˙ 2 g˙ ˙ R(1 − R2 ) 2 2 (1 − R )(1 + R ) 2 R − g + (h − l) = 0, R − A 1 + R2 g R3 1 + R2

which is of the same form as (2.17). This ODE has the Painlev´e property, that is, no movable singularities other than simple poles, so we can transform to a new variable U as follows, R(ξ) = (−U (ξ))1/2. then the function U has to satisfy the second order ODE 2 3 ¨ = ( 1 − 1 )U˙ 2 − g˙ U˙ + 2C (1 + U )(1 − U ) . U 2U 1−U g g2 U

Choosing a new variable



ξ

dt , g(t) = 0, g(t) ξ0 and Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis Applications, Nova Science Publishers, Incorporated, 2011. η=

236

Paul Bracken

in this, it is transformed into the following ODE with η as the independent variable 2 ¨ = ( 1 − 1 )U˙ 2 + 2C (1 + U )(1 − U )3 . U 2U 1−U U

(2.19)

If C = 0, the solution is given by U = tanh2 (K1 η + K2 ),

K1 , K2 ∈ R.

(2.20)

If C = 0, the equation can be found in Ince’s [19] classification, where β = −α = 2C 2 and γ = δ = 0, in Ince’s notation. This equation admits a first integral given by U˙ = −4C 2 U 4 + 4C1 U 3 + 8(C 2 − C1 )U 2 + 4C1 U − 4C 2 ,

C1 ∈ C,

(2.21)

where differentiation is with respect to η. Equation (2.21) can be written for C = 0 in equivalent form

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U˙ 2 = −4C 2 (U − U1 )(U − U2 )(U − U3 )(U − U4 ),

(2.22)

where Ui with i = 1, · · · , 4 denote the constant roots of the right-hand side of (2.21), and can be expressed in terms of B, C and C1 . The behaviour of the solutions of (2.22) depends upon the relationships between the roots of the quartic polynomial on the right-hand side of (2.22). It can be solved in terms of elliptic Jacobi functions or in terms of elementary algebraic functions with one or two simple poles, and trigonometric and hyperbolic solutions, which have been extensively tabulated [14,17], and will not be reproduced here. For example, when C = C1 = 0, (2.21) can be integrated easily to give √ U = D exp(± −2Bη). Given U , we can obtain R and φ thus giving ρ by means of (2.1), which can then be substituted into (2.3) to produce both ψ1 and ψ2 . These functions generate a surface by means of the integrals (1.39). When C1 = 0, the quadratic can be written in factorized form U˙ 2 = 4C1 (U − U1 )(U − U2 )U, where the roots are given by U1,2 = (n + 1) ±



n(n + 2).

with n = B/4C1 . We have summarized all possibilities in [14].

2.3. Multisoliton Solutions

Now the case in which some classes of solutions of GW system (1.37) can be obtained from transformation (2.3) such that the solutions of the sigma model (2.4) are invariant under the scaling transformation D = ρ∂ρ − ρ¯∂ρ¯ is considered. This means we subject the system (2.4) to the algebraic constraint (2.23) |ρ|2 = 1. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Generalized Weierstrass System Inducing Surfaces...

237

It follows from (2.23) that sigma model (2.4) can be written as ¯ − ρ¯∂ρ∂ρ ¯ = 0, ∂ ∂ρ and has a solution of the form

¯ ρ¯ − ρ∂ ρ¯∂¯ρ¯ = 0, ∂∂

(2.24)

ρ = eiϕ ,

(2.25)

where ϕ is any real harmonic function of z and z¯. From (2.4), we have the identity, ¯ − ρ¯∂ρ∂ρ ¯ = eiϕ (i∂ ∂ϕ ¯ − ∂ϕ∂ϕ) ¯ + eiϕ (e−iϕ¯ eiϕ ∂ϕ∂ϕ) ¯ = 0. ∂ ∂ρ Transformation (2.3) becomes ψ1 =

 iϕ/2 ¯ 1/2 (∂ϕ) , e 2

ψ2 =

 iϕ/2 (∂ϕ)1/2, e 2

 = ±1,

and satisfies GW system (1.37). In addition to solutions produced by the symmetry group discussed in the last section, the following two theorems give a straightforward way to produce multisoliton solutions to GW system (1.37). Theorem 2.1. Suppose that for any complex-valued function F of class C 1 , the function ρ satisfies the algebraic condition (2.23) and the differential constraints ¯ = −F¯ (¯ ∂ρ z )ρ.

∂ρ = F (z)ρ,

(2.26)

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Then the complex valued function ρ is a nonsplitting solution of (2.4). The associated surface has zero Gaussian curvature. Proof: From (2.26), we have ¯ = F (z)∂ρ. ¯ ∂∂ρ Thus system (2.4) is identically satisfied, since ¯ − ρ(F F (∂ρ) ¯ ρ)(−F ρ) = F (−F¯ ρ) − F F¯ ρ = 0, holds. Moreover, from (2.3) we get  ∂ψ1 = (F¯ ρ)1/2F, 4

¯ 2 = −  (F ρ)1/2F¯ , ∂ψ 4

p=

1 (F F¯ )1/2 , 2

 = ±1.

The GW system is satisfied identically. Moreover, we have ∂ ln p =

∂p ∂F = , p 2F

¯ which implies that ∂(∂p) = 0, since F is a function of only z. Thus, from the formula for K, the Gaussian curvature is zero. QED. There is also the possibility of constructing more general classes of solutions of GW system (1.37) which are based on nonlinear superpositions of N elementary solutions of sigma model (2.4). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Paul Bracken

Theorem 2.2. Suppose that for each i = 1, · · · , N the complex valued functions ρi satisfy sigma model system (2.4) as well as the conditions |ρi|2 = 1. Then the product of the functions ρi N  ρ= ρi , (2.27) i=1

is also a solution to system (2.4). The corresponding solution of the GW system (1.37) takes the factorized form, N

N

N

 1  ψ1 =  ρi(∂¯ ρ¯j )1/2, 2 i=1

ψ2 =

j=1

 1 ρj )1/2 . (∂ 2 j=1

Proof: It suffices to prove the theorem for N = 2, and then to invoke the process of induction to extend the result to a product with N functions. Suppose that ρ1 , ρ2 are solutions to (2.4). Substituting the function ρ = ρ1 ρ2 into (2.4), we obtain ¯ 1 )ρ2 +∂ρ1 ∂ρ ¯ 2 + ∂ρ ¯ 1 − ∂ρ ¯ 2 −ρ1 ρ¯2 ∂ρ2 ∂ρ ¯ 2 ¯ 1 ∂ρ2 +ρ1 (∂∂ρ ¯ 2 )− ρ¯1 ρ2 ∂ρ1 ∂ρ ¯ 1 ∂ρ2 −∂ρ1 ∂ρ (∂∂ρ

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¯ 1 − ρ¯1 ∂ρ1 ∂ρ ¯ 1 ) + ρ1 (∂∂ρ ¯ 2 − ρ¯2 ∂ρ2 ∂ρ ¯ 2 ) = 0, = ρ2 (∂∂ρ whenever the functions ρi satisfy sigma model equations (2.4). Thus the product of two solutions is a solution. Consequently, proceeding inductively from the product form of ρ given in (2.27), if the equation is satisfied for a solution with k factors, it is satisfied for a function with k + 1 factors. Substituting (2.27) into (2.3), we obtain ψ1 and ψ2 which satisfy system (1.37). Let us now discuss the calculation of an algebraic multi-soliton solution of (1.37) and associated surface based on these theorems. (i) First, we look for a particular class of rational solutions to the sigma model which admit simple poles at z¯ = a ¯j given by ρj =

z − aj , z¯ − a ¯j

aj ∈ C,

j = 1, · · · , N.

(2.28)

A more general class of rational solution to (2.4) admitting simple poles by Theorem 2 is given by N  z − aj ρ= . (2.29) z¯ − a ¯j j=1

¯ = 0 and |ρ|2 = 1 as well. The following algebraic multisoliton This function satisfies ∂ ∂ρ solution results by applying the theorems, ψ1 =

N N  1 z − ak 1/2   ) , ( 2 (¯ z−a ¯j ) z¯ − a ¯k j=1

k=1

N N  1 z − ak 1/2   ψ2 = ( ) , 2 (z − aj ) z¯ − a ¯k j=1

 = ±1.

k=1

(2.30) Moreover, p and current J given by (1.40) are calculated to be N

p=

1  1 |, | 2 z − aj

N

J=

1 1  )2 . ( 4 z − aj

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(2.31)

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239

For the case N = 1, the functions ψ1 and ψ2 can be substituted into relations (1.39) which give the coordinates Xi of a surface. The corresponding constant mean curvature surface is then given by the algebraic relation a2 2X3 a2 a2 e )(X12 + X22 ) + e2X3 X2 + 1 − e2X3 = 0. (2.32) 4 2 4 ¯ (ii) A large class of hyperbolic nonsplitting solutions, that is, ∂ ∂ρ = 0, of the sigma model equations (2.4) can be constructed when the function ρ satisfies the algebraic constraint |ρ|2 = 1. Consider a class of nonsplitting hyperbolic solutions of (2.4) ((X1 )2 + (X2 )2 )2 − (2 +

ρ=

N 

exp(cosh(z − ai ) − cosh(¯ z−a ¯i )).

(2.33)

i=1

In all of the solutions of (2.4) which are presented below, the ai will be arbitrary complex constants. The derivatives of ρ with respect to ∂ and ∂¯ are given by ∂ρ =

N 

sinh(z − ai )ρ,

¯ =− ∂ρ

i=1

N 

sinh(¯ z − ¯ai )ρ.

(2.34)

i=1

respectively. Substituting (2.34) into (2.3), we obtain the following solutions to GW system (1.37), N

ψ1 =

  sinh(¯ z−a ¯i ))1/2, (¯ ρ 2 i=1

N

ψ2 =

  sinh(z − ai ))1/2, (ρ 2 i=1

N

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1  p= | sinh(z − ai )|. 2

(2.35)

i=1

Note that these solutions do not admit any singularities. Now, solution (2.35) for N = 1 is substituted into the integrals in (1.39) and we obtain X1 + iX2 = i sinh(− cosh(z − a) + cosh(¯ z − ¯a)), X1 − iX2 = −i sinh(− cosh(¯ z − ¯a) + cosh(z − a)),

(2.36)

X3 = − cosh(cosh(z − a) − cosh(¯ z−a ¯)). Eliminating the z-dependent factors on the right-hand side (2.36), the following relationship between the Xi variables is obtained X12 + X22 + X32 = 1.

(2.37)

This represents a sphere of unit radius. Note that similar results hold when sinh is used in place of cosh in expression for ρ. (iii) Consider a class of hyperbolic nonplitting solutions of (2.4) which are obtained from the tanh function that satisfies the algebraic condition |ρ|2 = 1. This type of solution represents a kink-type solution, and is generated by ρ=

N 

exp(tanh(z − ai ) − tanh(¯ z−a ¯i )).

i=1 Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(2.38)

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Table 1. The Symmetry Reduction for System (2.7) Algebra and symmetry variable P1 + bΦ

y L3 + bΦ  x2 + y 2

One-Dimensional Orbits of subgroups

Coefficients of the reduction to ODEs ()

R = R(ξ)

g = g0 , h = 0, s = 0

φ = bx + F (ξ)

l = 1, m = 0

R = R(ξ)

1 g˙ = − , h = 0, s = 0 g ξ

1 φ = b sin−1  x2 + y 2

l=

b2 ξ2 ,

m=0

+F (ξ) D + bΦ

R = R(ξ)

x y

φ = b ln x + F (ξ)

D + aL3 + bΦ  ln x2 + y2 1 x + tan−1 a y

R = R(ξ) b x φ = − tan−1 a y a>0

+F (ξ)

g˙ 2ξ b =− , h = ξ(1+ξ 2) g 1 + ξ2 b b2 s=− 2 , l = ξ2 (1+ξ 2) ξ (1 + ξ 2 ) 4b m=− (1 + ξ 2 )2 g = g0 , h = − l=

2b 1 + a2

b2 , s = 0, m = 0 1 + a2

Reduction to second order ODE 2R ˙ 2 R 1 + R2 2 (1 − R )(1 + R2 )3 −A2 g02 R3 R(1 − R2 ) − =0 1 + R2 ¨− R

2R ˙ 2 1 ˙ R + R 1 + R2 ξ A2 C 2 (1 − R2 )(1 + R2 )3 − 2 ξ R3 2 b R(1 − R2 ) − 2 =0 ξ 1 + R2 ¨− R

2ξ ˙ 2R ˙ 2 R + R 1 + R2 1 + ξ2 A(ξ)2 C 2 (1 − R2 )(1 + R2 )3 − (1 + ξ 2 )2 R3 2 R(1 − R ) =0 − (1 + ξ 2 )2 (1 + R2 ) ¨− R

2R ˙ 2 R 1 + R2 2 (1 − R )(1 + R2 )3 −A2 g02 R3 R(1 − R2 ) 2 3 − a2 + b ( )=0 1 + R2 (1 + a2 )2 ¨− R

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241

The derivatives of ρ in (2.38) are N 

∂ρ =

sech2 (z − ai )ρ,

¯ =− ∂ρ

i=1

N 

sech2 (¯ z−a ¯i )ρ.

(2.39)

i=1

Substituting (2.39) into (2.3), we obtain the following multi-soliton solution ψi to GW system (1.37), N

  ψ1 = (¯ sech2 (¯ z−a ¯i ))1/2 , ρ 2 i=1

N   ψ2 = (ρ sech2 (z − ai ))1/2 , 2 i=1

N

1  p= | sech2 (z − ai )|. 2

(2.40)

i=1

Note that the functions ψi admit only simple poles.

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3. Conditional Symmetries and a Linear Spectral Problem The objective of this section is to derive a representation of a linear spectral problem for which the matrices are nonsingular, and find the corresponding Darboux and B¨acklund transformations for the GW system. A number of relevant points concerning the conditional symmetry approach for PDEs as developed in [20] will be given. This is a large subject and so this will be brief, however its application to the GW system will be considered at length. Consider the overdetermined system composed of a nondegenerate k-th order scalar PDE and a first order system of differential constraints Δ(x, u(k)) = 0,

Qi (x, u(k)) =

∂u − φi (x, u(k)) = 0, ∂xi

i = 1, · · · , p

(3.1)

in p independent variables x = (x1 , · · · , xp) which form some local coordinates in Euclidean space X. The compatibility conditions for (3.1) are given by (i) φ[i,j] + φ[j φi],u = 0,

i, j = 1, · · · , p,

(ii) Δ(x, u, φ(k−1)) = 0,

φ = (φ1 , · · · , φp ).

(3.2)

The brackets [i, j] denote the alternation with respect to the indices i and j, φ[i,j] = 2(φi,j − φj,i ),

φ[i φj],u = 2(φiφj,u − φj φi,u ).

Note that the second equation of (3.1) means that the characteristics Qi of a set of p−linearly independent vector fields defined on X × U Zi = ∂xi + φi (x, u)∂u, are equal to zero.

i = 1, · · · , p,

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(3.3)

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An Abelian Lie algebra L spanned by the vector fields Z1 , · · · , Zp is called a conditional symmetry algebra of the k-th order partial differential equation (3.1), if the vector fields Z1 , · · · , Zp are tangent to the subvariety S = SΔ ∪ SQ , where we associate the initial system Δ : J k → R and a first order system of differential constraints Qi : J 1 → Rp with the subvarieties of the solution spaces SΔ = {(x, u(k)) ∈ J k : Δ(x, u(k)) = 0}, SQ = {(x, u(1)) ∈ J 1 : Qi (x, u(1)) = 0, i = 1, · · · , p}, respectively. This definition means that the k-th prolongation of the vector fields Zi belongs to the tangent space to S at (x, u(k)), that is,

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pr (k) Zi |S ∈ T(x,u(k) )S,

i = 1, · · · , p.

(3.4)

A solution u = f (x) of the k-th order PDE (3.1) is called a conditionally invariant solution, if its graph {(x, f (x))} is invariant under an Abelian distribution of the vector fields Z1 , · · · , Zp satisfying conditions (3.4) It has been shown [20] that a nondegenerate k-th order partial differential equation (3.1) admits a p-dimensional conditional symmetry algebra L if and only if there exists a set of p linearly independent vector fields (2.3) for which the C k−1 functions φi satisfy the conditions (2.2). The graph of a solution of the overdetermined system composed of both equations in (3.1) is invariant under the vector fields Zi , 1 ≤ i ≤ p. Hence according to the above definition, this means that there exists a conditionally invariant solution of the equation in (3.1). It is known [21] that any equation (3.1) of the k-th order admits infinitely many compatible first order differential constraints. However, this statement shows only existence of such constraints, but does not provide a constructive method for finding the explicit form of these differential constraints. The construction then of conditional symmetries is reduced to the selection of such subsystems composed of an initial partial differential equation and differential constraints (3.1) for which conditions (3.2) hold. In general, system (3.2) is a nonlinear one and usually very difficult to solve. At this point we don’t consider further the general theory, but simply describe how it can be applied to the system of interest here and see what results can be derived. It will prove useful to change the dependent variables ψ1 and ψ2 to new dependent variables p = |ψ1 |2 + |ψ2 |2 , and the current (1.40) in order to simplify its structure. In terms of these new variables, we show that GW system (1.37) can be decoupled into a direct sum of the elliptic Sh-Gordon ¯ = 0. equation and the conservation of current condition ∂J In fact, differentiating the function p with respect to z and z¯, it is found that

¯ = ∂ψ ¯ 1 ∂ ψ¯1 +∂ψ ¯ 2 ∂ψ2 −p3 . ∂ ∂p (3.5) Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. ∂p = ψ1 (∂ ψ¯1 )+ψ¯2 (∂ψ2 ),

¯ = ψ¯1 (∂ψ ¯ 1 )+ψ2 (∂¯ψ¯2 ), ∂p

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Making use of conservation laws, the GW system (1.37) takes the decoupled form in the variables p and J, ∂ ∂¯ ln p =

|J|2 − p2 , p2

¯ = 0, ∂J

∂ J¯ = 0.

(3.6)

If we introduce the new dependent variable p = eϕ/2 , into equation (3.6), we then obtain an elliptic sinh-Gordon type equation of the form ¯ = −4 sinh ϕ − 2(1 − |J|2 )e−ϕ , ∂ ∂ϕ

¯ = 0. ∂J

(3.7)

In particular, if the modulus of the current J is different from zero, |J| = 0, then we can introduce new independent, dependent variables η, η¯ and ω dη = J 1/2 dz,

d¯ η = J¯1/2 d¯ z,

ω=

p2 , |J|

respectively, such that GW system (1.37) assumes the decoupled form

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(ln ω)ηη¯ = 2(

1 − ω), ω

¯ = 0, ∂J

∂ J¯ = 0.

Hence GW system (1.37) becomes a direct sum of the elliptic Sh-Gordon equation and the conservation of current J. Equation (3.6) has the Painlev´e property. Consequently, we obtain the result that the general solution p of (3.6) admits double poles with two residues of opposite sign p±2 = e±ϕ = ±χ−2 ,

(3.8)

where χ is the expansion variable of the Laurent series. According to the proposed procedure, we assume a specific form of the Darboux transformation for the case when (3.7) admits opposite residues φ1 ϕ − v = 2 ln . (3.9) φ2 The function v satisfies the initial equation (3.7) and φ1 and ψ2 are two entire functions. Introducing a new variable y = φ1 /φ2 and changing the variables in (3.9) according to p = eϕ/2 ,

q = ev/2 ,

(3.10)

we find that the Darboux transformation for (3.6) can be realized by the following expression p = qy, or equivalently, ϕ = v − 2 ln y.

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Paul Bracken

A first step on the way to constructing a B¨acklund transformation for (3.6) is to look for a conditional symmetry algebra L spanned by two vector fields Z1 , Z2 which have the characteristic equations of the form r

 ∂y α (x) = Ali (x) bαl(y(x)). ∂xi

(3.11)

l=1

A specific Lie algebra structure for the generators {ˆbl } must be assumed. We start the analysis with the lowest-dimensional case, namely that of the sl(2, C) algebra which admits the one-dimensional representation (∂y , y∂y , y 2 ∂y ) in terms of a coordinate y. This algebra comes up in the study of several completely integrable models [22]. In the case at hand, differential constraints (3.11) in one complex variable y take the form of coupled scalar Riccati equations with nonconstant coefficients ∂y = A01 (z, z¯) + A11 (z, z¯)y + A21 (z, z¯)y 2 ,

¯ = A0 (z, z¯) + A1 (z, z¯)y + A2 (z, z¯)y 2 . ∂y 2 2 2 (3.12) The zero curvature conditions for (3.12) are given by ¯ 0 − ∂A0 + A1 A0 − A0 A1 = 0, ∂A 1 2 1 2 1 2 ¯ 1 − ∂A1 + 2(A2 A0 − A2 A0 ) = 0, ∂A 1 2 1 2 2 1

(3.13)

¯ 2 − ∂A2 − A1 A2 + A2 A1 = 0. ∂A 1 2 1 2 1 2

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Substitution of the new variables (3.10) and the ansatz p = qy into equations (3.6) gives 1 ¯ 1 ¯ + ∂q∂y ¯ ¯ ¯ (∂ ∂q y + ∂q ∂y + q ∂∂y) − 2 (∂q y + q∂y)(∂q) qy q y −

2 1 ¯ − |J| + q 2 y 2 = 0. (∂q y + q ∂y) ∂y qy 2 q2 y2

(3.14)

Using equations (3.12) we can eliminate the derivatives of the complex variable y in (3.14). Next, we require that the coefficients of the successive powers of y in the equation so obtained vanish, to give the system (1) q 2 A01 A02 + |J|2 = 0, (2) ∂A02 − A11 A02 = 0, (3) ∂A12 +A22 A01 −A21 A02 +∂ ∂¯ ln q = 0, (4) ∂A22 + A22 A11 = 0, (5) A22 A21 + q 2 = 0,

q 2 A01 A02 + |J|2 = 0, ¯ 0 − A0 A1 = 0, ∂A 1 1 2 ¯ 1 +A2 A0 −A0 A2 + ∂∂ ¯ ln q = 0, (3.15) ∂A 1 1 2 1 2 ¯ 2 + A2 A1 = 0, ∂A 1 1 2 A21 A22 + q 2 = 0.

An overdetermined system is obtained composed of (3.13) and (3.15) for the unknown functions Aki . In this case, the system is consistent since the compatibility conditions are identically satisfied. This system has a nontrivial, unique solution for the Ak and it is Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. i

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briefly stated how this solution can be obtained. The second and fourth equations in the first column of (3.15) can be written in the form ∂ ln A02 = A11 ,

−∂ ln A22 = A11 .

Equating these equations, we can integrate to obtain A02 =

g¯(¯ z) , A22

(3.16)

where g¯ is a complex function of z¯. Similarly, from the second and fourth equations in the second column of (3.15), we obtain ∂¯ ln A01 = −∂¯ ln A21 , which can be solved to give, A01 =

h(z) , A21

(3.17)

where h is a complex function of z. Substituting these results into the first equation in (3.15), we find that h(z)¯ g(¯ z ) − |J|2 = 0. Thus, without loss of generality, one may take h(z) = J(z) and g¯(¯ z ) = J¯(¯ z ). From the first equation in (3.15), we can write

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A01 = −

|J|2 , q 2 A02

(3.18)

thus A01 is determined in terms of A02 . Equating equations (3.17) and (3.18), we eliminate A01 and we get, q2 A21 = − ¯ A02 . (3.19) J From this, we can substitute A21 into the fifth pair of equations in (3.15), and we obtain A22 in terms of A02 as J¯ A22 = 0 . (3.20) A2 Using the first column of (3.15), substitute the relation A11 = ∂ ln A02 from equation (3.152) as well as (3.18) through (3.20) into equation (3.15-3). In this way, we can eliminate the coefficients A01 , A11 , A21 and A22 from the differential equation (3.15-3). A partial differential equation for the function A02 is obtained in the form 2 2 ¯ 2 ¯ ln A0 − q (A0 )2 + |J| J + |J| − q 2 = 0. ∂∂ 2 q2 q 2 (A02 )2 J¯ 2

(3.21)

Moreover, if we introduce a new variable defined by qA02 Q = ¯1/2 , Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. iJ

(3.22)

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Paul Bracken

into (3.21), it is transformed into the following form 2 ¯ ln Q − |J| + Q2 = 0. ∂∂ Q2

(3.23)

This coincides with equation (3.6). Substituting (3.18)-(3.20) into the pair of Riccati equations (3.12), we obtain ∂y = −

q 2 A02 2 |J|2 0 + ∂ ln A y − y , 2 0 q 2 A2 J¯

¯ ¯ ¯ = A0 + ∂¯ ln( J )y + J y 2 , ∂y 2 q 2 A02 A02

¯ = 0, ∂J ∂ J¯ = 0.

(3.24)

The compatibility condition for (3.24) is satisfied identically whenever (3.6) holds. Note that Q is some solution to (3.6) which is related to q by (3.22). A particular form for A02 could be obtained by considering the case in which Q = q. Then (3.22) implies that  ¯ A02 = i J. Equation (3.6) is invariant under the transformation J → λJ,

(3.25)

provided that |λ|2 = 1. A B¨acklund parameter λ can be introduced into equation (3.24) by carrying out transformation (3.25). In this case, the pair of Riccati equations (3.24) takes the form

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∂y = iλ1/2 (

J J¯1/2 q2 2 − y ), q2 J¯1/2

¯ = 0, ∂J (3.26)

¯1/2 ¯ = iλ ¯ 1/2J¯1/2 + ∂¯ ln( J )y − iλ ¯ 1/2J¯1/2 y 2 , ∂y q2

|λ|2 = 1,

where q satisfies (3.6). Hence the differential constraints become an auto-B¨acklund for GW system (3.6), while the Darboux transformation is defined by (3.11). Furthermore, by linearizing the Riccati system (3.26), the associated linear spectral problem for (3.6) with spectral parameter μ is obtained, ⎛ ⎞ ¯1/2 J     1/2 J 0 iμ φ1 φ1 ⎜ q2 ⎟ ∂ =⎝ , ⎠ μ φ2 φ2 iq 2 ( ¯)1/2 0 J ⎛ ⎞ 1 ¯ J¯1/2 1/2 1/2     ¯ i¯ μ J ⎜ 2 ∂ ln( q 2 ) ⎟ φ1 φ1 ¯ ⎜ ⎟ ∂ =⎝ , (3.27) φ2 1 ¯ J¯1/2 ⎠ φ2 1/2 1/2 ¯ i¯ μ J − ∂ ln( 2 ) 2 q 2 ¯ where y = φ1 /φ2 , |μ| = 1 and ∂J = 0. The Lax pair (3.27) is based on a nondegen-

erate sl(2, C) representation. Note that for any holomorphic function J, the compatibility condition for (3.27) reproduces the system (3.6) in the variable q. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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4. Estimation of the Degree of Indeterminancy of the General Analytic Solution to the Weierstrass-Enneper System Using Cartan’s theorem on systems in involution [23], we estimate the degree of indeterminancy of the general analytic solution of GW system (1.37). For this purpose, we perform the analysis using the machinery of differential forms which are equivalent to the initial system. The problem is reduced to examining the Cartan numbers of these exterior forms and the number of arbitrary parameters admitted by the general solution of the system of polar equations. For computational purposes, it is useful to introduce the following notation x = (x1 , x2 ) = (¯ z, z),

¯ 1 , ∂ψ2, ∂ ψ¯1 , ∂¯ψ¯2 ), ξ = (ξ 1 , ξ 2, ξ 3 , ξ 4 ) = (∂ψ (4.1)

¯ 2 , ∂¯ψ¯1 , ∂ ψ¯2). u = (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (ψ1 , ψ2, ψ¯1 , ψ¯2 , ∂ψ1, ∂ψ It means that we interpret z and z¯ as independent coordinates x1 and x2 , respectively, in R2 space, and the coordinates (u1 , · · · , u8 ) as independent variables in R8 space. The quantity ξ represents a vector of all first derivatives of ψi which do not appear in the GW system. Under notation (4.1) the system becomes u5 = pu2 ,

u7 = pu4 ,

u6 = −pu1 ,

u8 = −pu3 ,

p = u 1 u3 + u 2 u4 ,

(4.2)

and the differentiation of p with respect to z and z¯ yields,

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∂p = u1 ξ 3 + u4 ξ 2 ,

¯ = u 3 ξ 1 + u2 ξ 4 , ∂p

(4.3)

¯ If whenever (1.37) holds. Note that the ξ s enter linearly into the expressions ∂p and ∂p. 1 4 we consider the variables u = (u1 , · · · , u8) and ξ = (ξ , · · · , ξ ) as unknown functions of x = (x1 , x2 ) then, in terms of (4.1) and (4.2), the GW system is equivalent to the system of differential one-forms ω1 = du1 − (ξ 1 dx1 + pu2 dx2 ) = 0, ω2 = du2 − (−pu1 dx1 + ξ 2 dx2 ) = 0, ω3 = du3 − (pu4 dx1 + ξ 3 dx2 ) = 0, ω4 = du4 − (ξ 4 dx1 − pu3 dx2 ) = 0,

(4.4)

¯ − p2 u1 ] dx1 − [u2 ∂p + pξ 2 ] dx2 = 0, ω5 = du5 − [u2 ∂p ¯ + pξ 1 ] dx1 − [u1 ∂p + p2 u2 ] dx2 = 0, ω6 = du6 − [u1 ∂p ¯ + pξ 4 ] dx1 − [u4 ∂p − p2 u3 ] dx2 = 0, ω7 = du7 − [u4 ∂p ¯ + p2 u4 ] dx1 − [u3 ∂p + pξ 3 ] dx2 = 0, ω8 = du8 − [u3 ∂p in two independent variables x1 , x2 which form some local coordinate system in the real space R2 . System (4.4) can be written in the abbreviated form ωs = dus − Gsμ (x, ξ, u) dxμ,

s = 1, · · · , 8,

μ = 1, 2,

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(4.5)

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Paul Bracken

where the functions Gsμ depend only on (x, ξ, u) and where ξ enters linearly into Gsμ , due to (4.3). We are interested in the evaluation of the degree of freedom of the most general analytic solution of (4.4) which can be expressed by us = us (x1 , x2 ),

ξ r = ξ r (x1 , x2 ),

s = 1, · · · , 8,

r = 1, · · · , 4.

According to Cartan’s Theorem on Systems in involution [23], we can formulate the following Proposition. Proposition 4.1. If the system of differential one-forms (4.4) is in involution at a regular point (x0 , ξ0 , u0 ) and if it is an analytic system in some neighborhood of (x0 , ξ0 , u0 ), then the general analytic solution of (4.4) with independent variables x1 , x2 exists in some neighborhood of a regular point (x0 , ξ0 , u0 ) and it depends on four arbitrary real analytic functions of one real variable. Proof. Under the notation (4.1) and relations (4.3), the exterior differentiation of system (4.4) leads to the following system of 2-forms whenever system (4.4) holds ¯ − p2 u1 ] dx1 ∧ dx2 , Ω1 ≡ dω1 = dx1 ∧ dξ 1 − [u2 ∂p Ω2 ≡ dω2 = −[u1 ∂p + p2 u2 ] dx1 ∧ dx2 + dx2 ∧ dξ 2 , Ω3 ≡ dω3 = [u4 ∂p − p2 u3 ] dx1 ∧ dx2 + dx2 ∧ dξ 3 , ¯ + p2 u4 ] dx1 ∧ dx2 , Ω4 ≡ dω4 = dx1 ∧ dξ 4 + [u3 ∂p Ω5 ≡ dω5 = −u2 u3 dξ 1 ∧ dx1 + u22 dx1 ∧ dξ 4 − u1 u2 dξ 3 ∧ dx2 − (p + u2 u4 ) dξ 2 ∧ dx2 2 2 ¯ ¯ +[u1 u2 (u4 ∂p−p2 u3 )−(p+u2 u4 )(u1 ∂p+p2 u2 )−u2 u3 (u2 ∂p−p u1 )+u22 ((u3 ∂p+p u4 )] dx1 ∧dx2,

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Ω6 ≡ dω6 = −(p + u1 u3 ) dx1 ∧ dξ 1 − u1 u2 dx1 ∧ dξ 4 − u1 u4 dx2 ∧ dξ 2 −u21 dx2 ∧ dξ 3 + [−u21 (u4 ∂p − p2 u3 ) + u1 u4 (u1 ∂p + p2 u2 )

(4.6)

¯ + p2 u4 ) + (p + u1 u3 )(u2 ∂p ¯ − p2 u1 )] dx1 ∧ dx2 , −u1 u2 (u3 ∂p Ω7 ≡ dω7 = u3 u4 dx1 ∧ dξ 1 + (p + u2 u4 ) dx1 ∧ dξ 4 + u24 dx2 ∧ dξ 2 +u1 u4 dx2 ∧ dξ 3 + [u1 u4 (u4 ∂p − p2 u3 ) − u24 (u1 ∂p + p2 u2 ) ¯ − p2 u1 ) + (p + u2 u4 )(u3 ∂p ¯ + p2 u4 )] dx1 ∧ dx2 , −u3 u4 (u2 ∂p Ω8 ≡ dω8 = −u23 dx1 ∧ dξ 1 − u2 u3 dx1 ∧ dξ 4 − u3 u4 dx2 ∧ dξ 2 −(p + u1 u3 ) dx2 ∧ dξ 3 + [u3 u4 (u1 ∂p + p2 u2 ) − (p + u1 u3 )(u4 ∂p − p2 u3 ) ¯ − p2 u1 ) − u2 u3 (u3 ∂p ¯ + p2 u4 )] dx1 ∧ dx2 . +u23 (u2 ∂p

In this case, using (4.5), all 2-forms (4.6) can be clearly expressed in the form Ωs ≡

4 8   ∂Gsμ r ∂Gsμ ∂Gsμ μ dξ ∧ dx + ( (Glν )+ ) dxν ∧ dxμ , r l ∂ξ ∂u ∂xν r=1

s = 1, · · · , 8.

l=1

(4.7)

Let Yμ be linearly independent vector fields defined on R14 Y = (a1 ∂ , a2 ∂ , b1 ∂ , · · · , b4 ∂ , c1 ∂ , · · · , c8 ∂ ),

μ = 1, 2

1 1 μ ξ 1 Analysis and ξ 4 μ uNova u8 μ x2 : μ μ xand μ Applications, Partial Differential Equations: Theory, Analysis Applications Theory, Science μ Publishers, Incorporated, 2011.

(4.8)

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such that they annihilate systems (4.4) and (4.6), composed of the 1-forms ωs and the 2forms Ωs , respectively, that is ωs Yμ  = 0,

Ωs Y1 , Y2  = 0,

s = 1, · · · , 8,

μ = 1, 2

(4.9)

at some regular point (x, ξ, u) ∈ R14 . The above system is called a system of polar equations [23]. The set of vector fields yμ satisfying this system depends on a certain number N of free parameters. In our case, the solution of (4.9) is given by Y1 = ∂x1 +

4  r=1

¯ − p2 u1 ]∂u br1 ∂ξ r + ξ 1 ∂u1 − pu1 ∂u2 + pu4 ∂u3 + ξ 4 ∂u4 − [u2 ∂p 5

¯ + pξ 1 ]∂u − [u4 ∂p ¯ + pξ 4 ]∂u + [u3 ∂p ¯ + p2 u4 ]∂u , +[u1 ∂p 6 7 8 and Y2 = ∂x2 +

4  r=1

br2 ∂ξ r + pu2 ∂u1 + ξ 2 ∂u2 + ξ 3 ∂u3 − pu3 ∂u4 − [u2 ∂p + pξ 2 ]∂u5 (4.10)

+[u1 ∂p + p2 u2 ]∂u6 − [u4 ∂p − p2 u3 ]∂u7 + [u3 ∂p + pξ 3 ]∂u8 ,

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where

b21 = −(u1 ∂p + p2 u2 ),

b31 = u4 ∂p − p2 u3 ,

¯ − p2 u1 , b12 = u2 ∂p

¯ + p2 u4 ). b42 = −(u3 ∂p

To simplify formulae (4.10) we have used notation (4.3). Solution (4.10) contains four arbitrary parameters b11 , b41 , b22 , b32 , hence we have N = 4.

(4.11)

According to the definition of the first Cartan character [23], we have  ∂G   1μ X μ · · · ∂G1μ X μ   ∂ξ 1  ∂ξ 4   . .   .. · · · .. s1 = max rank(X 1,X 2 )∈R2    ∂G8μ μ 8μ μ   ∂ξ 1 X · · · ∂G X ∂ξ 4 at a regular point (x, ξ, u) ∈ R14 . The nonvanishing elements of the 8 × 4 matrix (asr ) = ∂G μ ( ∂ξsμ r X ) are a11 = X 1 ,

a22 = X 2

a52 = (p + u2 u4 )X 2 , a62 = −u1 u4 X 2 , a72 = u24 X 2 ,

a33 = X 2 ,

a44 = X 1 ,

a53 = u1 u2 X 2 ,

a54 = u22 X 1 ,

a63 − u22 X 2 ,

a64 = −u1 u2 X 1 ,

a73 = u1 u4 X 2 ,

a82 = −u2 u3 X 1 ,

a74 = (p + u2 u4 )X 1 ,

a83 = −(p + p1 u3 )X 2,

a51 = u2 u3 X 1 , a61 = −(p + u1 u3 )X 1, a71 = u3 u4 X 1 , a81 = −u23 X 1 ,

a84 = −u3 u4 X 2 ,

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Paul Bracken

since the function Gsμ depends linearly on ξ. Hence, the maximal rank of the matrix (asr ) is s1 = 4. In that case, the second Cartan character is given by s2 = n − s1 = 0, where n = 4 is the number of coordinates ξ. Taking into account the definition [23] of the Cartan number Q, we have Q = s1 + 2s2 = 4. (4.12) Thus, from (4.11) and (4.12), we get Q = N = 4 and, according to Cartan’s Theorem, system (4.4) is in involution at the regular point (x0 , ξ0 , u0 ). So, its general analytic solution exists in some neighborhood of this regular point and depends on four arbitrary real analytic functions of one real variable. ♣ Let us note that, since system (4.4) is equivalent to GW system (1.37), Proposition 1 implies the existence of the general analytic solution of (1.37). This solution depends on two arbitrary complex analytic functions of one complex variable and their complex conjugate functions, since z and z¯ are interpreted as coordinates on C and ψi and ψ¯i as complex conjugate functions on C.

5. Inducing Surfaces in Higher Dimensional Spaces

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5.1. Surfaces in R4 Generalizations of the Weierstrass representation for surfaces immersed into the multidimensional Euclidean and Riemann spaces can be given. The cases of inducing can be extended to higher dimensional spaces, in particular, 4-dimensional Euclidean space and Minkowski spaces. This was first proposed by Konopelchenko and Landolfi [24]. They consider a first order nonlinear system of two-dimensional Dirac-type equations in terms of four complex valued functions ψα and ϕα , with α = 1, 2. This system can be written as follows ¯ α = −pψα , α = 1, 2, ∂ψα = pϕα , ∂ϕ (5.1) √ p = u1 u2 , uα = |ψα|2 + |ϕα|2 , as well as the complex conjugate equations of (5.1). The system (5.1) possesses several conservation laws, such as ¯ αϕβ ) = 0, ∂(ψαψβ ) + ∂(ϕ

¯ αψ¯β ) = 0. ∂(ψαϕ¯β ) − ∂(ϕ

(5.2)

As a consequence of these conserved quantities, there exist four real-valued functions Xi (z, z¯), i = 1, · · · , 4 which can be interpreted as the coordinates of a surface immersed in Euclidean 4-space. The coordinates of the position vector X = (X1 , X2 , X3, X4 ) of a constant mean curvature surface in R4 are determined by the integrals  i X1 = [(ψ¯1ψ¯2 + ϕ1 ϕ2 ) dz  − (ψ1 ψ2 + ϕ¯1 ϕ ¯2 ) d¯ z  ], 2 Partial Differential Equations: Theory, Analysis and Applications Γ : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Generalized Weierstrass System Inducing Surfaces... 251  1 X2 = [(ψ¯1ψ¯2 − ϕ1 ϕ2 ) dz  + (ψ1 ψ2 − ϕ¯1 ϕ ¯2 ) d¯ z  ], 2 Γ  1 X3 = − [(ψ¯1 ϕ2 + ψ¯2 ϕ1 ) dz  + (ψ1 ϕ ¯2 + ψ2 ϕ¯1 ) d¯ z ], (5.3) 2 Γ  i X4 = [(ψ¯1ϕ2 − ψ¯2 ϕ1 ) dz  − (ψ1 ϕ¯2 − ψ2 ϕ ¯1 ) d¯ z  ]. 2 Γ In (5.3), Γ is any contour in the complex plane. The integrals depend only on the endpoints of the contour on account of conservation laws (5.2). Many results for the 4-dimensional case have been found and presented in [25] in a novel way from that of [24]. These functions X i (z, z¯) can now be treated as the coordinates of a surface in R4 . The components of the induced metric are determined from gzz =

4  i=1

(Xzi )2 = g¯z¯z¯,

gz¯z =

4  (Xzi Xz¯i ),

(5.4)

i=1

and one obtains gzz = gzz ¯ = 0,

(5.5)

and

1 gz¯z = (|ψ1 |2 + |ϕ1 |2 )(|ψ2 |2 + |ϕ2 |2 ). 2 Further, two normal vectors N1 , N2 are   |ϕ1 |2 |ϕ2 |2 |ϕ1 |2 |ϕ2 |2 N1 = (A), N2 = (A), u1 u2 u1 u2

(5.6)

(5.7)

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where

uk = (|ψk |2 + |ϕk |2 ), k = 1, 2, ψ¯1 ψ¯2 ψ1 ψ¯2 ψ1 ψ¯2 ψ1 ψ¯2 A = [i( − ), − + ,1− , −i(1 + )]. ϕ ¯ 1 ϕ2 ϕ1 ϕ2 ϕ¯1 ϕ2 ϕ¯1 ϕ2 The mean curvature vector H = Xz¯z /gz¯z is given by 2p H= [−i(ψ1ϕ2 +ψ2 ϕ1 ), (ψ1ϕ2 +ψ2 ϕ1 ), (ψ1ψ¯2 −ϕ1 ϕ ¯2 ), i(ψ1ψ¯2 −ϕ1 ϕ ¯2 )]. u1 u2 If we write H = h1 N1 + h2 N2 , the components h1 , h2 of H along N1 and N2 are ¯2 − ϕ ¯1 ϕ2 ϕ1 ϕ¯2 + ϕ¯1 ϕ2 ϕ1 ϕ h1 = −p  , h2 = ip  . 2 2 u1 u2 |ϕ1 | |ϕ2 | u1 u2 |ϕ1 |2 |ϕ2 |2  The mean curvature H = h21 + h22 is equal to √ 2p H=√ . u1 u 2

(5.8) (5.9)

(5.10)

(5.11)

(5.12)

The Gaussian curvature is given by 2 K=− [log(u1 u2 )]z¯z . (5.13) u1 u2  Finally, the Willmore functional W = H2 dS, which is of physical interest, is given by  W =4 p2 dxdy. (5.14)

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Paul Bracken

5.2. Surfaces in Multi-Dimensional Spaces Any solution (ψ1 , ψ2 ) = (ψ, ϕ) of system (1.37) gives rise by means of (1.39) to the three coordinates X 1 , X 2 and X 3 . Given a pair of solutions (ψ1 , ϕ1 ) and (ψ2 , ϕ2 ), there are two possibilities. The first is to generate four coordinates by using (5.3), the second is to get six coordinates, first X 2 , X 2 , X 3 via (1.39) with the solutions (ψ1 , ϕ1) and X 4 , X 5 , X 6 again by (1.39) using (ψ2 , ϕ2 ). In the second case, one has the conformal immersion of a surface into R6 with the induced metric ds2 = (u21 + u22 ) dz d¯ z.

(5.15)

Moreover, introducing four coordinates X 7 , X 8 , X 9 , and X 10 using (3.5) one can get an immersion into R10 . The induced metric in this case is ds2 = (u21 + u1 u2 + u22 ) dz d¯ z.

(5.16)

In such a manner one can apparently get an immersion of a surface into Euclidean space RN with N = 3n + 4m where m and n are arbitrary integers with induced metric 2

ds = (

n  a=1

u2a

+

m 

ua ub ) dz d¯ z,

(5.17)

a=b

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where m is equal to the number of pairs a, b where (a = b). The Weierstrass representation for immersion of surfaces into the complex spaces are defined analogously. Let (ψa, ϕa) and (ψb, ϕb) be any two solutions of the system (1.37), that is, ψaz = pϕa ,

ϕa¯z = −pψa,

ψbz = pϕb ,

ϕb¯z = −pψb ,

(5.18)

where p is a complex valued function. The system (5.18) implies (ψaψb )z = −(ϕa ϕb )z¯. Thus the integral α

X =

 a,b

Aαab

(5.19)

 Γ

(ψaψb d¯ z − ϕa ϕb dz),

(5.20)

where Aαab are arbitrary constants does not depend on the contour of integration Γ. Treating the N functions X α(z, z¯) (α = 1, · · · , N ) as the coordinates in CN , one gets an immersion of a surface into CN .

6. Physical Applications to String Theory Involving GW Inducing Representations

The formulation of a quantum theory of gravity which is renormalizable seems to require a point of view which differs in very significant ways from the traditional field theoretic approach. Although originally developed to account for the strong interaction, the occurence Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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The Generalized Weierstrass System Inducing Surfaces...

253

of a massless spin 2-particle in the spectrum of states of certain string theories suggests that these theories would be more applicable in the study of gravitation. Thus string theories seem to always include gravity, and so string theory is a very natural setting for gravitation as well as gauge fields. In the string and brane approaches, elementary particles are thought of as strings or surfaces in a higher dimensional space, which in turn manifest some dynamics in space-time. Thus, string theory provides a framework for studying the Kaluza-Klein mechanism. We would like to make use of this general method for constructing surfaces in multidimensional spaces by means of solutions to GW type systems. The same type of ideas that have been considered thus far have been extended to Minkowski space as well. Here, a simple model for a string theory which is based on a Nambu-Goto action is considered [26]. The equations of motion for the string variables will be reformulated in terms of the functions which are solutions of a system of first order equations whose solutions can be used to generate surfaces in Minkowski space. In particular, we will look for complex exponential solutions of this first order system which are also solutions to the equations of motion which are obtained from the Nambu-Goto action. It is not meant to be inferred that this is the only group of solutions to this system, but they are particularly useful to examine in this context. The dynamics of the object will be found to be given by the wave equation with the string coordinates as the dependent variables. It will be shown that a common collection of solutions to the first order system of equations and the dynamical equation can be found. Since they are solutions to this first order system, these solutions can be used to calculate surfaces in the associated space. The complex exponential solutions can then be used to construct general solutions of this wave equation, which is the reason for the interest. This is not to say that sums of the first order system are also solutions of this same system, since this first order system will be nonlinear in general. Using these results, a quantization procedure can be introduced. The action can be written by introducing a world-sheet metric gαβ (ξ, η) and the action for the relativistic string is written as  1 √ SP = − (6.1) d3 x gg αβ ∂α X μ ∂β X ν ημν , 4πα where ημν is the Minkowski metric and g = − det(gαβ ). The world sheet metric gαβ (ξ, η) has signature (+, −), such that α = 0, 1 refer to ξ and η, respectively. This form of the string action is the starting point of the path-integral quantization procedure of Polyakov [27], which will be discussed in the next section. The gauge symmetries are the twodimensional world-sheet reparametrization invariances and conformal invariance. It will be useful to choose a gauge by considering the gauge condition gαβ = e2φ(ξ,η)gˆαβ , the conformal gauge in which gˆαβ = ηαβ . In this conformal gauge, the Polyakov action simplifies to  1 SP = − (6.2) d2 η αβ ∂α X μ∂β Xμ . 4πα Varying the action with respect to X μ with the appropriate boundary conditions for an open or closed string, the following equations of motion are obtained (∂ 2 − ∂ 2 )Xμ = 0.

η ξ and Applications, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis Nova Science Publishers, Incorporated, 2011.

(6.3)

254

Paul Bracken

For a closed string, there is the condition X μ(ξ, η + 2π) = X μ(ξ, η) and for an open string, Xημ |η=0,π = 0. Closed strings will be of interest here. Equation (6.3) is that of a two-dimensional massless wave equation. Hyperbolic time-like surfaces with plus minus signature appear naturally in pseudoEuclidean spaces. For time-like surfaces, minimal lines are real. To obtain a surface parametrized by minimal lines, we will construct a model based on the following representation defined by solutions to the linear Dirac equations which contain two real independent variables ξ and η as proposed by Konopelchenko and Landolfi [28]. Theorem 6.1. The formulae  1 1 X = [(ϕ¯1 ϕ2 + ϕ1 ϕ ¯2 )dξ + (ψ¯1 ψ2 + ψ1 ψ¯2 ) dη], 2 Γ  i 2 X = [(ϕ¯1 ϕ2 − ϕ1 ϕ¯2 ) dξ + (ψ¯1 ψ2 − ψ1 ψ¯2 ) dη], (6.4) 2 Γ  1 3 X = [(ϕ1 ϕ ¯1 − ϕ2 ϕ¯2 ) dξ + (ψ1 ψ¯1 − ψ2 ψ¯2 ) dη], 2 Γ  1 4 X = [(ϕ1 ϕ ¯1 + ϕ2 ϕ¯2 ) dξ + (ψ1 ψ¯1 + ψ2 ψ¯2 ) dη], 2 Γ where the functions ψα, ϕα, α = 1, 2 satisfy the linear system ψα,ξ = pϕα ,

ϕα,η = p¯ψα,

α = 1, 2,

(6.5)

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and Γ is a contour of integration in R2 , define an immersion in Minkowski space of a generic time-like surface parametrized by the minimal lines ξ = c1 , η = c2 . The Gaussian and squared mean curvature are given respectively by 2 ¯ ∂ ∂[log(|ϕ 1 ψ2 − ψ1 ϕ2 |)], |ϕ1 ψ2 − ψ1 ϕ2 |2

K=

H2 =

4|p|2 . |ϕ1 ψ2 − ψ1 ϕ2 |2

(6.6)

The determination of explicit solutions to generalized Weierstrass systems has received considerable attention recently [29]. As an example of an instanton or soliton solution to (6.5), suppose a and b are real constants and define the functions ψα =

(ξ − a)1/2 , (−η + b)

ϕα =

i . (−η + b)1/2

Differentiating these functions, it can be seen that system (6.5) is satisfied, that is ∂ξ ψα =

−i 2(−η +

∂η ϕα =

b)1/2(ξ

− a)1/2

i = pϕα, (−η + b)1/2

(ξ − a)1/2 = p¯ψα, 2(−η + b)1/2(ξ − a)1/2 (−η + b) i

provided that p is defined in the obvious way.

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255

Suppose the Xμ satisfy wave equation (6.3), then a system of equations can be obtained which must be satisfied by the functions ϕα , ψα for the Xμ which are defined by (6.4) to satisfy the wave equation. Differentiating the Xμ in (6.4) twice with respect to ξ and then η, the following set of equations is obtained after substituting into (6.3), ∂ξ (ϕ¯1 ϕ2 + ϕ1 ϕ¯2 ) − ∂η (ψ¯1 ψ2 + ψ1 ψ¯2 ) = 0, ∂ξ (ϕ¯1 ϕ2 − ϕ1 ϕ¯2 ) − ∂η (ψ¯1 ψ2 − ψ1 ψ¯2 ) = 0, (6.7) ∂ξ (ϕ1 ϕ ¯1 − ϕ2 ϕ¯2 ) − ∂η (ψ1 ψ¯1 − ψ2 ψ¯2 ) = 0, ∂ξ (ϕ1 ϕ ¯1 + ϕ2 ϕ¯2 ) − ∂η (ψ1 ψ¯1 + ψ2 ψ¯2 ) = 0, Adding the first two equations in (6.7) gives ¯ 2 ) − ∂η (ψ¯1 ψ2 ) = 0, ∂ξ (ϕϕ

(6.8)

and adding the last pair of equations in (6.7) we obtain that ∂ξ (ϕ1 ϕ¯1 ) − ∂η (ψ ψ¯1) = 0.

(6.9)

Conversely, it is easy to see that we can proceed in the reverse direction to obtain (6.7) by taking linear combinations of both (6.8) and (6.9). Consider a particular class of solutions to system (6.5), and simultaneously of (6.8) and (6.9), in particular the following set of complex harmonic type functions ϕ1 = u1 eic1 (ξ−η) , ψ1 = v1 eib1 (ξ−η) , Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

(6.10) ϕ 2 = u2

eic2 (ξ−η) ,

ψ2 = v2

eib2 (ξ−η) .

Here ui , vi are complex constants and ci and bi are real constants. Upon differentiating the functions ψα and ϕα with respect to ξ and η, the derivatives in (6.5) can be written in the form ∂ξ ψα = iei(bα −cα )(ξ−η)ϕα, ∂η ϕα = −ie−i(bα −cα )(ξ−η) ψα.

(6.11)

These will be consistent provided that cα = 1/bα, the case of interest here. The complex function p is taken to be p = iei(bα −cα )(ξ−η). It now remains to see under which further conditions these functions satisfy the equations of motion (6.7) and (6.8). The second equation is satisfied automatically since ϕ¯1 ϕ1 and ψ¯1 ψ1 are both real constants. Moreover, (6.8) is given by −i¯ u1 u2 (c1 − c2 )e−(c1 −c2 )(ξ−η) − i¯ v1 v2 (b1 − b2 )e−(b1 −b2 )(ξ−η) = 0. Replacing cα = 1/bα and selecting b2 = −1/b1 reduces to the following constraint u ¯1 u2 + v¯1 v2 = 0.

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(6.12)

256

Paul Bracken

Its complex conjugate also holds. Subject to these auxiliary conditions, the functions ϕα and ψα are given by i

ϕ 1 = u1 e b 1

(ξ−η)

ψ1 = v1 eib1 (ξ−η),

,

(6.13) − bi (ξ−η)

ϕ2 = u2 e−ib1 (ξ−η), ψ2 = v2 e and the function p is given as

−1

p = iei(b1 −b1

)(ξ−η)

1

,

.

Substituting (6.13) into (6.6), the Gaussian curvature is zero and the mean curvature is constant [26]. Given that the functions in (6.13) satisfy system (6.5), the coordinates of the corresponding surface can be calculated using the solution functions. If we set n = b1 +b−1 1 and use constraints (6.12), we obtain X 1 = ni [¯ u1 u2 e−in(ξ−η) − u1 u ¯2 ein(ξ−η) ], X 2 = − ni [¯ u1 u2 e−in(ξ−η) + u1 u ¯2 ein(ξ−η) ], 3

X =

1 2 2 [(|u1 |

2

2

(6.14)

2

− |u2 | )ξ + (|v1 | + |v2 | )η],

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X 4 = 12 [(|u1 |2 + |u2 |2 )ξ + (|v1 |2 + |v2 |2 )η]. Thus each of these solutions individually generates a surface in Minkowski space. Eliminating the terms containing ξ and η on the right-hand side of the equations in (6.14), an equation describing a surface in terms of the X μ would be obtained. The subscripts r and i denote real and imaginary parts, respectively. A consistent value of p is found as well. The X μ obtained from Theorem 6.1 can be used to produce the general solution of the wave equation which is compatible with the periodicity condition X μ (ξ, η + 2π) = X μ (ξ, η) which is a linear combination of right and left moving solutions as follows: X μ(ξ, η) = XRμ (ξ − η) + XLμ(ξ − η).

(6.15)

In this expression, XRμ and XLμ are given by 1 1 μ i  1 μ −in(ξ−η) XRμ (ξ − η) = xμ + , p (ξ − η) + √ αn e 2 4πT 4πT n=0 n μ

XL (ξ + η) =

1 μ 1 μ i  1 μ −in(ξ+η) , x + p (ξ + η) + √ α ¯ ne 2 4πT 4πT n=0 n

(6.16)

with arbitrary Fourier modes αμn and α ¯μn , since the ui are arbitrary. A Hamiltonian can then be expressed in terms of oscillators and is written ∞

H=

1 μ (α αμn + α ¯μ−n α ¯μn ). 2 −∞ −n

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(6.17)

The Generalized Weierstrass System Inducing Surfaces...

257

The quantity T = 1/2πα leads to the fundamental length l = (πT )−1/2 and from the action, we can identify the canonical momentum conjugate to X μ P μ (ξ, η) = T

∂X μ . ∂ξ

A simple quantization scheme would then follow naturally in the following way. Equal ξ canonical commutation relations may now be imposed, [X μ(ξ, η), X ν (ξ, η )] = [X˙ μ(ξ, η), X˙ ν(ξ, η )] = 0,

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and

[P μ (ξ, η), X ν (ξ, η )] = T [X˙ μ(ξ, η), X ν(ξ, η )] = iδ(η − η  )η μν .

(6.18)

This is one way to perform quantization, a very direct type of quantization based on the general expression in (6.16), deforming the coefficients which appear in (6.14). Another approach, a path integral approach, will be discussed next. A different approach has been developed based on the description of the string world sheets by way of the Gauss map for surfaces conformally immersed into Euclidean spaces. Within this approach, both the Nambu-Goto and Polyakov actions can be written in terms of a constrained K¨ahler σ-model action. The use of the Gauss map makes the calculations of quantum effects induced by extrinsic geometry easier. Nonlinear constraints associated with the Gauss map for nonminimal surfaces give rise to serious computational difficulties. An approach based on the generalized Weierstrass representation which has been the subject thus far will be introduced. Any surface in R3 can be generated by means of this representation provided a system of two linear equations is solved. Using the GW representation, the Nambu-Goto action and in particular, the Polyakov action have a very simple form. This permits the calculation of the one loop correction to the background for the full Polyakov action exactly. The propagators of the fields can be found and their infrared behavior analyzed. Moreover, quantum corrections to the classical Nambu-Goto and spontaneous curvature actions can be evaluated perturbatively. To this end let us express the GW system in a slightly more general form from that in (1.37)-(1.39) which is suited to nonconstant mean curvature systems [30]. The representation for a surface conformally immersed in R3 (X(z, z¯) : C → R3 ) is given by   1 2 2  2  1 2 ¯ X + iX = i (ψ dz − ϕ¯ d¯ z ), X − iX = −i (ψ 2 d¯ z  − ϕ2 dz  ), Γ

X3 = −



Γ

(ϕψ¯ dz  + ψ ϕ ¯ d¯ z  ),

(6.19)

Γ

where z, z¯ ∈ C and the complex valued functions ψ, ϕ obey the system of equations ∂ψ = pϕ,

¯ = −pψ ∂ϕ

(6.20)

where p = p(z, z¯) is a real-valued function. The formulae (6.19)-(6.20) define a conformal immersion of a surface into R3 with the induced metric z. ds2 = (|ψ|2 + |ϕ|2 ) dzd¯

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(6.21)

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Paul Bracken

The Gaussian and the mean curvature are given by K=−

4 ∂ ∂¯ log(|ψ|2 + |ϕ|2), (|ψ|2 + |ϕ|2)2

H=

2p . |ψ|2 + |ϕ|2

(6.22)

Any surface in R3 can be represented in this form. At p = 0 one gets a minimal surface, H = 0, and the formulae (6.19)-(6.20) are reduced to the classical Weierstrass formulae for minimal surfaces. The Polyakov action  SP =

H 2 dS,

where dS is the area element, takes a very simple form by applying (6.22)  SP = 4 p2 [d2 z],

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where [d2 z] = in this context

i 2

(6.23)

 dzd¯ z. The Nambu-Goto action SN G = α0 dS takes the following form  SN G = α0 (|ψ|2 + |ϕ|2 )2 [d2 z]. (6.24)

The extrinsic Polyakov action has been known for a while under a different name, namely, as the Willmore functional. The GW representation gives the possibility of defining an infinite class of integrable deformations of surfaces generated by the modified VeselovNovikov hierarchy. A characteristic feature of these deformations is that they preserve the extrinsic action. One-loop corrections for the Polyakov action have been studied, however, nonlinear constraints associated with the Gauss map has not allowed the calculation of the one-loop corrections for the full Polyakov action. However, the generalized Weierstrass representation enables the difficulties encountered here to be overcome. A somewhat deeper understanding of the results from the geometrical point of view is offered. The classical action is given by   2 2 2 2 S = α0 (|ψ| + |ϕ| ) [d z] + β0 p2 [d2 z], (6.25) where α0 and β0 are the tension and extrinsic coupling constant, respectively. The next step is to take into account the system (6.20) which relates the fundamental fields ψ, ϕ and p. Introducing complex Lagrange multiplier fields and requiring the action to be real, the following constraint term is required  ¯ + pψ) + λ( ¯ ∂¯ψ¯ − pϕ) ¯ Sc = [d2 z][λ(∂ψ − pϕ) + σ(∂ϕ ¯ +σ ¯ (∂ ϕ¯ + pψ)]. (6.26) This has to be added to the action (6.25). Since (6.26) is supplying constraints, it has been known since the start of the study of gauge theory quantization, once constraints are introduced into the generating functional of Green’s functions  Z = [DΠ] exp(−S), (6.27)

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259

then the correct definition of the measure [DΠ] requires the evaluation of a specific Faddeev-Popov determinant. In this case, path integral (6.27) serves to quantize the theory and can provide physical information. In the case of the GW system, the fundamental fields are constrained by the set of Dirac equations (6.20), which in matrix form is      ψ ∂ −p ψ = 0. (6.28) = L ϕ p ∂¯ ϕ To evaluate det(L), the heat kernel procedure is used. The Faddeev-Popov action term is defined as follows, SF P = − log[det(L) ] = [

d  1 d  ζ (s|L)]s=0 = [ ζ (s|A)]s=0 , ds 2 ds

where A = L+ L = LL+ and, 1 ζ (s|A) = tr [(A) ] = Γ(s) 



−s



∞

ts−1 tr  [exp(−tA)] dt.

(6.29)

(6.30)

0

Here, the Riemann zeta function ζ is constructed from eigenvalues of the operator A and the prime in (6.30) means that the contribution of zero modes is omitted. The heat kernel of A is defined as Kt(z, z  |A) = exp(−tA)(z, z  ), ∂t Kt(z, z  |A) + AKt (z, z  |A) = 0,

(6.31)

lim Kt (z, z  |A) = δ(z − z  ).

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t→0+

The small t expansion of A looks like Kt(z, z  |A)|z=z =

1  kn (z)tn , 8πt

where the kn (z) are matrix valued functions. As a result, one finds that  d  1   [ ζ (s|A)]s=0 = γζ (0|A), ζ (0|A) = [d2 z]tr  [k1 (z)], ds 4π

(6.32)

(6.33)

where γ is the Euler-Masheroni constant. It is known that for the operator A, one has [31] tr  [k1 ] = −2p2 , so the Faddeev-Popov contribution to the action is  γ SF P = − [d2 z]p2 . 4π

(6.34)

Thus by comparing with (6.23), the effect of the Faddeev-Popov determinant is reduced to a redefinition of the extrinsic coupling constant β0 . Such a new coupling constant will be denoted as β, and we assume that β > 0. The total action is the sum of (6.25), (6.26) and (6.34) ST = S + Sc + SF P . (6.35) Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

260

Paul Bracken

The corresponding action in one-loop approximation can be derived by employing the background field method. If the fields are split into slow and fast components, for example p = p0 + p1 , respectively, and retaining only terms quadratic in the fast, or 1 variables, the following one-loop action follows  (2) ¯ 1 +p0 ψ1 +p1 ψ0 )+p1 (ψ1 σ0 −ϕ1 λ0 )+c.c.] S = [d2 z][λ1(∂ψ1 −p0 ϕ1 −p1 ϕ0 )+σ1 (∂ϕ  +β

[d2 z] p21 .

(6.36)

The Fourier space version of this can be worked out and is given in [32]. As usual, Fourier ¯ of fast fields are defined using components f (k) = f (k, k)  1 i ¯ f (z, z¯) = [d2 k] f (k) exp[− (k¯ z + kz)], 2π 2 ˜ < |k| < Λ. Exact propagators can where the integration is performed over the domain Λ be calculated.

7. Non-Constant Mean Curvature Surfaces

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Consider next the case in which the mean curvature H is not constant [33]. Up to this point, constant H with factor 1/2 has been absorbed into the spatial coordinates. Putting the numerical factor in the coordinates (z, z¯), the system of equations satisfied by the ψα which determine a surface with mean curvature function H become ∂ψ1 = pHψ2 ,

¯ 2 = −pHψ1 , ∂ψ

∂¯ψ¯1 = pH ψ¯2 ,

∂ ψ¯2 = −pH ψ¯1 ,

(7.1)

p = |ψ1 |2 + |ψ2|2 . The function H(z, z¯) denotes the mean curvature the surface will have. Versions of Propositions 2.1-2.2 can be obtained based on system (7.1). Proposition 7.1. If ψ1 and ψ2 are solutions of the system (7.1) and ρ is given by (2.1), then ψ1 and ψ2 are obtained from ρ by means of the equations ψ1 = ρ

(∂¯ρ¯)1/2 , H 1/2(1 + |ρ|2 )

ψ2 = 

(∂ρ)1/2 , H 1/2(1 + |ρ|2 )

 = ±1.

(7.2)

Moreover, ρ is a solution to the following second order system, ¯ − ∂ ∂ρ

2¯ ρ ¯ = ∂(ln ¯ H)∂ρ, ∂ρ∂ρ 1 + |ρ|2

∂ ∂¯ρ¯ −

2ρ ¯ ∂ ρ¯∂ ρ¯ = ∂(ln H)∂¯ρ¯. 1 + |ρ|2

(7.3)

The converse of Proposition 7.1 holds as well. Note the close similarity between (7.3) and (2.4). Moreover, (7.1) implies that the ψα satisfy the set of conservation laws (1.38). Surfaces can be induced by using solutions of (7.1) and then substituting into (1.39), or Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Generalized Weierstrass System Inducing Surfaces...

261

alternatively, given a solution of (7.3), it can be put in (7.2) to obtain the ψα, which are then used in (1.39). Several results concerning this system and details concerning the proofs can be found in [33]. Proposition 7.2. (i) If ψ1 and ψ2 are solutions of (7.1) given in terms of ρ in (2.1) by (7.2), then J defined by (1.40) in terms of the function ρ takes the form J(z, z¯) = −

∂ρ ∂ ρ¯ . H(1 + |ρ|2 )2

(ii) Let J be defined by (1.40), then the quantity J defined by  z¯ p2 (z, τ )∂H(z, τ ) dτ, J =J+

(7.4)

z¯0

is conserved under differentiation with respect to z¯, ¯ = 0. ∂J

(7.5)

Proof: (i) Substituting ψα from (7.2) into (1.40) and differentiating using the product rule, we obtain J = −¯ ρ

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+¯ ρ =

∂ρ (∂ρ)1/2 (∂ρ)1/2 ∂H + ρ ¯ ) ∂( 2H 2 (1 + |ρ|2)2 H(1 + |ρ|2 ) 1 + |ρ|2

∂ρ (∂ρ)1/2 ρ¯(∂ρ)1/2 ∂H − ) ∂( 2H 2 (1 + |ρ|2)2 H(1 + |ρ|2) 1 + |ρ|2

(∂ρ)1/2 ∂ ρ¯(∂ρ)1/2 (∂ρ)1/2 (∂ρ)1/2 ) − − ρ ¯ ∂( )] [¯ ρ ∂( H(1 + |ρ|2) 1 + |ρ|2 1 + |ρ|2 1 + |ρ|2 =−

∂ρ ∂ ρ¯ . H(1 + |ρ|2 )2

(ii) A proof can be found in [33]. Proposition 7.3. With p defined in (7.1), and J in (1.40), then p satisfies a second order differential equation which involves p, J and the mean curvature function H. The equation is given by |J| ∂ ∂¯ ln p = 2 − H 2 p2 . (7.6) p It has been shown [34] that when H is constant, there is a connection between the timeindependent Landau-Lifshitz equation, which can be expressed as ¯ = 0, [S, ∂ ∂S]

(7.7)

and the two-dimensional nonlinear sigma model. The matrix S will be referred to as the spin matrix. In terms of the sigma model quantity ρ, the matrix S is given by   1 1 − |ρ|2 2¯ ρ S= . (7.8) 2 2ρ −1 + |ρ|2 + |ρ| Partial Differential Equations: Theory, Analysis and Applications :1 Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Paul Bracken

Define f and f¯ to be the ρ-dependent factors on the left-hand side of the sigma model equations given in (2.4), so in fact (2.4) can be written in the form f = 0, f¯ = 0. In terms of f and f¯, the matrix generated by (7.7) is of the form   4 ρ¯f − ρf¯ ρ¯2 f − f¯ ¯ [S, ∂ ∂S] = . (7.9) ρ2 f¯ − f ρf¯ − ρ¯f (1 + |ρ|2)2 These results can be summarized as follows. Proposition 7.4 If ρ is a solution of the nonlinear sigma model system (2.4), then the spin matrix S defined by (7.8) is a solution of the Landau-Lifshitz equation (7.7). Proposition 7.4 and equation (7.7) can be modified to include the case in which the mean curvature is not constant. Define the matrices R and H as follows, ⎛ ⎞   ¯ ln(H) ρ¯∂¯ ln(H) ∂ ¯ 4 −¯ ρ ∂ρ ρ∂ ρ¯ ⎠ . (7.10) 1 B= , H=⎝ ∂ ln(H) ∂ ln(H) ∂ρ −ρ2 ∂¯ρ¯ (1 + |ρ|2 )2 ρ The matrix B depends only on the variable ρ. The following generalization of Proposition 7.4 can be formulated. Proposition 7.5. If ρ is a solution of the sigma model equations (7.3) and the matrices B and H are defined in (7.10), then spin matrix S given by (7.8) is a solution of the nonhomogeneous Landau-Lifshitz equation ¯ + BH = 0, [S, ∂ ∂S]

(7.11)

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modulo (7.3).

References [1] B. G. Konopelchenko and I. Taimanov, J. Phys. A 29, 1261 (1996). [2] B. G. Konopelchenko, Stud. Appl. Math., 96 9 (1996). [3] D. Nelson, T. Piran and S. Weinberg. Statistical Mechanics of Membranes and Surfaces , World Scientific, Singapore, (1992). [4] R. Carrol and B. G. Konopelchenko, Int. J. Mod. Phys. A 11, 1183 (1996). [5] D. G. Gross, C. N. Pope and S. Weinberg, Two-dimensional Quantum Gravity and Random Surfaces, World Scientific, Singapore, (1992). [6] O. Zhong-Can, L. Ji-Xing and X. Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Physics, World Scientific, Singapore, (1999). [7] B. G. Konopelchenko, Phys. Letts. B 414, 58-64 (1997).

[8] S. S. Chern, Surface Theory with Darboux and Bianchi, Miscellanea Mathematica, 59-69, Springer, Berlin (1991). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[9] C. Rogers and W. F. Schief, B¨acklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [10] A. Das, Integrable Models, World Scientific Notes in Physics, Vol. 30, World Scientific, Singapore, (1989). [11] K. Weierstrass, Fortsetzung der Untersuchung u¨ ber die Minimalfl¨acher, Mathematische Werke, Vol. 3 (Verlagsbuchhandlung, Hillesheim), 219-248, (1866). [12] I. Taimanov, Modified Novikov-Veselov Equation and Differential Geometry of Surfaces, Transactions of Amer. Math. Soc., Ser. 2, 179, 133-150 (1997). [13] P. Bracken, A. M. Grundland and L. Martina, J. Math. Phys., 40, 3379-3403 (1999). [14] P. Bracken and A. M. Grundland, J. Math. Phys. 42, 1250-1282 (2001). [15] P. Bracken and A. M. Grundland, J. Nonlinear Math. Phys., 6, 294-313 (1999). [16] P. Bracken and A. M. Grundland, J. Nonlinear Math. Phys., 9, 229-247 (2002). [17] P. Bracken and A. M. Grundland, J. Nonlinear Math. Phys., 9, 357-381 (2002). [18] P. Bracken and A. M. Grundland, Inverse Problems, 16, 145-153 (2000).

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[19] E. L. Ince, Ordinary Differential Equations, Dover, NY, (1956). [20] A. M. Grundland, L. Martina and G. Rideau, Partial Differential Equations with Differential Constraints, Lecture Notes CRM 11, Providence, 135-154 (1997). [21] P. J. Olver, Proc. Roy. Soc. London, A 444, Nr. 1922, 509-523 (1994). [22] P. Winternitz, Lie Groups and Solutions of Nonlinear Differential Equations, in Lecture Notes in Physics, Vol. 189, Editor: K. B. Wolf, Springer, Berlin, 262-329, (1983). [23] E. Cartan, Sur la Structure des Groupes Infinies de Transformation, Chapitre I, Les Syst´eme Differential en Involution, Gauthier-Villars, Paris, 1953. [24] B. G. Konopelchenko and G. Landolfi, J. of Geometry and Physics, 29, 319-333 (1999). [25] P. Bracken and A. M. Grundland, J. Nonlinear Math. Phys. 9, 357-381 (2002). [26] P. Bracken, Phys. Letts. B 541, 166-170 (2002). [27] D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory, Institute of Physics, Bristol, 1994.

[28] A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, London, 1987. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[29] B. G. Konopelchenko and G. Landolfi, Stud. Applied Math. 104, 129-169 (1999). [30] P. Bracken amd A. M. Grundland, Czechoslovak J. Physics, 51, 293-300, (2001). [31] B. G. Konopelchenko and G. Landolfi, Phys. Letts. B 459, 522-526, (1999). [32] E. Elizalde, S. D. Odintosov, A. Romeo, A. A. Bytsenko, S. Zerbini, Zeta regularization techniques with applications, World Scientific, Singapore, (1994). [33] B. G. Konopelchenko and G. Landolfi, Phys. Letts. 444, 299-308, (1998). [34] P. Bracken, Acta Applicandae Math., 92, 63-76 (2006).

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[35] I. Taimanov, Modified Novikov-Veselov Equation and Differential Geometry of Surfaces, Translations of Amer. Math. Soc., Ser. 2, 179, 133-150 (1997).

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 c 2011 Nova Science Publishers, Inc. 

Chapter 10

N ATURAL C ONVECTION AND I TS E FFECT ON D IFFUSION M EASURED WITH N UCLEAR M AGNETIC R ESONANCE Aleˇs Mohoriˇc ∗ Faculty of mathematics and physics, University of Ljubljana

PACS 44.25. +f, 47.27. Te, 92.60. Ek, 87.61. -c Keywords: NMR, self-diffusion, natural convection, NMR imaging, chaos, Lorenz model, Benard convection.

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1.

Abstract

The chapter presents PDA’s governing free convection and nuclear spin dynamics in an experiment measuring the distribution of self-diffusion with magnetic resonance imaging.

2.

Introduction

Natural convection is an effect occurring in fluids where temperature gradient causes flow on a macroscopic scale if the fluid’s density changes significantly. If the gradient is strong enough, the convection turns chaotic. Equations governing the flow dynamics are non-linear Navier-Stokes, continuity, and heat conduction equations which in the first approximation reduce to the Lorenz system [1] of equations. The equations describe the Lorenz oscillator and the long-term behavior of the oscillator corresponds to a fractal structure known as the Lorenz attractor. The system was introduced to describe convection roles in the atmosphere [2]. Chaotic velocity fluctuations can be observed in magnetic resonance images of self-diffusion constant distribution. To describe specific measurements [3], a model of natural convection in a horizontally oriented cylinder, cooled from above, can be derived. The Lorenz model of natural convection is derived for a free boundary condition, so its validity is of a limited value for the natural no-slip boundary condition. It can be

∗ E-mail address: [email protected] Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Aleˇs Mohoriˇc

shown, that even a slight temperature gradient can cause measurable enhancement of the apparent self-diffusion constant of the liquid. Self-diffusion is a process of particle random motion causing the spread in statistical distribution of these particles in space. Nuclear magnetic resonance is one of the best suited methods for measuring self-diffusion since it non-invasively measures particle motion in magnetic field gradient through attenuation of spin echo [4].

3.

Natural Convection

The convection in a voxel of liquid at hight z will not occur - the liquid is in stable mechanical equilibrium - as long as a small adiabatic rise by ξ brings liquid to a position with a smaller specific density: w  (p , s ) − w(p, s) > 0 , (1) where w, p and s are respectively specific volume, pressure and entropy of the voxel at ds position z and primed values refer to position z + ξ. With the series expansion s − s = dz ξ and a few thermodynamic identities we arrive at the lower limit of the temperature gradient over which liquid is mechanically stable: dT gT =− dz cp w



∂w ∂s



=− p

gT β . cp

(2)

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Here g is gravitational acceleration, β is volumetric coefficient of thermal expansion and cp is isobaric specific heat. In the case of water, this gradient at room temperature is dT 10 m/s2 300 K 0.00013 K−1 =− ≈ −1 K/m . dz 4200 J/kgK

(3)

When the temperature gradient in liquid drops below the value in Eq. 2, the motion convective flow - starts. Newton’s second law for the liquid inside a voxel a = F/m states for incompressible liquid in the low velocity gradient regime ∂v ∇p + (v · ∇)v = f − + ν∇2 v , ∂t ρ

(4)

where lefthand side represents acceleration - substantial (material) derivative of the velocity of the liquid mass m inside the voxel, and the terms on the righthand side respectively correspond to external forces (in case of gravity f = g), forces caused by spatial changes in pressure and frictional (viscous) forces. ν = η/ρ is kinematic and η dynamic viscosity. The real no-slip boundary condition v(r) |∂ = 0 cannot be satisfied in the first approximation, as will be discussed later. As fluid moves the mass in a voxel increases with inflow and decreases with outflow. If the inflow exceeds the outflow, the density of liquid will increase. This states the continuity equation: ∂ρ + ∇ · (ρv) = 0 . (5) Partial Differential Equations: Theory, Analysis and Applications : Theory, ∂t Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Natural Convection and NMR

267

The system is irreversible because of viscosity and thermal conduction. The temperature change of liquid inside the voxel follows the energy conservation law: ∂T + v · ∇T = χ∇2 T . ∂t

(6)

χ is thermometric conductivity. In derivation of this equation we made Boussinesq approximation [5]: heat produced by friction, effect of pressure on density, and temperature dependence of all material properties (viscosity, thermal conductivity and specific heat) except density are neglected. This also implies weak flow, small velocity gradient regime, and small temperature difference (not necessarily temperature gradient). Convection is characterized with two dimensionless numbers: the Prandtl number P r = 3 ν and the Grashof number Gr = gβlν 2ΔT . l is the typical length of the system (in our case χ it is the diameter of the cylinder), ΔT is the temperature difference between the bottom and the top. At low Gr natural convection does not contribute significantly to heat transfer. In convection flow the laminar and turbulent regimes are not characterized by Reynolds number but rather by Gr. For Gr on the order of 106 and more the flow becomes turbulent. 3 Natural convection starts if temperature difference is such that the product GrP r = gβlνχΔT is on the order of or larger than 103 . Now, let us turn our attention to the convection inside a horizontal cylinder of diameter 2R and infinite length, to simplify the analysis. The derivation closely follows Saltzman’s procedure [6] for plane geometry. Temperature on the boundary falls linearly with height: 

1 r·n T (r = R, ϕ, t) = T0 + ΔT 1 − 2 R



,

(7)

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where n is the unit vector in the vertical direction, as shown in Fig. 1.

Figure 1. Crosssection of infinite, horizontal cylinder of diameter 2R. At the boundary of the cylinder, temperature decreases linearly with height and is ΔT higher at the bottom. Gravitational acceleration g and vertical unit n are also indicated. In the Boussinesq approximation, only the density changes with temperature ρ(r) = ρ0 (1 − β[T (r) − T0 ]) .

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Aleˇs Mohoriˇc

The dynamics of liquid is described by Eqs. 4, 5 and 6. The continuity equation is further simplified to ∇ · v = 0. Thus spatial variation of density is neglected. This equation is automatically solved in cylindrical geometry with no axial flow assumed (two-dimensional flow), if we define flow velocity with the stream function ψ as vr = −

1 ∂ψ(r, ϕ) ∂ψ(r, ϕ) , vϕ = . r ∂ϕ ∂r

(9)

Streamlines are given by stream function as ψ = const.. The temperature field in the liquid is described with a small deviation θ(r, ϕ, t) from the linear profile: 

1 r·n T (r, ϕ, t) = T0 + ΔT 1 − 2 R



+ θ(r, ϕ, t) .

(10)

We substitute velocity and temperature fields in Eqs. 4 and 6 with Eq. 9 and 10, apply the curl (rotor) to Eq. 4 (thus we get rid of the pressure term) and get Eqs.: ∂θ 1 ∂(ψ, θ) ΔT =− + ∂t r ∂(r, ϕ) 2R





∂ψ 1 ∂ψ − sin ϕ − cos ϕ + χ∇2 θ , ∂r r ∂ϕ

   2  1 ∂(ψ, ∇2 ψ) 1 ∂θ ∂θ ∂∇2 ψ 2 =− + ν∇ ∇ ψ + gβ − sin ϕ − cos ϕ , ∂t r ∂(r, ϕ) ∂r r ∂ϕ

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where

∂(a,b) ∂(r,ϕ)

=

∂a ∂b ∂r ∂ϕ

−

(11) (12)

∂a ∂b ∂ϕ ∂r .

The boundary condition for temperature deviation is θ(r = R) = 0 and the deviation is finite inside the cylinder. In a bounded space the solution for θ can be found by series expansion in a proper base. In cylindrical geometry the natural candidates are harmonic and cylindrical Bessel functions. In the first approximation only the first terms, giving the non-trivial solution, are kept. Due to symmetry reasons temperature deviation can be in the first approximation expressed as Ra θ(r, ϕ, t) = b02 (t)J0 (ξ02 r/R) + b11 (t)J1 (ξ11 r/R) cos ϕ . ΔT

(13)

Here Jn is the cylindrical Bessel function of the n-th order and ξnm is the m-th zero of Jn . This form already satisfies the boundary conditions. Ra is the Rayleigh number Ra =

βgΔT 8R3 νχ

(14)

and is the control parameter of the system. The term J0 (ξ01 r/R) is absent because the average of temperature deviation is zero. The boundary conditions for the stream function cannot be fulfilled in the first approximation. The tangential velocity at the boundary cannot be set to zero with only two terms in the series expansion. With this approximation we are unable to describe heat transfer,  but the velocity profile is described quite well except for a thin boundary layer [7] δ ∼ νl/v0 . The first term in the series expansion of the stream function is ψ(r, ϕ, t) = χa11 (t)J1 (ξ11 r/R) sin ϕ .

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Natural Convection and NMR

269

Figure 2. The velocity field of the convection flow inside an infinite horizontal cylinder. It is given by the stream function Eq. 15. Fluid rises in the center and falls at the edges. The direction of flow can be reversed if convection becomes chaotic.

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The velocity field corresponding to Eq. 15, is shown in Fig. 2, and the temperature deviation Eq. 13 in Fig. 3. Expansion coefficients a11 , b02 and b11 are functions of time and by inserting 13 and 15 into 11 and 12 we get the Lorenz system 2 b˙ 02 = c2 a11 b11 − ξ02 b02 − Ra c1 a11 ,

(16)

2 b˙ 11 = −c3 a11 b02 − ξ11 b11 ,

(17)

2 a˙ 11 = −σξ11 a11 − σc4 b02 ,

(18)

Dot is partial derivation over time τ = numerical constants are ξ11

c1 =

c2 =

ξ11

χ t, R2

dimensionless parameter σ = ν/χ, and

1

J0 (ξ11 x)J0 (ξ02 x)xdx ≈ 0.80  4 01 J02 (ξ02 x)xdx

0

1

− J2 (ξ11 x))J1 (ξ11 x)J0 (ξ02 x)dx ≈ 2.6  2 01 J02 (ξ02 x)xdx

0 (J0 (ξ11 x)

1

c3 = ξ02 c4 =

0

ξ02

J1 (ξ02 x)J1 (ξ11 x)J1 (ξ11 x)dx ≈ 3.8 1 2 0 J1 (ξ11 x)xdx

1

J1 (ξ02 x)J1 (ξ11 x)xdx ≈ 0.019 1 2 2 8ξ11 0 J1 (ξ11 x)xdx 0

The system of Eqs. 16, 17 and 18 can be solved numerically. We used computational software program Mathematica with the following command:

atr=NDSolve[ A11’[t]==- 95 A11[t]- 0.13 B02[t], B02’[t]==2.6 A11[t] B11[t]-Ra 0.8 A11[t]-31 B02[t], B11’[t]==-3.8A11[t] B02[t] - 15 B11[t], Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Aleˇs Mohoriˇc

Figure 3. The temperature deviation θ (given in Eq. 13) from the linear temperature profile. Bright parts indicate increased temperature and the dark parts indicate lower temperature as is in the case of linear dependence. Liquid is heated at bottom and warmer liquid rises in the center, thus making that part warmer. Eventually, liquid cools at the walls and flows down at the edge. This flow can be reversed if convection becomes chaotic.

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A11[0]==0,B02[0]==1,B11[0]==0, {A11,B02,B11},{t,0,upbound},MaxSteps->maxsteps]

Here upbound and maxsteps set the upper limit in normalized time and the number of iterative steps taken in solving the problem. The solution (for the right control parameter Ra) is the Lorenz attractor. A case with Ra = 350000 is shown in Fig. 4. When measuring diffusion with magnetic resonance, only the velocity plays part and we will limit our further discussion to the variable a11 . Solutions of a11 for various temperature differences (parameter Ra) reveal following: there is no convection for small values of Ra, at higher values, flow settles at constant value. With further increase of temperature difference stable oscillations occur, which diverge into chaotic oscillations for even larger Ra. If the convection-driving temperature difference disappears, the flow will die out exponentially corresponding to: 

∂a11 /∂t = −ν

ξ11 R

2

a11

(19)

with characteristic time for water in the 10 cm diameter cylinder at 20o C and σ = 6.3, χ = 1.4 × 10−7 m2 /s, ν = 9 × 10−7 m2 /s and β = 2.2 × 10−4 /K: τK =

R2 (0.05)2 m2 ≈ ≈ 150 s . 2 νξ11 10−6 m2 /s 16

In constant external temperature gradient the velocity can change chaotically. In the center of the cylinder velocity v = 1 × 10−6 m /s corresponds to the value a11 = 1 and t = 3.5 × 104 s corresponds to τ = 1. This gives us an idea how slow the flow and its change rate are. We can observe the evolution of convection towards chaotic motion in Fig. 5. More on flow instability and bifurcations can be found in [8]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Natural Convection and NMR

271

Figure 4. The Lorenz attractor in phase space of variables representing temperature deviation and the amplitude of velocity.

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4.

Nuclear Magnetic Resonance and Diffusion

Nuclear magnetic resonance (NMR) can be observed in atoms without electron angular momentum and with finite nuclear angular momentum Γ. Such nucleus also possesses magnetic moment μ = γΓ and γ is gyromagnetic ratio. Spin is a common moniker for all: nucleus, angular momentum and magnetic moment and will be used indiscriminately. Many spins in the sample are partially aligned in external magnetic field B which results in macroscopic magnetization M . The evolution of magnetization in the magnetic field B is described by Bloch equation [9]. Random motion (diffusion) of spins in liquid also causes change in magnetization, and a corresponding term must be added to the equation [10] to get Bloch-Torrey equation: ∂M = γ(M × B) + D∇2 (M − M0 ) , ∂t

(20)

where D is the diffusion constant, M0 is equilibrium magnetization and relaxation is neglected. NMR is especially efficient in measuring diffusion, if uniform external magnetic field is temporarily spoiled by short magnetic field gradient pulses [4] and pulsed-gradient spin-echo (PGSE) has long been used to investigate correlated and uncorrelated motion in a number of systems. The effect of natural convection on the measurements is well known [11, 12, 13, 14]. Reader interested in the basics of NMR measurements is referred to [15]. One can combine spin-echo measurement with magnetic resonance imaging (MRI) to study the distribution of diffusivity [3, 16] and the effects of convection [17, 18]. In [3] we measured diffusion in a horizontally oriented cylinder, heated from below. Natural Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

272

Aleˇs Mohoriˇc

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Figure 5. Amplitude of the velocity vs. time for different temperature differences. a) Small Ra - liquid is in mechanical equilibrium - there is no flow. b) Onset of stable convection at Ra > 2.5 × 104 . c) Periodical fluctuations and d) chaotical fluctuations for large enough temperature difference.

convection has been thoroughly studied in certain geometries such as plane geometry [19, 20, 7] and vertically oriented cylinder [21, 22]. Another way of dealing with diffusion is to use only the Bloch equation, find its solutions and only then take motion of spins into account [23]. In semi-classical solution of Bloch equation, spin is precessing around external magnetic field with the frequency proportional to the magnitude of the magnetic field: E = E0 eiθ

(21)

and the phase increases with time as the integral of the precession frequency. E can be understood as the component of magnetic moment lying in the plane perpendicular to the axis of precession which points along uniform magnetic field. The sum of this components over all spins is proportional to the signal induced in a coil surrounding the sample. If the magnetic field is not uniform, because magnetic field gradient G is present, the phase depends on the position of spin: 

θ(r, t) = γ

0

t

G(r, t ) · r dt .

(22)

Here, we have discarded the phase accumulated in the uniform field (γBt), which is the same as describing the spins in a frame rotating with frequency γB. Even if the phase is different for different spins, it can be refocused for stationary spins, by changing the sign of the gradient. Total magnetization after refocussing - during the spin echo - is the Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Natural Convection and NMR

273

same as in the beginning. If the spins move, complete refocussing is not possible and the echo is attenuated. This can be described in the following manner: the phase is described by the velocity of spins along the direction of magnetic field gradient and not by their position θ = 0t q(t )v dt and the accumulated gradient phase q = γ 0t G(t ) dt . The transversal components must be added to get the quantity measured in NMR (transversal magnetization) and the sum is evaluated statistically (position and velocity of spins are stochastic variables) with an average. Average is then calculated using cumulant expansion [24]: S= eiθ(rj ) = neiθ(r)  = S0 eiϕ−β . (23) j

Here the sum encloses the n spins of a pixel and ... is an ensemble average over the trajectories of different spins belonging to the pixel. S0 is the normalized amplitude. The phase shift of the signal in the pixel due to the net flow is 

t

q(t )v(t )dt ,

(24)

q(t1 )∂v(t1)∂v(t2 )q(t2 )dt1 dt2 .

(25)

ϕ=

0

while the signal attenuation is  t

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β=

0

0

t

If velocity fluctuation ∂v = v−v is considered variable of the Gaussian stochastic process a truncation of the cumulant expansion to the second order when used for the phase averaging is possible [23]. Therefore the process is defined only by the variance ∂v(t)∂v(0), i.e. the velocity correlation function. In the case of molecular thermal motion an immense number of spins, each experiencing weak phase fluctuations, adds to the induction in the detection coil. This assures that the spin-echo phase fluctuation can be treated as a Gaussian process. But for the velocity variation of non-stationary or turbulent flow, the associated spin phase fluctuations may in general not justify the truncation of higher terms in the cumulant expansion. With an appropriate selection of timing and intervals of the signal acquisition with respect to the speed and pace of the flow, one can enhance the Gaussian assumption to allow the use of Eq. 25 for the flow fluctuation as well. Namely, in the case of convection, a slow flow vc is superposed on the fast molecular Brownian motion vm . The average velocity of the Brownian motion is vm  = 0 and this motion does not contribute to the phase ϕ of the signal. Almost stationary flow vc effects only the signal phase Eq. 24 but not the amplitude in Eq. 23 during the short interval of measurement. The change of the convection flow velocity vc , that occurs within each signal acquisition, is small but the velocity is different for every frame of acquisition and so are the phase shifts ϕ at the pixel position. Convection can thus be directly measured through the phase of the signal. However if many subsequent signals are added together for the purpose of noise reduction, changes in phase may be significant enough to cause the attenuation akin to attenuation caused by diffusion. This additional attenuation is



N N 1 1 βc (r) = ϕ2n (r) − ϕm (r)ϕn(r) . 2N n=1Analysis and Applications, N m=1 Partial Differential Equations: Theory, Analysis and Applications : Theory, Nova Science Publishers, Incorporated, 2011.

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274

Aleˇs Mohoriˇc

Here N is the number of frames added in the averaging process, and ϕn is the phase accumulated in the n-th signal. In the limit of many accumulations βc gets the form of Eq. 25, where the velocity variation of the pixel embraces both the molecular motion and the convection flow fluctuation, ∂v = ∂vm + ∂vc . (27) Whenever one can neglect the mutual correlation between the flow and the molecular motion the contribution of both to the spin-echo attenuation can be separated. For the Brownian diffusion the velocity correlation time is short compared to the interval of acquisition, allowing to assume the velocity fluctuation along the applied magnetic field gradient as ∂vm (t)∂vm(0) = 2Dδ(t). This provides the spin-echo attenuation from Eq. 25 as 

βm = D

t

0

q 2 (t ) dt

(28)

with D being the self-diffusion constant. The attenuation βm does not depend on the position of the pixel. For a slow velocity variations of the non-stationary convection with respect to the duration of the acquisition, the resulting attenuation follows from Eq. 26 and the definition of the phase Eq. 24 and, since q is periodic with the period ∂t (time between two successive acquisitions) and the change of the velocity ∂vc during the time of acquisition is small we can express the attenuation caused by convection as βc =

1 ∂vc2 c 2

 0

t

q(t )dt

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2

, 

(29) 

2 where we have replaced the sum with an integral → dt ∂t . Here ∂vc c is the mean squared velocity fluctuation of flow projected on the direction of the magnetic field gradient at the position of pixel and is defined as:

∂vc2 c =

1 t2m

 0

tm

 0

tm

∂vc (t )∂vc (t )dt dt ,

(30)

where the time of the measurement is tm = N ∂t. Note that the average is taken over a much longer time tm (several minutes) in ...c then in ... (less then a second) average for a single acquisition. For a usual PGSE sequence (details in [4]) with δ long pulses separated by Δ, the total attenuation of the pixel is δ 1 β = βm + βc = γ 2 G2 δ 2 D(Δ − ) + γ 2G2 (δ 2 + δΔ)2 ∂vc2 c . 3 2

(31)

Two aspects of convection flow contribute to the positional dependance of the attenuation: the direction of flow (this is shown in Fig. 2) with respect to the direction of magnetic field gradient and the amplitude and rate of convection. A case of fluctuations is shown in Fig. 5 d). As Fig. 6 shows, the results of the experiment measuring the diffusion distribution with NMR imaging match quite well the predictions of the theory based on Eq. 31. One can observe that, besides uniform attenuation brought about by self-diffusion, there is Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

Natural Convection and NMR

275

also additional attenuation in the form of dark patches. We can determine from the attenuation of pixels on the MR image (Fig. 6) that the velocity changes during the course of frame accumulation should be on the order of ∂vc ≈

3 ≈ 0.1mm/s γΔδG0.5

(32)

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These velocity fluctuations of 0.1 mm/s signify the changes in a11 of ∂a11 = 1 × 102 . Such velocity fluctuations are represented in Fig. 5 d) which corresponds to Rayleigh number of 1.4 × 106 . In our case this means a temperature difference ΔT on the order of 0.01 K, just enough for the onset of convection.

Figure 6. the results of the experiment measuring the diffusion distribution with NMR imaging match quite well the predictions of the theory based on Eq. 31. One can observe that, besides uniform attenuation brought about by self-diffusion, there is also additional attenuation in the form of dark patches.

References [1] H.G. Schuster, Deterministic Chaos: an introduction, VCH, Weinheim, 1988. [2] E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963). [3] A. Mohoriˇc, J. Stepiˇsnik, M. Kos, and G. Planinˇsiˇc, J. Magn. Reson. 136, 22-26 (1999). [4] E. O. Stejskal, J. Chem. Phys. 43, 3597 (1965); E. O. Stejskal and J. E. Tanner, ibid. 42, 288 (1965). [5] J. Boussinesq, Theorie Analytique de la Chaleur, 161, 2, p. 172 Gauther-Villars, Paris (1903). [6] B. Saltzman, J. Atmos. Sci. 19, 329 (1962). [7] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1987).

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN: 978-1-61122-858-8 c 2011 Nova Science Publishers, Inc.

Chapter 11

PARTIAL D IFFERENTIAL E QUATIONS AS A T OOL FOR E VALUATION OF THE C ONTINUOUS WAVELET T RANSFORM∗ Eugene B. Postnikov Humboldt Univ., Berlin/Kursk Univ. Germany

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Abstract The presented review is dedicated to the consideration of the Continuous Wavelet transform from the diffusion signal and image processing point of view. Such an approach is based on the consideration of diffusion smoothing via the solution of proper partial differential equations. Within this group of methods the real and complex wavelet transform with the wavelets of Gauss and Morlet families are considered. Especial attention is concentrated on the variety of numerical examples considering the processing of regular and irregular (random samples, chaotic ODE solutions etc.) signals. All of them are graphically illustrated.

1.

Introduction

Since the time of publication of Grossman and Morlet first paper [1] almost a quarter of century ago, the continuous wavelet transform (CWT) has been an burstly developing area of the harmonic analysis, equally interesting for pure and applied mathematicians, as well as for researchers who use its methods in natural science and engineering. This fact is connected with the unique properties of the transform determining its powerfulness among the others [2] for the signal processing. Historically, CWT was introduced for the improvement of the analysis of geophysical data. However, recently one can find the specialized reviews describing its application in physics and astrophysics [3], chemistry [4], geosciences [5, 6], biology and medical imaging [7], bioinformatics [8] and many more. Naturally, the citied ∗

A version of this chapter also appears in Mathematical Physics Research Developments, edited by Morris B. Levy, published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort in encourage wider dissemination of research

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278

Eugene B. Postnikov

items represent extremely small part of books dedicated to the applied signal processing and they are selected as the comprehended topical reviews. Today there exist a lot of books, which could play the role of introductory courses and reviews of the continuous wavelet transform, for example a classical monograph by S. Mallat “Wavelet Tour in Signal Processing” [9]. But generally they consider the continuous wavelet transform from the point of view of generalization of the continuous Fourier transform, i.e. within the “integral transform ideology”. The main goal of this review is another one. It can be shown that CWT may be constructed as some kind of diffusive smoothing process, especially if a wavelet consists of the Gaussian as a factor. In this approach, the partial differential equations play the leading role. For these reasons, the presented review does not contain usual introductory facts comparing the wavelet and the Fourier transform from the spectral and function theory point of view. This information could be found elsewhere, say in [9], and only basic definitions will be given. But the organization of this work is based on the review of the diffusion signal and image processing via PDE solving with the following generalization on the cases of various real and complex wavelet transforms.

1.1.

Basic Definitions

The continuous wavelet transform of a function f (t) with the wavelet ψ(ξ) is defined as: Z

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w(a, b) = C(a)

+∞ −∞

f (t)ψ

∗





t−b dt, a

(1)

where the asterisk means the complex conjugation. Here t and b are the time variables, and a is called a scale. The norm factor is denoted with C(a). The dependence of the scale variable a is depended on the type of a norm. Two most popular ones are the following: • Quadratic (energy) norm is defined by the condition Z

+∞ −∞

|ψ(ξ)|2 dξ = 1.

It leads to C(a) ∼ a−1/2 . This norm means that every basis function on each scale saves the equal energy. It is analogous to standard commonly used for the Fourier transform since in this case the Parseval equality is fulfilled. • Linear (amplitude) norm is defined by the condition Z

+∞ −∞

|ψ(ξ)| dξ = 1,

and in this case C(a) ∼ a−1 . This condition defines the conservation of the area bounded by a basis wavelet for all scales.

The most typical (and most important!) property of wavelets as the basis functions is the following: the function ψ must be well localized in the time domain and its Fourier Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

279

1 0.8

ψ(ξ)

0.6 0.4 0.2 0 −0.2 −0.4 −5

0

ξ

5

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Figure 1. The DoG wavelet.

transform ψˆ must be well localized in the frequency domain. This combination of properties provides a local filtering in time (any wavelet function cuts the part of a signal localized around the point with the co-ordinate b, thus b is a shift) and in the frequency (the result of transform with the scale a consists of mainly the details with the typical lengths (or periods) in comparison with its value. For detailed consideration of these time-frequency properties see [9], including the discussion connected with the size and the shape of Heisenberg uncertainty window comparing with the Fourier, Gabor, and Shennon transforms. Additionally, if one needs to define a basis function as a wavelet in the strict sense, just one more condition is necessary, namely the admissibility condition: Z

+∞ −∞

ψ(ξ)dξ = 0.

(2)

It allows to evaluate the inverse transform. Nonetheless, a practical calculation could be evaluated with the violation of (2). In this case a realistic condition be the properly small value of the mean of wavelet function over the domain of function.

1.2.

The Simplest Wavelets Based on the Gaussian: The DoG Wavelet and the Morlet Wavelet

As it was mentioned above, the wavelet function is assumed to be well localized in both the time and the frequency domains. Is is well known that the function, which provides the best localization of such kind is the Gaussian, ϕ(ξ) = exp(−ξ 2 /2). Thus, it seems to be a good building block for the construction of the wavelet. However, the Gaussian completely does not admit the admissibility condition (2): Z +∞ √ ξ2 e− 2 dξ = 2π.

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280

Eugene B. Postnikov

Figure 2. The Morlet wavelet with various central frequencies. But the situation can be fixed if one uses the combination of two Gaussian instead of one: ξ2 1 ξ2 ψ(ξ) = e− α − e− 2 , (3) α where α is some number. The example of (3) with α = 0.5 see in Fig. 1.2.. This is the example of real-valued wavelets. But the complex-valued wavelets are also very important, since they are widely used for the local spectral analysis. Obviously, the most natural window, which bounds the initially infinite harmonic function evaluating the spectral decomposition is also the Gaussian. And the standard Morlet wavelet with the central frequency ω0 is simply: 1 2

ψ(ξ) = eiω0 ξ e− 2 ξ .

(4)

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Note, that in this form the admissibility condition is also not fulfilled, Z

+∞ −∞

1 2

eiω0 ξ e− 2 ξ dξ =

√

2πe−

2 ω0 2

.

Nevertheless, the right-hand-side is a fast decaying function. Let us consider some numerical examples. If ω0 = 5 then r.h.s is equal to 9.34 · 10−6), thus the condition ω0 ≥ 5 is a working standard for usage of this kind of wavelet. However, even ω0 = π gives exp(−ω02 /2) = 0.018. This value is small enough for the large variety of the signal analysis problems, which do not require the exact inverse transform. The norm factor for the standard Morlet wavelet is especially simple for the amplitude case: 1 C(a) = √ . a 2π If one needs to use the admissibility condition (2) strictly, there exists the modification, which is called the exact Morlet wavelet: ψ(ξ) =

e

iω0 ξ

−e

−

ω2 0 2

!

1 2

e− 2 ξ

Conversely to the standard Morlet wavelet, the expression for the energy norm 1 √ , C(a) = √ 4 π Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, a Nova Science Publishers, Incorporated, 2011.

(5)

PDE as a Tool for the Evaluation of CWT

281

Figure 3. The WAVE and MHAT (Marr’s) wavelets is simpler than for the amplitude one in this case. The shapes of exact Morlet wavelet for ω0 = π and ω0 = 2π are represented in the Fig. 2a and Fig. 2b correspondingly. In both pictures, the real part is drawn with the solid line, and the imaginary part – with dashed one. Note, that the shape of the standard Morlet wavelet is practically the same for these parameters.

1.3.

The Gaussian and Morlet Families of Wavelets with Higher Vanishing Moments

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The considered wavelets admit only the basic requirements such as time-frequency localization and the admissibility condition (2). However, some problems require additional conditions of orthogonality. Namely, the higher vanishing moments: Z

+∞ −∞

ξ k ψ(ξ)dξ, 0 ≥

k ≥ n.

First of all, the most developed areas of application of this kind of wavelets are the regularized numerical differentiation of noisy signals [10], image segmentation [11], and, especially, characterization of singularities [12]. The last application has found an outlet as a very powerful tool for processing fractal data and extracting the H¨older exponent and the multifractal spectrum [13]. The most direct way to generalize the considered case of wavelets consisting of the Gaussian as a factor is to replace the Gaussian with its derivative. The desired functions be ψn (ξ) =

dn ψ0 (ξ), dxn

where ψ0(ξ) = e−

ξ2 2

(6)

(7)

for the case of real wavelets (the Gaussian family) and ψ0(ξ) = eiω0 ξ e−

x2 2

.

for the complex progressive wavelets (the Morlet family).

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(8)

282

Eugene B. Postnikov

Figure 4. The first two members of the progressive Morlet wavelet family.

Figures 3 and 4 show the examples of first two wavelets from these families. The first derivative of the Gaussian is known as WAVE-wavelet (Fig. 3a), and the second one as MHAT-wavelet (“Mexican hat” wavelet – after its form) or Marr’s wavelet (Fig. 3b). Fig. 4a is the first derivative of (8) and 4b is the second one. In both figures the real part is drawn with the solid line and the complex one – with the dashed.

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2.

Real Gaussian Wavelets and the Image Processing

One of the precursors of the wavelet transform was the theory of diffusion image smoothing founded in the early 80-s by David Marr. His seminal book “Vision. A computational Investigation into the Human Representation and Processing of Visual Information” [14] created the fundamental background for the modern developed theory of image processing widely based on the general theory of diffusion processes. The fundamental conjecture about recognizing the static images posed by Marr were the following: one needs to define the contours of objects from the variations of their light intensity and to consider the set of these contours at different scales. As it will be shown below, the most natural way in this direction is to consider the diffusion process. Since the gradient of a function determines speed of its variation, it will acquire the maximum at the sharp border of a signal or an image; see the example on the Fig. 2.. Thus, the modulus maximum of the first signal’s derivative could be used for the identifications of borders. This algorithm is known as the Canny Edge Detection. However, it must be pointed out that this method works correctly only if the processed function is smooth enough. In the opposite case, the peaks of noise will lead to the extremely large values of the gradient’s modulus and, correspondingly, to the artifacts masking actual edges. Additionally, processing of such explicit discontinuities is hard work for numerical algorithms. For this reason, one uses the preliminary smoothing. The most popular filter is the

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PDE as a Tool for the Evaluation of CWT

283

Figure 5. The step function, its smoothed variant, the first and the second derivatives of the latter one (without the actual scaling).

Gaussian one,

Z

dx0, (9) 4πt where I0 (x) and I(x) are initial and smoothed signals correspondingly and t is a smoothing parameter determining a width of the bell-shaped averaging curve. I(x) =

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(x−x0 )2 4t

e− I0 (x0) √

+∞

+∞

The choice of this kernel in (9) is based on the following reasons: • The Gaussian effectively blurs the signal wiping out all structures at scales much smaller then t. • This kernel is smooth and well localized in both spatial and frequency domains, and, moreover, act simultaneously in both ones. • It is easy to generalize this filter on 2D case without any problems connected with the choice of filter’s direction: since the exponential of Gaussian function is squared coordinate, in the multidimentional case there will be the sum of squared co-ordinates; thus this operator is orientational-independent. This kernel tends to the Dirac delta-function (

δ(x) =

∞, if x = 0, 0, otherwise

if t is equal to zero. In this case I(x) = I0 (x) correspondingly to the definition: Z

I(x) =

+∞

I0 (x)δ(x − x0 )dx0.

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284

Eugene B. Postnikov

Growth of t leads to the averaging over the wider interval. Thus, in 1D case, the point x0 will be detected as an edge if the expression   (x−x0 )2 Z +∞ − 4t ∂ 0 e 0  = √ I (x ) dx 0 ∂x 4πt −∞

reaches an extremum (maximum) for the a priory fixed smoothing parameter t. It should be pointed out that this method has some complications in the direct implementations. The first one is that the finding a maximum directly is often a complicated numerical procedure. The second one is the following: the numerical calculation of the first difference approximating the first derivative has lower robustness comparing with the second difference representing a second derivative. For this reason, it is easier to use the fact that the second derivative is equal to zero if the first one reaches a maximum. Corresponding criterion is the Marr-Hildreth Edge detector: a point x0 belongs to an edge of an initial signal I(x) if the second derivative of the smoothed one I(x) is equal to zero. Since the output co-ordinate x appears only in the Gaussian in (9), only the kernel will be differentiated: ∂I(x) =− ∂x

Z

+∞ −∞

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(x − x0 )2 I0(x ) 1 − 4t 0

!

(x−x0 )2 4t

e− √

4πt

dx0 ,

(10)




Today the kernel ψ(ξ) = 1 − ξ 2 exp −ξ 2 /2 is known as the Marr’s wavelet. In the paper [16] it was noted that since the equation (9) is a solution of the simple diffusion equation ∂I ∂ 2I = , ∂t ∂x2

then the signal/image processing could be govern by iterated local processing schemes. Note also that such an approach also has a clear physical explanation [10]. Namely, this diffusion equation describes the evolution of the one-dimensional temperature profile T (x) = I(x). The right-hand-side is a speed of heat loss, and this quantity is the largest if the left-hand side acts strongest that takes place at the sharp corner. The original Marr’s conjecture is that the image will be completely recognized if one determines the sequence of lines defined by the set of zero-crossing of (10) for the discrete sequence of various t. This original proposition is not correct in general, see the counterexample and the discussion in [15]. This fact as well as some others led one of wavelet theory founders St`epane Mallat to consider more generalized version of this method [11]. This version uses not only the Marr’s wavelet but also WAVE wavelet. The maximum value of this wavelet transform in a set with localization of zero-crossing of the wavelet transform with the MHAT-wavelet could reach Marr’s goal for almost all real finite images (there exist examples, see [15], that Mallat’s conjecture is also not general, but the counterexamples are very specific artificial functions with infinite support). √ It is obvious that if we consider the replacement of independent variable a = 2t and analyze 1D signal in the co-ordinates (a, x), then the first and the second derivatives of the Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

285

smoothed function, 

w1(x, a) = a

∂   ∂x

and

Z



w2(x, a) = a2

2

∂   ∂x2

+∞ −∞

Z

I0 (x0)

+∞ −∞

e

I0 (x0)

−

(x−x0 )2 2a2

a

e

−

(x−x0 )2 2a2

a

 

dx0

 

dx0  ,

coincide with the definition of the real wavelet transform with the WAVE (Fig. 3 a), ψ1(ξ) = −ξe−

x2 2

(11)

and the Marr’s (or Mexican hat wavelet) wavelet (Fig. 3b) 



ψ1(ξ) = 1 − ξ 2 e−

x2 2

.

(12)

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The approach based on the differentiation of the solution to the diffusion equation can easily be extended to two [11] (the image processing) and three [17] (the structure of threedimensional turbulent motion) dimensions. In this case, the diffusion equation takes a form ∂I 1 = ∇2 I ∂τ 2 taken at the time τ = a2 or ∂I = τ ∇2 I ∂τ

(13)

taken at the time τ = a. In the both cases the transformed function I0 (x) plays a role of initial conditions. Thereafter, the decided wavelet transform be wn (a, b) = an ∇2 I(a, b).

As a final remark, it should be noted that in the area of image processing via solutions of diffusion differential equations, the significant progress was archived on the way of usage of non-linear diffusion coefficients. It started from the article of P. Perona
and J. Malik [18] who proposed to use diffusion coefficient in the form exp −|∇a I|2/K , where ∇a I is a smoothed gradient considered above and K is a coefficient. This kind of algorithms allows to proceed images in accordance with their local morphology. Recently it was found that these methods have a close correspondence with the wavelet shrinkage algorithm (see very comprehensive discussion in [19]). However, this topic is beyond this review dedicated to the linear partial differential equations. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

286

3.

Eugene B. Postnikov

Complex Continuous Wavelet Transform with the Morlet Wavelet as a Cauchy Problem for PDE

Note that from the point of view of the theory of partial differential equations, the possibility to proceed the diffusion image smoothing is based on the property of the Gaussian to be reduced to the Dirac delta-function in the limit of zero scale. It makes possible to use transformed function as an initial value for the corresponding Cauchy problem. At the same time, the constant value of the integral Z

+∞ −∞

ξ2

e− 2a2 √

dξ 2πa2

=1

for all a, strictly does not admit the admissibility condition (2), thus the pure Gaussian never can be a wavelet. On the other hand, the standard Morlet wavelet (4) due to its oscillating character has so small inconsistence with the admissibility condition that it could be used in the wavelet transform. Moreover, it is the most important among the practically used wavelets for local spectral analysis. But non-zero mean value of this wavelet, as it was shown in the paper [20], allows to consider the continuous wavelet transform with the standard Morlet wavelet as a solution of Cauchy problem for the complex partial differential equation of the diffusion kind. The present section is dedicated to the detailed proof of this fact and to the discussion of the advantages of this new approach to the practical wavelet analysis.

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3.1.

Haase Equation for CWT with the Morlet Wavelet

It is interesting but the groundbasing result of Maria Haase – the CWT with the Morlet wavelet admits the partial differential equation – was received in [21] without any connections with the diffusion signal processing. Namely, this PDE was found as an intermediate step on the way to the parametric ODE describing the motion along the maxima lines of the given wavelet transform amplitude. Since this PDE turns out one of bases for a new method, let us provide its detailed derivation. It could be simply evaluated by comparison of the first and the second derivatives of the complex-conjugated standard Morlet wavelet ψ ∗(ξ) = exp(−iω0ξ − ξ 2 /2) (the complex conjugated form is used since it is included in the definition of the complex wavelet transform (1)): ξ2

ψ ∗0(ξ) = eiω0 ξ− 2 (−iω0 − ξ) = −ψ ∗(ξ)(iω0 + ξ), ψ ∗00(ξ) = −ψ ∗0 (ξ)(iω0 + ξ) − ψ ∗(ξ). Correspondingly, the partial derivatives for the independent variables of the wavelet transform, a and b, be ∂ψ ∗ ∂ξ t−b ξ = ψ ∗0 = ψ ∗0 = −ψ ∗0 , 2 ∂a ∂a −a a

∂ψ ∗ ∂ξ 1 = ψ ∗0 = −ψ ∗0 , Partial Differential Equations: Theory, Analysis and Applications : Theory, Nova Science ∂bAnalysis and Applications, ∂b a Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

287

 ∂ 2ψ ∗ 1 ∂ψ ∗0 1  ∗0 ∗00 1 ∗ = − = ψ = − ψ (ξ)(iω + ξ) + ψ (ξ) . 0 ∂b2 a ∂b a2 a2 Note that the final calculation should also include a partial derivative of the norm factor. As it will be shown in the next Subsection, only an amplitude norm provides the convergence of the wavelet transform in the limit a → 0 to the function, which can be used as an initial condition. Thus, the scaling factor a−1 should be used and differentiated: 

∂ 1 ∂w = ∂a ∂a a 1 − 2 a

+∞ Z

dt f (t)ψ ∗(ξ) √  = 2π

−∞

dt 1 f (t)ψ (ξ) √ + 2π a

−∞



+∞ Z

+∞ Z

∗

−∞



ξ f (t)ψ (ξ) − a ∗0



dt √ . 2π

Substituting the expression derived above into the second derivative of the wavelet transform, we get ∂ 2w ∂b2

 +∞  +∞ Z Z 1 dt dt =− 3 f (t)ψ ∗0 (ξ)(iω0 + ξ) √ + f (t)ψ ∗(ξ) √  ,

a

2π

−∞

2π

−∞

Multiplying both sides by a and grouping, a

∂ 2w iω0 =− 2 2 ∂b a

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|

1 a2

|

Z

+∞ −∞

Z

+∞ −∞

dt f (t)ψ ∗0(ξ) √ − 2π {z

}

iω0 ∂w ∂b

Z

∗

f (t)ψ (ξ)dt + {z

∂w ∂a

+∞ −∞



dt f (t)ψ (ξ)ξ √ . 2π ∗0

}

Finally, the Haase equation takes a form: ∂w ∂w ∂ 2w = a 2 − iω0 . (14) ∂a ∂b ∂b The construction of this equation has a simple mathematical explanation. The first term in right-hand-side coincides with (13), i.e. it describes the diffusion smoothing of the transformed function. The emergence of the second term is based on the presence of the harmonic factor exp(−iω0 ξ) in the definition of the Morlet wavelet. Moreover, at a = t the equation (14) reduces to the well-known Cauchy-Riemann condition for the derivative of complex function.

3.2.

Initial Value for the Cauchy Problem Evaluating CWT with the Standard Morlet Wavelet

To formulate the Cauchy problem now we need to find the initial conditions for the differential equations (14). First of all, let us rewrite the complex conjugated standard Morlet wavelet 2 ∗ −iω0 ξ − ξ2 ψ (ξ) = e e . Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

288

Eugene B. Postnikov

in the equivalent form. The transforming of the exponential be:  ξ2 1 1 ω2 − iω0 ξ = − ξ 2 + 2iω0 ξ + (iω0)2 − (iω0 )2 = − (ξ + iω0 )2 − 0 . 2 2 2 2 As a result, this kernel will be rewritten as

−

ψ ∗(ξ) = e−

ω2 0 2

2

1

e− 2 (ξ+iω0 ) .

Finally, the continuous wavelet transform with the standard Morlet wavelet could be presented in the following explicit form:

w(a, b) = e

−

2 ω0 2

Z

+∞

√

dt. (15) 2πa2 As it was described above for the case of the diffusion smoothing of images, it is natural to use the known expression for the Dirac delta-function as a limit passage of the Gaussian: −∞

f (t)

((t−b)+iω0 a)2 2a2

e−

ξ2

e− 2a2 δ(ξ) = lim √ . a→0 2πa2 The matter is, the imaginary component of the exponential in (15) will be equal to zero if a = 0. Moreover, as it could be easily shown, the value of this integral does not depend on this completely). For this reason Z

+∞

e−

((t−b)+iω0 a)2 2a2

√

Z

+∞

f (t)δ(t − b)dt = f (b). −∞ 2πa2 Therefore, this limit passage defines the value of the continuous wavelet transform at a = 0 as 2 lim

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a→0 −∞

f (t)

dt =

w(0, b) = f (b)e−

ω 0 2

.

(16)

Since the w(a, b) for all a > 0 admits the Haase equation (14), the formula (16) represents the initial value for this partial differential equation. Thus, the Cauchy problem, the solution of which is the desired complex continuous wavelet transform with the standard Morlet wavelet is formulated completely . Note, that in the practical calculations, the small factor exp(−ω02 /2) sometimes could lead to the declining of an exactness. However, since the transform is linear, one can easily use the transformed function as the initial value and then to multiply only the result of PDE solution by this factor.

3.3.

Advances of the Numerical Evaluation with the Usage of PDE-Based Algorithm

This subsection includes the discussion of advantages of the continuous wavelet transform numerical evaluation based on the described PDE approach with the test examples. Today there exist various approaches for the numerical calculations of the transform with the Morlet wavelet. Such a variety is inducted by the fact that the direct computation of this highly oscillating integral is a complicated task. Below for the comparison with the proposed new method, will be considered two algorithms, which now are de facto the standard of the computational wavelet transform now. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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3.3.1. Review of the Standard Methods of CWT Calculations Usage of filters. In this method the integral (1) should be discretized and replaced with the discrete convolution ˜ i) w(ai , bj ) = C(a

X

f (tk )ψ˜i (tk − bj ).

k

Here by ψ˜ be the one-dimensional set (filter) representing the values of a wavelet in the predefined point of grid; ψ˜i be its part, which is located within the characteristic length of filter for the given (i−th) scale. This algorithm has the following advantages: 1. The filter is a direct discretization of the integral transform’s formula without any additional suppositions; 2. Since the filter is pre-computed, the speed of a calculations is quite high because the sampled transformed function is multiplying on the initially given set of number for all scales. 3. The direct discretization of the integral implies the possibility to calculate the transform for almost arbitrary scale a.

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Due to its simplicity and directness, the algorithm has the following defects: 1. Since the filter is pre computed and could not be rescaled (one uses simply the subsamples of the initial sample for various scales), the sample of the analyzed function must be defined in equispased points. 2. The local wavelet discretization completely does not depend on the local behaviour of the transformed function. 3. The usual simple realization of the filter for the wavelet defined over all number line leads to undamped boundary effects near the boundaries of the sample. Typically they consist of the decrease of the wavelet transform amplitude and corruption of its form.

This method is realized in Wavelet Toolbox of MATLAB [22] almost in the simplest form described above. The unique modification is the usage of the integral of the wavelet as a filter source function with a posterior differentiation of the result of a convolution. There are some possible ways of the elimination of the described defects. For example, it is possible to calculate the filter in actual points of a non-equispaced sample discretization. It leads to a loss of speed’s advantage, which provides a pre-computed filter. Moreover, the rescaling of a single-shot calculate non-equispaced filter does not damp the errors connected with the various local behavior o the transformed sample at the various scales. Taking this into account needs the recalculation of the filter at every scale step, that is to say, a fast algorithm of the filtration will changed to a usual numerical calculation of the integral. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Eugene B. Postnikov

Usage of the Fourier transform as an intermediate step This method is based on the following fact: the Fourier transform of a convolution is equal to the product of the convolved function’s Fourier transforms. Applying this to the wavelet transform (1), we get Z

w(a, b) = C(a)

+∞ −∞

fˆ(k)ψˆ∗(ak)eikb dk,

(17)

where Fourier-images are denoted with ˆ. Usually (for example, it is realized in a very powerful and popular package WaveLab of Stanford University [23]; particularly, this package was used for illustrative calculation in a classical book [9]) one uses Fast Fourier Transform (FFT) for this goal. The main advantage of this algorithm is • the speed of calculation due to the FFT. However, FFT determines the disadvantages: 1. The sample must contain 2N nodes and be equispaced. 2. FFT periodizes the sample; therefore, the transform of functions with sufficiently different properties at the left and right boundaries of a sample will exhibit artifacts connected with periodization. These artifacts grow with the scale.

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3. FFT algorithm is best adapted to the logarithmical output frequency scale. As a result, the wavelet transform will be calculated for logarithmically distributed a (octave representation). These difficulties are mostly connected with FFT algorithm and are sufficient for some applications of real signal data processing. Naturally, they are can be eliminated by usage of other algorithms for evaluation of the Fourier transform. There exist a large variety of methods of Fourier transform calculation including Non-equispaced FT. But they are more complicated comparably with FFT and, consequently, annuls its preferences in speed and small complexity. 3.3.2. PDE Usage

The advantages of the presented algorithm will be considered below by examples. To show the advantages of the proposed method via the examples in the following subsection, the partial differential equation (14) with the initial value (16) is solved by the standard MATLAB solver pdepe. It provides a high precision of the result and simple programming. However, it is useful to describe the inner structure of discretization with the goal to explain the algorithm and to estimate the calculation cost. The explanation is based on the paper [24]. To find a numerical calculation scheme for (14), let us consider the semidiscrete approximation of (14) via replacing the partial differential equation by the ODE system with the independent variable a. This approximate method is the method of lines. Let Nb nodes bj subdivide the interval, where b is defined into Nb − 1 subintervals [bj , bj+1],Theory, where j = 1, 2, ..., Nb − Analysis 1. and Applications, Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Analysis and Applications : Theory,

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Let us rewrite (14) as the following equivalent system:   

∂w ∂b

=

v,

∂v ∂b

=

1 a

 



∂w ∂a

(18)



+ iω0 v .

We integrate the first equation of the system (18) over each subinterval: Z

w(bj+1 ) − w(bj ) =

tj+1 tj

v(b)db

(19)

Then, after integration of (18) with the variable upper and bottom limits, we get v(t) inside the interval [bj , bj+1]: Z

1 a

v(b) = v(bj ) +

b bj

∂w iω0 db + [w(b) − w(bj )] , ∂a a

(20)

Z

1 bj+1 ∂w iω0 db − [w(bj+1) − w(b)] , a b ∂a a The next step is to substitute (20) and (21) into (19) and to use the interpolant v(b) = v(bj+1) +

w(a, b) ≈ wj (a)

(21)

bj+1 − b b − bj + wj+1 (a) bj+1 − bj bj+1 − bj

after that we get (a truncation error is omitted):

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wj+1 − wj = vj βj +

βj2 6a

wj+1 − wj = vj+1 βj +



βj2 6a

∂wj+1 ∂wj+1 +2 ∂a ∂a 

2∂wj+1 ∂wj + ∂a ∂a



+ iω0

βj (wj+1 − wj ) , 2a

(22)

− iω0

βj (wj+1 − wj ) , 2a

(23)



where βj = bj+1 − bj . To exclude vj let us write (23) for the previous layer (line), subtract (22) from such a form of (23) and group the similar terms. Finally, the desired semidiscrete form of (14) is represented as the system of ordinary differential equations for inner nodes: βj−1 ∂wj−1 βj−1 + βj ∂wj βj ∂wj+1 + + = 6 ∂a 3 ∂a 6 ∂a a βj−1

iω0 + 2

!

wj−1 −

a βj−1

a + βj

!

wj +

a βj−1

iω0 − 2

!

wj+1 .

(24)

The system (24) also can be written in the matrix form M ({βj })

∂w(a) = F (a, {βj }) w(a). ∂a

(25)

From the numerical implementation point of view, it is especially important that the matrices (M and F) are three-diagonal. There are many well-developed speedy and memorysaving algorithms for the processing of such kind of matrices. For example, one can numerically solve (25) using Adams-Crank-Nikolson method, which is always numerically stable Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

292

Eugene B. Postnikov

and convergent, and an error is quadratic over a time step. Namely, we can get a recurrent equation by integrating (25) over the interval [an , an+1 ] and by the usage of the trapezoidal rule for the approximate integration: 







1 1 M − Fn+1 ∆a wn+1 = M + Fn ∆a wn , 2 2

(26)

where the interval of change of a is subdivided into the equal subinterval with the length ∆a. Both sides’ matrices are three-diagonal, therefore one can use the LU-decomposition (Thomas algorithm) and to calculate wn+1 with the backward substitution. The initial value determines w0 . Since all matrices in (26) depend on a linearly, all iterations are very simple: one needs to add   1 1 ∆ M − Fn+1 ∆a wn+1 = − (∆a)2 2 2 and





1 1 ∆ M + Fn+1 ∆a wn+1 = − (∆a)2 2 2

to the current values correspondingly. Therefore, one needs to keep in the computer’s memory only two matrices: 

βj−1 βj−1 + βj βj M = tridiag , , 6 3 6 and

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A = tridiag

1 βj−1

,−

1 βj−1

1 1 − , βj βj



!

.

Obviously, one can save the layers containing a desired value of the transform on the hard disk during the work. This keeps the temporary storage and allows to accelerate calculations using an asynchronous input-output. Such parallel method is the most effective for multicore/multiprocessor systems with the processes or threads scheduling by OS and/or a programmer. We can point out that this kind of processors is constantly getting cheaper and modern hardware/software producers recently have drawn their attention to these processors even for PC. Since the matrix (26) is three-diagonal, the time of calculations is proportional to O (Nb ) if one uses Thomas algorithm with the forward and backward substitutions. The number of the iterations is Na, therefore, the full time is proportional to O (Na Nb) for our scheme (26). But the direct calculation of the integral takes O (Nb ) for one point and O (Na Nb) for the complete mesh of Na Nb nodes. The calculation of the wavelet transform using FFT is faster, but it takes time O (NaNb log Nb ) . Thus, the proposed algorithm (26) is the fastest one.

Numerical examples Note that there are no functions with an infinite support in the real computations, all samples are finite. Therefore, one needs to define the behaviour of the numerical solutions at the boundaries of the interval. In other words, one needs to solve the boundary problem instead of the initial problem. Naturally, the solutions will be not equivalent in general, but due to the locality of the wavelets this difference takes place only Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

293

near the boundary points and grows to the center of the interval with the growth of the scale. There are some methods of elimination of the boundary influence, see [25]: padding with zeros before/after the beginning/end of sample, multiplying a simple by the function, which tends to zero in the interval’s ends, periodization and anti-periodization of the sample, etc. The particular supremacy of PDE-based method is in its allowance to choose the boundary conditions independently for both ends of a sample, and to tune them to signal’s specific properties. For the calculations, let us introduce the real u(a, b) and the imaginary v(a, b) parts of the wavelet transform: w(a, b) = u(a, b) + iv(a, b). Then the equation (14) will be written as the system ! ! ! ∂ ∂2 ∂ u u v =a 2 + ω0 v ∂a v ∂b ∂b −u or in the divergent form ∂ ∂a

u v

!

a ∂u ∂b + ω0 v a ∂v ∂b − ω0 u

∂ = ∂b

!

.

The practical experience shows that the boundary disturbances could be eliminated by a sharp increasing of the inward/outward flows of real or imaginary flows at the both ends of the interval. Corresponding boundary conditions (which are exact for a = 0) are

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u(a, ·) v(a, ·)

!

a + −n 10 + an

b1 0 0 b2

!

a ∂u ∂b + ω0 v ∂v a ∂b − ω0 u

!

=

u(0, ·) v(0, ·)

!

.

(27)

Here parameter n is a number, from 4 to 8, that provides the best results for the samples on the interval [0, 1] (for other cases the interval can be easily rescaled to this). The value of b1 and b2 are {b1 = 1, b2 = 0} or {b1 = 0, b2 = 1} depending on the behaviour of the real and imaginary parts of the given transformed function at every end of the interval. Simple complex harmonic oscillation Let us consider the wavelet transform of the simplest complex harmonic function f (t) = exp(iωt). Let we have the sample over the interval [0, 1]. If this function were defined over a full number line, there exists the exact explicit expression for its wavelet transform with the Morlet wavelet: ω

w(a, b) = ei a b e−

(ω0 −ωa)2 2

.

Note that the corresponding calculation is the same as for equivalent to the derivation of the initial conditions presented above. Correspondingly, the transform modulus be |w(a, b)| = exp −(ω0 − ωa)2/2. It has a maximum, which is equal to one. This maximum line with the ordinate amax = ω/ω0 is placed horizontally and parallel to the abscissa. The real and imaginary parts of the transformed signal with ω = 20π are represented in Fig. 6 as well as the results of the evaluation of the transform with the central frequency ω0 = π by the three described algorithms. The value of the given function in the point 1 is a periodic continuation of its value in the point 0 for this combination sample support and signal frequency. On this account Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

294

Eugene B. Postnikov

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Figure 6. Simple compplex harmonic with periodical boundary values and its wavelet transform calculated by three numerical methods.

Figure 7. Simple complex harmonic with non-periodical boundary values and its wavelet transform calculated by three numerical methods. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

295

both algorithms, FFT and PDE usage, provide a very good result coinciding with the exact solution for the modulus. At the same time, usage of the filter leads to noticeable boundary defects. Moreover, one can detect the artificial maximum even in the center region. Let us consider now the transform of the signal with the frequency ω = 25π on the same interval, see. Fig. 7). The condition of periodicity of the sample is detuned now. As a consequence, the method based on the FFT usage gives the significant boundary errors – the splitting of the maxima line. Boundary defects of the filter calculation are still the same as before. But the picture of the wavelet modulus evaluated with the ODE algorithm shows that the boundary influence is successfully damped. Moreover, let us explore in detail the crosssection of the modulus surface with the plane b = 0.5, i.e. the point which is the farthest from the boundaries, Fig. 8. It is obvious that all three methods (PDE: thick solid line; Filter: dashed line; FFT: thin solid line) allow to detect the point of maximum in the right place. Also they coincide in the nearest vicinity of this point. However, one can see the deviation at larger scales. Only the line calculated with the help of PDE tends to zero as the Gaussian must do. Other two lines exhibit the opposite behaviour. Especially it is clear in the left picture, where the region a > 0.25 is shown. The most irregular line corresponds to the usage of the pre-computed filter. This irregularity is a consequence of the absence of adaptation to the details of the sample. Thus, if the transform of such simple signal contains the significant defects, the result for more complicated functions will be worse.

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3.4.

Composite Non-stationary Signal

Let us consider the decaying harmonic oscillation f (f ) = e−2t cos(10πt) with the additional simple harmonic signal cos(50πt), which exists only on the interval [0.048, 0.39], see Fig. 9. Fig. 3.4. depicts a wavelet modulus calculated via the three described ways. It is clear that the methods based on the filter and FFT usage do not give a fully adequate description of the signal behaviour in the wide region closed to the left boundary. In this case, the periodization of FFT calculation generates a very big artificial maximum there. The method based on PDE solution due to independency of left and right boundary conditions allows to overcome both boundary effects and get the correct picture. As just one more example, let us analyze the modulus maximum corresponding to the decaying harmonic along the line of maximum far from the boundaries. Fig. 11 presents this as a semilogarithmic graph: scales of the abscissa axis is linear, the axis of ordinates has an logarithmical scale. For such a scaling the line of maximum must be a constant slope line. The taken interval is positioned in the middle part of the sample, thus one can neglect the boundary effects. All three methods reproduce the slope and its value correctly, but the algorithm based on the usage of filter gives the line oscillating around mean (correct) value. As before this fact is connected with non-adaptivity of pre-computed filter to the sample. PDE-based method works quite correctly.

In a conclusion, Fig. 12 represents the transform of the same signal sampled in 197 random uniformly distributed points from [0, 1]. Both a pre-calculated equispaced filter and FFT are not completely applicable in this situation. At the same time, the usage of PDE easPartial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

296

Eugene B. Postnikov 1

0.9 0.025 0.8

0.7 0.02

|w(a,0.5)|

|w(a,0.5)|

0.6

0.5

0.015

0.4 0.01 0.3

0.2 0.005 0.1

0

0

0.04

0.1

0.15

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a

0.2

0.25

0.25

0.3

0.35

0.4

0.45

0.5

a

Figure 8. The crossection of the surface representing wavelet transform of the harmonic with frequency ω = 25π .

ily solved on the non-equispaced grid with the same simplicity as on the equispaced allows to find the modulus and the phase of the wavelet transform with the Morlet wavelet. And this result provides an considerably detailed picture to detect both characteristic maxima and their localization even for this complicated case.

4.

The Continuous Wavelet Transform with the Wavelets of Gaussian and Morlet Families as a Superposition of PDE Solutions

Due to the admissibility condition (2) existence, it is impossible to define a Cauchy problem for any exact wavelet. The explanation of this fact is the following. In the limit a → 0 the exact continuous wavelet transform tends to zero for all b. Accordingly, there are no unique initial conditions for partial differential equation, which admit the current wavelet. However, since the wavelet transform is linear, the authors of [26] have proposed to use the difference between two diffusion smoothed images, where each one could be found as a Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

297

1

0

−1

0

0.1

0.2

0.3

0.4

0.5

+

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

0

−1

f(t)

2

||

0

−2

t

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Figure 9. The construction of the considered composite signal.

Figure 10. The comparison of signal’s wavelet transform calculated by three methods.

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298

Eugene B. Postnikov PDE

|w(0.1,b)|

0.278

0.185 0.3

0.32

0.34

0.36

0.38

0.4 Filter

0.42

0.44

0.46

0.48

0.5

0.32

0.34

0.36

0.38

0.4 FFT

0.42

0.44

0.46

0.48

0.5

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

|w(0.1,b)|

0.278

0.185 0.3

|w(0.1,b)|

0.278

0.185

0.3

b

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Figure 11. Tracing of the decaying wavelet modulus maximum from Fig. 3.4.

Figure 12. The wavelet transform of composite signal (Fig. 9) sampled in the random points. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

299

solution of the diffusion differential equation.

4.1.

Evaluation of CWT with the DoG and the Exact Morlet Wavelets

Let is consider the continuous wavelet transform with the DoG wavelet (3) in the explicit form:   Z +∞ (t−b)2 1 − (t−b)2 2 dt √ w(a, b) = C(a) f (t) e αa − e− 2a2 , (28) α −∞ 2πa2 where C(a) is a norm factor. The expression (28) could be presented via the difference of two integrals with the Gaussian as a kernel, w(a, b) = C(a) (W1 (a, b) − W2(a, b)), where Z

W2 (a, b) =

+∞ −∞

Z

W2 (a, b) =

f (t)e

−

(t−b)2 √ (a α)2

+∞

f (t)e

−∞

−

dt , √ 2π(a α)2

q

(t−b)2 a2

√

dt 2πa2

.

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Both these integrals are the solutions of the equation ∂W1,2 1 ∂ 2W1,2 = ∂τ 2 ∂b2 but taken in the moments τ1 = a2 α and τ2 = a2 correspondingly. The authors of the paper [26] have constructed the two-layered optical network realizing this transform. The principle of its work is the following: every layer diffusively smooths an input image, but the second one has some time delay, since the “time variable” τ in the classic diffusion equation could be identified with the scale variable of the continuous wavelet transform. This method can be transferred [27] also to the case of the exact Morlet wavelet (5), which has an explicit form w(a, b) =

√ 4



4πa2e

ω2 − 20

 

Z

+∞ −∞

f (t)

e

−

((t−b)+iω0 a)2 2a2

√

2πa2

dt −

Z

+∞ −∞

−

e f (t) √

(t−b)2 2a2

2πa2

 

dt ,

which is very close to (28). Here an energy norm is used, since in the case of the exact Morlet wavelet the correspondence between the scale localization of the wavelet modulus maxima and Fourier frequencies is simpler as for an amplitude norm. Therefore, like the case of DoG wavelet, 2 √ ω2 4 w(a, b) = 4πa2e− 2 (W1 (a, b) − W2(a, b)) , where W1 (a, b) is a solution of the Haase equation ∂W1 ∂W1 ∂ 2 W1 − iω0 =a , 2 ∂a ∂b ∂b and W2 (a, b) is a solution of usual diffusion equation

∂W2 1 ∂ 2 W2 = . ∂a 2 ∂b2 Thus, there exists the system of two PDE, the superposition of their Cauchy problem solutions with solutions W1 (0,: Theory, b) =Analysis W2 (0, b) = f (b) will be desired wavelet transform. Partial Differential Equations: Theory, Analysis and Applications and Applications, Nova Science Publishers, Incorporated, 2011.

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Eugene B. Postnikov

4.2.

Evaluation of CWT with the Wavelets of Gaussian and Morlet Families

In the similar way one could consider the wavelets of the Gaussian and Morlet families with the higher vanishing moments (6) – (8). Let us remark that in the form (15), 8 differs from (7) only by imaginary shift in the exponential. Thereafter, it is possible to make all
2 calculations with the function ϕ(ζ) = exp −ζ /2 . Using Rodrig formula, 2 dn − ζ 2 − ζ2 2 = He e e , n dζ n

(−1)n with the Hermit polynomials

[n/2]

Hen (ζ) =

X

(−1)mn! ζ n−2m m (n − 2m)! m!2 m=0

(here the square brackets denote an integer part of number) we get the following representation of the transform with this kind of wavelets as a series: wn (a, b) = Cn e

−

ω2 0 2

n/2 X

(−1)m n! m!2m (n − 2m)! m=0

Z

+∞ −∞

f (t) (t − (b + iω0a))

n−2m

e

−

(t−(b+iω0 a))2 2a2

an−2m+1

√

dt . 2π

The second term could be expanded with the Newtonian binomial formula: n−2m

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(t − (b + iω0 a))

=

n−2m X

n − 2m l

l=0

!

(−1)n+l−2m (b + iω0 a)n−2m−l tl

As a result, wn (a, b) takes a form of superposition of the following integrals Z

Wl (a, b) =

+∞ −∞

f (t)tl e

−

(t−(b+iω0 ))2 2a2

dt √ . a 2π

Each integral is the Cauchy problem solution for the partial differential equation ∂Wl ∂Wl ∂ 2 Wl − iω0 =a 2 ∂a ∂b ∂b with the initial value Wl (0, b) = f (t)tl . The explicit form of the wavelet transform with two first members of these families, √   ζ2 e 1 − ζ2 ψ1(ζ) = ζe 2 , ψ2(ζ) = 1 − ζ 2 e− 2 , be 2 4 r

w1 (a, b) =

ω2 0

π e− 2 (W1(a, b) − (b + iω0a)W0 (a, b)) , 2 a w (a, b) =

2 Applications, Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and

(29)

PDE as a Tool for the Evaluation of CWT r

301

ω2

0   πe e− 2  2 2 a − (b + iω a) W (a, b) + 2(b − iω a)W (a, b) − W (a, b) . 0 0 0 1 2 8 a2

(30)

Here the amplitude norm is used. As the first example, let us consider the evaluation of this kind or real ( ω0 = 0) wavelet transform (29) – (30) as for the power function. It is known that the corresponding expressions could play a role of the smoothed derivatives. For f (t) = t3 the analytical results be r r 3 9π 2 π + 3b a; (31) w1(a, b) = a 2 2 r

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w2(a, b) = −6b

πe a. 8

(32)

Fig. 13 represents the result of the transform for the cube-power function 13a by the method (29). Let us take the sample over the interval [−1, 1]. For simplicity, the null value was used as a boundary condition for both boundaries. Fig. a2w1 (a, b) depicts the graph of the function a2 w1(a, b), i.e. the difference in the brackets in (29) multiplied by the norm factor. One can see that the function grows from the null value and its cross-sections at all a = const > 0 are parabolas. Some of these equispaced cross-sections of w1(a, b) starting from a = 0.32 are represented in Fig. 13. Obviously they are the parabolas nonequidistantly shifted along the ordinate-axes. The last effect is an influence of the first term in (31)). This is shown in more detail in Fig. 13d, where logarithmical scale is used. The resulting plot of w1(a, 0) is a linear one and the tangent allows to determine this power coefficient. Fig. 14 represents the analogous calculation using (30) for the same transformed function. Clearly that the cross-sections of a2 w2(a, b) are the straight lines for all given scales, and the tangents of these lines depend on the scale’s values. As the next example, let us consider the real and complex wavelet transforms of a signal consisting of various harmonics within various intervals, and their superposition in the middle, Fig. 15e:   

sin(16πt), 0 ≤ t < 13 , f (t) = sin(16πt) + sin(64πt), 13 ≤ t < 23 ,   2 sin(64πt), 3 ≤ t ≤ 1. Only the module of the complex wavelet transform are shown in Fig. 15a-b since it is enough to extract the periods within various intervals. The boundary conditions (27) are used. Note that this family of wavelets has just one more useful property in comparison with the exact wavelet transform. Due to the fact that these wavelets are constructed by multiplication of a polynomial and the Gaussian, there is the possibility to find the close form expression for the relation between the value of modulus maximum and the frequency of pure harmonic (at least for the wavelets with the small number of vanishing moments). Moreover, it is true also for the rescaled combination used above in the examples:

√ (W (a, b) − (b − iω )aW (a, b)) , = iω 2πa2 exp −(ω0 − aω)2/2 . 1 0 0 Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(33)

302

Eugene B. Postnikov 1 0.25 0.5 aw1(a,b)

f(x)

0.2 0

0.15 0.1

−0.5 0.05 −1 −1

−0.5

0 x

0.5

0 −0.4

1

−0.2

0 b

0.2

0.4

0.05 0.04 0.03 0.02 aw1 (a,b)

aw1(a,b)

0.03 0.02

0.01

0.01 0.2 0 −1

−0.5

0.1 0

0.5

1

0

0.05

a

0.1

0.15

a

b

0.04 0.03 0.02 a2w2(a,b)

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Figure 13. The wavelet transform with the WAVE-wavelet of cube power function.

0.01 0.2

0 −0.01

0.15

−0.02 0.1

−0.03 −0.04 −1

0.05 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

b

0.6

0.8

0

a

1

b

Figure 14. The wavelet transform with the MHAT-wavelet of cube power function. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 15. The harmonic function with the variable frequency and the amplitude of its wavelet transform with two first members of the Gaussian and the Morlet families.

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The resulting expression connecting the scale of the maximum point and the frequency for (33) takes the form s ! ω0 8 amax = 1+ 1+ 2 . 2ω ω0 The results of the real wavelet transform using the formulae (29) – (30) are presented in Fig. 15c-d. As above, they are multiplied by a for the wavelet, which is the first derivative of Gaussian function, and by a2 if the derivative is the second. As it must be expected, corresponding pictures of the transform are the smoothed first and second derivatives of this harmonic signal for every scale. Naturally, this statement is valid up to the constant factor determined by the scale value.

5.

The Cauchy Problem for the CWT with the Variable TimeScale Resolution

As it has been described above, the calculation method, which uses the reduction of the integral CWT with the Morlet wavelets to the solution of PDE system is very powerful. However, it has some limitation if one evaluate the transform of a finite sample. This restriction is connected with the boundary influence and will be sufficient for the large values of the central frequency. Let us consider just one more time the wavelet transform of the pure complex harmonic f (t) = exp(iωt). In the ideal case of an infinite line the corresponding result is (ω0 −ωa)2

w(a, b) = eiωb e− 2 . Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(34)

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304

Eugene B. Postnikov

As it was considered previously, the amplitude has one maximum corresponding to the maximum of the Gaussian in (34) and the phase does not depend on the scale a. But, let consider the result represented in Fig. 16: the numerical calculation of wavelet phase for the finite interval. One can clearly see that the picture of phase has a regular form of (34) only if a ≤ T , where T = 2π/ω is a period of the signal. For bigger scales the phase maxima lines tend to converge in the direction of the middle of interval. Obviously, it is error determined by the finiteness of the interval, i.e. by the presence of boundaries. Note that this error is not unique for the PDE-based method of calculation, two other algorithms (FFT- and filtered-based) reproduce the same picture (moreover, filter-based calculation produce also a very significant level of artificial noise for large scales; its background has been discussed above). Why this phenomenon is so sufficient for PDE-based algorithm? The answer is simple: two other methods work with every scale independently: to get numerically the transform’s values for the sequence of ak one multiplies the transformed sample by some filter (actually, numerical Fourier transform is also a kind of filter) calculated only for this ak . Numerical errors for all ak are independent. But for a numerical solution of PDE system the result for every layer ak depends on the value of ak−1 , down to the initial sample. Since the phase determines the relation of imaginary and the real part of the calculated function, the large phase errors lead to the accumulation of errors. What is a criterium for algorithm’s applicability? The maximum line for the pure harmonic corresponds to amax = ω0 /ω or, taking into account the relation between a frequency and a period, amax = (ω0 /2π) T . Thus, if ω0 ≤ 2π, then am ax ≤ T and it is placed below the phase error’s region. Here, as it is demonstrated in the previous section, the PDE-based method has supremacy in an exactness over others. But if ω0 > 2π it is not applicable. At the same time some problems need to find the wavelet transform for central frequencies larger then 2π. First of all, these are the problems, where one needs better frequency resolution than time (spatial) one. Also there exist some problems, which require a set of wavelet transforms with various central frequencies for estimation of the signal’s parameters, see examples in the paper [28]. Fortunately, a diffusion approach is also applicable in this situation [29].

5.1.

The Wavelet Transform with the Fixed Scale as a Cauchy Problem

Note that Morlet wavelet transform +∞ Z

f (t)eiω0

w(a, b) = −∞

t−b a

2 2a2

e−f rac(t−b)

dt √ 2πa

involves not only the scale a and the time shift b , but also the center frequency ω0 , which is known to determine both time and scale-space resolution. In particular, it was demonstrated in [28, 30] that the Morlet wavelet transform provides a better detection tool for isolated pulses and short pulse trains when ω0 is low and better resolves long signals (which can be represented by Fourier series). Let us analyze in some detail the dependence of detection of chaotic synchronization on the center frequency. To facilitate interpretation, a normalized inverse scale ν is defined Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

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f(b)

1 0 −1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

PDE

a

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5 Filter

a

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5 FFT

a

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

b

Figure 16. Wavelet phase of pure complex harmonic calculated for the finite sample.

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by the relation πν = ω0 /a. In terms of ν, the transform is rewritten as

w(ν, b) = e

iπνb

+∞ Z

f (t)e −∞

− −iπνt

e

q

(t−b)2 4ω 2 0

1 2π 2 ν 2

4πω02 2π12 ν 2

dt.

(35)

The desired transform now has the form w(a, b) = u(ν, b) exp(iπνb); i.e., it can be interpreted as harmonic oscillation with frequency ν, and time-dependent complex amplitude. The corresponding u(τ, b) is an integral transform with a diffusion kernel; i.e., it is the solution to the diffusion equation ∂u  2 2 −1 ∂ 2u , = 2π ν ∂τ ∂b2

(36)

taken at the “instant” τ = ω02 for each value of ν in the initial condition u(0, b) = [Re (f (b)) cos(πνb) + Im (f (b)) sin(πνb)] + + i [Im (f (b)) cos(πνb) − Re (f (b)) sin(πνb)] .

(37)

Equation (36) is an equivalent to the system of two equations for the real and imaginary parts of u(τ, b) under amplitude normalization, whereas energy normalization makes the Cauchy problem ill-posed because the square root of the scale in the denominator implies that the transform vanishes on zero scale. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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The modulus and phase of the wavelet transform are defined in terms of the solution to the corresponding differential equation as follows: |w(a, b)| = |u| = φ(a, b) = arctg

5.2.

q

Re(u)2 + Im(u)2 ,

Re(u) sin(πνb) + Im(u) cos(πνb) . Re(u) cos(πνb) − Im(u) sin(πνb)

(38)

Example of Application: The Chaotic Phase Synchronization

As an example, let us consider from the wavelet point of view the study of the system of two coupled R¨ossler oscillators x˙ 1,2 = −ω1,2y − z1,2 + (x2,1 − x1,2), y˙1,2 = −ω1,2x1,2 + a0 y1,2 − z1,2 + (y2,1 − y1,2),

(39)

z˙1,2 = p + z1,2(x1,2 − c)

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with ω1 = 0.98, ω2 = 1.03, a0 = 0.22, p = 0.1 c = 8.5,  = 0.05. Large recent scientific interest to this object is closely connected with the wavelet analysis. One of nonlinear phenomena, which is under consideration during last two decades is the chaotic synchronization, see [31]. That effect means that the phases difference of two oscillating solutions of coupled strongly non-linear systems could be bounded despite these solutions are chaotic. It should be pointed out that the phases of the solutions of ODE (39) determined by standard conventional methods (like a velocity-to-displacement ratio or by means of the Hilbert transform) are not synchronized for the presented values of the parameters. At the same time, as it was firstly noted in [32], a synchronization can be detected if one consider the complex wavelet transform w(a, b) = |w(a, b)| exp(iφ(a, b)), where φ(a, b) is the desired phase. The corresponding condition for synchronization of chaotic systems with phases φ1 (a, b) and φ2 (a, b) on a scale a0 is |φ1(a0 , b) − φ2 (a0, b)| < const for any b; i.e., the absolute value of the instantaneous phase difference must be a bounded function of time. This method is applicable no matter whether a phase is well- or ill-defined (by conventional methods). It a more circumstantial paper [33], the authors argue this detection as an example of some general phenomena titled by them “time-scale synchronization”. But in the presentations of this method in [32, 33] and further articles, it is neither explained why the phase of a chaotic signal can be regularized by means of the wavelet transform nor pointed to the actual regularization parameter. Recently in the paper [29], such a parameter has been determined (it is namely ω0 ), and the method for calculating its critical value was described. Furthermore, the relationship between synchronization detection methods using conventional and wavelet-based definitions of phase has been elucidated. This question will be considered in more detail below, especially basing on the background that the described method of the wavelet transform with the fixed scale as a Cauchy problem provides a quite clear explanation. Let the analyzed signals be the complex combination f1,2(t) = x1,2 (t) + x˙ 1,2, see Fig. 17a, where the real and the complex parts are drown with the solid and the dashed lines correspondingly. It be used as the initial conditions for appropriate PDE. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 17. The chaotic osclilltion in the coupled R¨ossler system and its wavelet transforms for ω0 = π and ω0 = 2π.

To sketch the main idea about the background of the wavelet regularization, let us find the wavelet transform of he function f (t) with the method described in the Sec. 3 for ω0 = π (the modulus and the phase are depicted in the 17b and 17b) and for ω0 = 2π (Figs. 17d,e). One can clearly see the difference. In the first case the modulus is quite irregular, the most significant line of maximum at a = 5.25 is not straight, there are a lot of vertical cones corresponding to quick transitions from the high-frequency periodicities (the short horizontal lines at the smaller scales) to this one. When the central frequency is bigger, these small-period oscillations practically are not detected. And the main maximum line (naturally shifted up to doubled scale) is almost straight. The phase picture is also more regularized. Now let us apply the method (36) for the fixed scale corresponding to the main line of maximum. Since the discussed question is the phase behaviour, Fig. 18 shows the phase portraits depicting the dynamics of f1,2 (t) = x1,2(t)+ x˙ 1,2 in the wavelet phase space. This line is calculated for the fixed reduced scale ν and variable central frequencies. By the definition of Cauchy problem, the wavelet phase corresponding to the limit of ω0 = 0, is determined by substituting (37) into (38):

x˙ φ|ω0 =0 = arctg , Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova x Science Publishers, Incorporated, 2011.

308

Eugene B. Postnikov 4 2

0

Im w

Im w

100

−100

−2

ω0=0 −200 −30

−20

−10

0

10

20

0

ω0=0.5π

−4 −2

30

Re w

0

2

2

2

4

Re w

Im w

Im w

1 0

−1

ω =π

−2 −1

0

1

2

−2 −2

3

0

1

2

Re w 1

Im w

1

Im w

0

−1

Re w

2

0 −1

0 −1

ω =3π

ω0=4π

0

−1

0

Re w

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ω =2π

0

−2

−2 −2

0

1

2

−1

0

1

Re w

Figure 18. Phase portrets of the R¨ossler oscillator dynamics in the wavelet space for various central frequencies.

which agrees with one of the standard conventional definitions. In the Fig. 18 it is simply the polar angle of the radius-vector directed to the instant point of the phase curve, which is also a usual phase portrait of the R¨ossler oscillator in these variable. With increasing ω0 the wavelet support has not more than zero length (as it took place for Dirac delta-function if ω0 ). Accordingly, the phase of a wavelet transform with finite ω0 is defined in terms of a Gaussian smoothed product of harmonic and chaotic signals (see kernel in expression (35)). Thus the high harmonics are eliminated, and the phase portrait tends to more regular shape, which are almost elliptic (with a slow change of the radiusvector’s amplitude, which coincides with the modulus of the wavelet transform). Naturally, the motion along these “almost-ellipses” leads to an increasingly regular change of the polar angle, i.e. wavelet phase. As ω0 → ∞, the Morlet wavelet transform becomes the Fourier integral transform, with infinite time domain and zero frequency/scale band. In this limit, two signals can be compared in terms of the presence of certain scales (harmonics), but neither instantaneous phase nor instantaneous scale can be well defined; i.e., time-scale synchronization is out of question. Thus, the center frequency ω0 plays the role of a regularization parameter used to introPartial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Figure 19. Time evolution of phase difference calculated for the center frequency values ω0 : 0.5π, π, 1.5π, 2π, 2.5π, 3π, 3.5π, 4π.

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duce a well-defined phase, which is not a real phase of the chaotic oscillator but its filtered version. Naturally, after the elimination of short high-frequency bursts, the syncronization of the smoothed phase could be detected. Obviously, of interest is the minimum value of ω0 that corresponds to such a regularization. To look at this, let us draw the explored absolute value of phase difference, Fig. 19. It is calculated from the sample of the solution of (39) starting from the moment t = b = 1812. This time shift is necessary to exclude the influence of the initial conditions. Since the system is chaotic it is completely enough to blow them over. The evolution of the phase difference for center frequencies varying from 0.5π to 4π with a step of 0.5π is presented. The lower frequency part of the graphs (ω0 < 2.5π) demonstrates a lack of synchronization: the phase difference is a monotonically increasing function. At higher central frequencies (ω0 ≥ 2.5), the phase difference is a bounded function for the particular value of the normalized scale. Thus, the synchronization threshold lies somewhere between 2π and 2.5π. In the other words, before this threshold the continuous wavelet transform with the Morlet wavelet transform detects pulses of width comparable to the oscillation period. And, since the signals are chaotic (even if synchronized on some scale), the localization of pulses in x1,2 (t) fluctuates, and the phase difference accumulates accordingly. With increasing ω0 time resolution deteriorates, the wavelet length spans an increasing number of cycles, and phase difference fluctuations are increasingly smoothed by diffusion averaging. As a result, time scale synchronization is detected. Thus, the method can be used to determine the minimum smoothing window size required to regular or “quasi-regular” parts of the signals from two coupled chaotic systems and to specify the lower limit of applicability of Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Eugene B. Postnikov

the wavelet-based method for detecting chaotic synchronization of these parts.

6.

Conclusion and Future Perspectives

Thus, as it was demonstrated, the theory of partial differential equations provides a new and very powerful point of view on the continuous wavelet transform, both real and complex. Its genetic connection with the developed theory of diffusion image processing allows to understand some significant results such as a regularization of a wavelet image and phasespace representation, etc. Let us shortly review some perspectives, which are opened by this group of methods. First of all, the just obtained representation could serve as a background for the development of new numerical applications. There are two well-promising items for this:

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• It has been shown that in the discretization of PDE evolving CWT theoretically provides the fastest possible numerical algorithm (up to the constant numerical factor) for the ideal case of functions with an infinite support. Naturally, in the real-life case, the stiff ODE system in the implemented method of lines as well as problems with the boundary conditions for finite samples make this estimation worse. Nonetheless, the algorithm will be still fast, and it additionally provides a high exactness . • The second numerical advantage of the method is its locality. To discretize the operators of the second and the first derivatives, one needs only few nodes of a sample, three as a minimum. Thus, it is possible to create a tool for a fast processing (a local spectral analysis) of the stream data: every three new received indications on input can be processed, the result will be transferred in the next layer for the consequential calculations and so one. Also, this locality makes it possible to construct the parallel algorithms for processing of extremely large samples. Now it may be useful not only for clusters and parallel supercomputers but even for the modern PC with two-kernel processors. Secondly, the approach based on the consideration of a wavelet transform set with the variable central frequency may be a background for further more detailed researches in the analysis of the multiscale properties of chaotic dynamical systems. The most direct question is the dependence of the topology of a system’s phase portrait on the variable resolution in a general case. What are the tasks for the future theoretical researches? One of them is connected with the generalization on the multidimensional case for the CWT with the wavelets of the Gaussian and Morlet families. Namely, in (36) both terms consist of the differential operators, which are simply one-dimensional forms of the Laplacian and the gradient. To reach a scalar form in the right-hand-side, there must be a dot product of this gradient with the central frequency vector Ω: ∂w ~ · ∇w = a∇2 w + Ω ∂a

(40)

This is a natural way to the definition of the directional Morlet wavelet. See, for example, the clear and comprehensive review theirs properties and usage in [34] for 2D case. Namely, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

PDE as a Tool for the Evaluation of CWT

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if Ω = (ωx , ωy ), then α = atan (ωy /ωx ) is the angle determining the direction of the wavelet. Additionally, the form (40) provides an opportunity to apply the method in the case of non-Cartesian grids, say on a sphere. In this case one needs only to represent the Laplacian and the gradient operators in the spherical co-ordinates. Also, it seems to be real to apply the approach of the reducing of the CWT to a Cauchy problem for PDEs in the case of other wavelets, which are not from the Gaussian and the Morlet families. It is known that very weak restrictions on the functional properties have made possible to construct a large variety of wavelets adjusted for a processing of specific signals [10]. Thus, it is an interesting task to try to find partial differential equations and initial conditions for the continuous wavelet transform with some of these specific wavelets. Finally, a very new and relatively poor explored area is interconnection between the continuous wavelet transform and the non-linear diffusion image processing, see [19]. There is a strong probability that integration of the methods reviewed in the cited paper and in this one will lead to new results interesting from mathematical physics’ point of view as well as for the practical signal and image processing.

References [1] Grossman, A.; Morlet J., SIAM J Math Anal, 1984, 15, 723-736. [2] Poularikas, A.D. The Transforms and Applications Handbook ; CRC Press, 2000. [3] van den Berg, J. C. (Ed.). Wavelets in Physics; Cambridge University Press, 1999.

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[4] Walczak, B. Wavelets in Chemistry; Elsevier Science & Technology, 2000. [5] Klees, R.; Haagmans, H.N. (Eds.). Wavelets in the Geosciences; Spinger, 2000. [6] Keller, W. Wavelets in Geodesy and Geodynamics; de Gruyter, 2004. [7] Aldroubi, A.; Unser, M.A. (Eds.). Wavelets in Medicine and Biology ; CRC Press, 1996. [8] Li`o, P. Bioinformatics, 2003, 19, 2-9. [9] Mallat, S. A Wavelet Tour of Signal Processing ; Academic Press, 1999. [10] Holschneider, M. Wavelets. An Analysis Tool. Oxford University Press, 1995. [11] Mallat, S.; Zhong, S. IEEE T Pattern Anal, 14, 710-732. [12] Mallat, S.; Hwang, W.L. IEEE T Inform Theory, 1992, 38, 617-643. [13] Bacry, E.; Muzy, J.F.; Arneodo A. J. Stat. Phys., 1993, 70, 635-674. [14] Marr, D. Vision. A computational Investigation into the Human Representation and Processing of Visual Information. W.H. Freeman and Company, 1982. [15] Jaffard, S.; Meyer, Y.; Ryan, R.D. Wavelets. Tools for Science & Technology. SIAM , 2001 Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[16] Koenderink, J.J. Biol Cybern, 1984, 50, 363-370. [17] Kestener, P.; Arneodo A. Phys Rev Lett, 2003, 91, 194501. [18] Perona, P.; Malik J. IEEE T Pattern Anal, 1990, 12, 629-639. [19] Mr´azek, P.; Weickert, J.; Steidl, G. In: Lecture Notes in Computer Science. 2695. Springer, 2003, 101-116. [20] Postnikov, E.B. Comp Math Math Phys, 2006, 46, 73-78. [21] Haase, M. In: Paradigms of Complexity. M.M.Novak, M.M., Ed.: World Scientific, 2000, pp. 287-288. [22] http://www.mathworks.com/products/wavelet/?BB=1 [23] http://www-stat.stanford.edu/ wavelab/ [24] Kiseliov, R.V.; Postnikov E.B. In: Proceedings of the XIX Session of the Russian Acoustical Society. Acoustical Measurements and Standartization. Nizhny Novgorod, 2007, pp. 233-235. [25] Torrence, C.; Compo, G.P. B Am Meteorol Soc, 1998, 79, 61-78. [26] Cho, C.S.; Ha, S.-W.; Kim, J.H.; Yon T.-H.; Nam, K.G. Opt Eng 1997, 36, 34713475.

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[27] Postnikov,E.B. Vychislitelnye metody i programmirovanie (Numerical Methods and Programming), 2008, 9, 84-89 (in Russian). [28] De Moortel, I.; Munday, S.A.; Hood, A.W. Sol Phys 2004, 222, 203-. [29] Postnikov, E.B. J Exp Theor Phys, 2007 105, 652-654. [30] Addison, P.S.; Watson, J.N.; Feng T. J Sound Vib 2002, 254, 733-762. [31] Pikovsky, A.; Rosenblum, M; Kurths, J. Synchronization: An Universal Concept in Nonlinear Sciences. Cambridge University Press, 2001. [32] Hramov, A.E.; Koronovskii, A.A. JETP Lett, 2004, 79, 391395. [33] Hramov, A.E.; Koronovskii, A.A. Physica D, 2005, 206, 252-264. [34] Antoine, J.-P.; Murenzi, R., Vandergheynst, P., Ali, S.T. Two-Dimensional Wavelets and Their Relatives. Cambridge University Press, 2004.

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In: Partial Differential Equations Editor: Christopher L. Jang

ISBN 978-1-61122-858-8 c 2011 Nova Science Publishers, Inc. 

Chapter 12

T HE B LOWUP M ECHANISM IN N ONLINEAR PARTIAL D IFFERENTIAL E QUATIONS - S CALING AND VARIATION∗ Takashi Suzuki† Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyamacho 1-3, Toyonakashi, 560-8531, Japan

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Abstract We study the blowup mechanism of the solution to nonlinear partial differential equations derived from their scaling and variational properties. Then this method is applied to several mean field equations provided with the free energy decreasing and conservation laws. Among them is a Smoluchowski-Poisson equation describing the kinetic motion of self-gravitating fluids which exposes two different blowup patterns.

1.

Blowup of the Solution

It occurs often that the solution to the nonlinear evolution equation is not able to extend global in time. This phenomenon is called the blowup of the solution. So far, the question when and how the blowup of the solution occurs has been studied in details. This article is focused on the equations with self-similarity and variational structure associated with several conservation laws and the laws in thermodynamics. First, we illustrate the study on the semilinear parabolic equation to clarify what happens to the solution, including new results and their proof. This study concerning the blowup of the solution is evoked by fruitful mathematical structures of the equation, that is, the ODE part, the linear part, stationary solutions, their stability, variational structure, scaling properties, the backward and the forward self-similar transformations, self-similar solutions, blowup rates, blowup sets, critical ∗

A version of this chapter also appears in Mathematical Physics Research Developments, edited by Morris B. Levy, published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort in encourage wider dissemination of research. † E-mail address: [email protected]

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314

Takashi Suzuki

exponents, thresholds, and so forth. Having clarified these profiles, we will turn to several quasilinear parabolic equations derived from mean field theories, and describe their blowup mechanisms emphasizing physical and biological significances. The first object is the ordinary differential equation du = |u|p−1 u, dt

u|t=0 = u0

(1)

with 1 < p < ∞ standing for the exponent. The unique stationary solution u = 0 is unstable in (1). More precisely, if u0 > 0 it holds that limt↑T u(t) = +∞ in a finite time T > 0, while u0 < 0 implies limt↑T u(t) = −∞ similarly. Such a solution is given by, explicitly,   1 p−1 1 − 1 (T − t) p−1 (2) u(t) = ± p−1

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1 and hence T = p−1 |u0 |−p+1 . The blowup of the solution, as is described, indicates the phenomenon that the solution is not able to extend after a finite time. Thus we can say that any non-trivial solution blows-up in finite time in (1). The blowup of the solution is certainly made clear based on the well-posedness of the equation. Given a nonlinear partial differential equation of evolution, we try to establish its local in time well-posedness in a suitable function space X. This attempt is achieved by generating a local semi-flow (or semi-group) in X, so that the solution is strongly continuous in X both in time t and the initial value u0 . In the case of the autonomous equation where t is not involved explicitly, it occurs often that the existence time T ∈ (0, +∞] of the solution is bounded from below by u0 X . If this property is the case, we obtain limt↑T u(t)X = +∞ when T < +∞. The semilinear parabolic equation

∂u − Δu = |u|p−1 u in Rn × (0, T ), ∂t

u|t=0 = u0 (x)

(3)

is a natural perturbation of (1). In this case, X = L∞ (Rn ) can cast such a function space. Thus, (3) generates a local in time semi-group in L∞ (Rn ) and limt↑T u(t)∞ = +∞ holds when T < +∞. Then the blowup set defined by S = {x0 ∈ Rn ∪ {∞} | ∃ tk ↑ T, ∃ xk → x0 , |u(xk , tk )| → +∞} is not empty. In contrast that the ordinary differential equation (1) is the spatially homogenuous part of (3), its linear part is given by ∂u − Δu = 0 in Rn × (0, T ), ∂t

u|t=0 = u0 (x).

(4)

Any solution to (4) defined suitably in a function space exists global in time. For example, (4) is well-posed global in time in X = Lq (Rn ), 1 ≤ q ≤ ∞. What we can expect is that if u0 ∞  1, (3) will be close to (4) and consequently, the solution u = u(·, t) will exist global in time. This property, however, is the case only for p 1. In fact, (3) is close to Applications, 1. In this case, the solution for u   1 (1) if 0 < p − 1  1 even for u0 ∞ and Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis Nova Science Publishers, Incorporated, 2011. 0 ∞

The Blowup Mechanism in Nonlinear Partial Differential Equations

315

(and consequently any non-trivial non-negative solution to (3) by the comparison theorem,) blows-up in finite time. The threshold case of p is called the Fujita exponent which is equal to pf = 1 + n2 . Thus we have the following theorem, see also [37, 71]. Theorem 1 ([15, 26]) If p > pf and u0 ∞  1 the solution to (3) is global in time, while if 1 < p ≤ pf any non-trivial non-negative solution to (3) blows-up in finite time. A heuristic argument of E. Yanagida says that the solution u(t) =

1 − 1 (−t) p−1 p−1

1 defined for t < 0 indicates the unstabilizing rate p−1 of the ODE part (1) as t ↓ −∞, while the Gauss kernel |x|2 n (5) G(x, t) = (4πt)− 2 e− 4t

is associated with the stabilizing rate n2 of the linear part (4) as t ↑ +∞. They are thus competing at p = pf , that is p − 1 = n2 . Here it will be worth mentioning that Theorem 1 concerning the part dominated by the linear equation (4) follows from the Duhamel principle  t

u(t) = etΔ u0 +

0

e(t−τ )Δ (|u|p−1 u)(τ )dτ

combined with the estimate n 1

1

etΔ u0 s ≤ (4πt)− 2 ( q − s ) u0 q ,

1 ≤ q ≤ s ≤ ∞,

while the other part dominated by the ODE equation (1) is detected by showing Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.



u0 (0, t) ≤ for

1 p−1



1 p−1

1

t− p−1

(6)



u0 (x, t) =

Rn

G(x − y, t)u0 (y)dy

using Jensen’s inequality which gives a contradiction by the decay order of the Gauss kernel (5) as t ↑ +∞. It is true that the above critical exponent is related to the well-posedness in Lq (Rn ), 1 ≤ q < ∞, of (3). The profiles are summarized as follows, see [69, 70, 32, 33] for the proof. 1. The sub-critical case is Lq (Rn ), q > n2 (p − 1). Equation (3) is well-posed even in the weak topology of Lq (Rn ) which results in the property that the existence time of the solution is estimated from below by u0 q . 2. This equation is ill-posed in Lq (Rn ), 1 < q < n2 (p − 1). There is a case that even local in time solution is not admitted in this function space.

3. In the critical case of q = n2 (p − 1), local in time well-posedness of (3) is actually true but is restricted to the strong topology of Lq (Rn ). The existence time of the solution is not estimated from below by u0 q , and, consequently, there is the case T ↓ 0 when |u |q converges weakly to a measure with u0 q conserved. Partial Differential Equations: Theory, Analysis and0 Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Takashi Suzuki

We see that p = pf arises with q = n2 (p − 1), q = 1. The above described local in time well-posedness of (3) in L1 (Rn ) suggests limt↑T u(t)1 = +∞ for any non-trivial non-negative solution in the case of 1 < p < pf . This property is true in some sense, see (6). The reason why L1 -norm is selected to define the Fujita exponent pf is associated that (3), p > 1, is taken in the whole space Rn . If Rn is replaced by the bounded domain and the Dirichlet boundary condition is imposed, then the L2 -norm takes place of it because of the Poincar´e inequality. Then, a part of the role of the Fujita exponent shifts to pf = 1 + n4 , see [31]. Here we note that this exponent p = pf satisfies 2 = n2 (p − 1).

2.

Variational Structure

Generally, multiple stationary solutions to the evolution equation make the dynamics complicated. In the ordinary differential equation du = u(1 − u), dt

u|t=0 = u0

for example, there are two stationary solutions u = 0 and u = 1. The former is unstable and the latter is stable. The solution is global in time if u0 > 0 and approaches u = 1 in infinite time, while u0 < 0 implies limt↑T u(t) = −∞ with T > 0 finite. The role of stationary solutions to the dynamics becomes stronger under the presence of the Lyapunov function. Equation (3) actually has the variational structure. Since it is the gradient system (7) ut = −δJ(u) Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

defined for the functional J(u) =

1 1 ∇u22 − up+1 p+1 2 p+1

(8)

its global in time dynamics is under the control of the stationary solutions −Δu = |u|p−1 u

in Rn .

(9)

To see the roles of stationary solutions in the gradient system, we take a simple abstract argument here. We assume that t ∈ [0, +∞) → u(t) ∈ X is a continuous orbit in a Banach sapce X provided with the decreasing property of t → J(u(t)), where J : X → R is a continuous functional. We define the ω-limit set of this orbit by ω(u(0)) = {u∞ ∈ X | ∃ tk ↑ +∞, u(tk ) → u∞ in X}. Taking ui∞ ∈ ω(u0 ), i = 1, 2, we find tik ↑ +∞, i = 1, 2, such that u(tik ) → ui∞ . Passing to a subsequence, we obtain t1k < t2k < t1k+1 , k = 1, 2, · · · which implies J(u(t1k )) ≥ J(u(t2k )) ≥ J(u(t1k+1 )) and hence

J(u1 ) = J(u2 )

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis ∞ and Applications,∞ Nova Science Publishers, Incorporated, 2011.

(10)

The Blowup Mechanism in Nonlinear Partial Differential Equations

317

by the continuity of t ∈ [0, +∞) → u(t) ∈ X and u ∈ X → J(u) ∈ R. Hence J is invariant on ω(u(0)). Next, we put u(t + s) = Tt (u(s)) and obtain a semi-flow {Tt}t≥0 defined on the orbit O = {u(t)}t≥0 satisfying that (t, u0 ) ∈ [0, +∞) × O → Tt(u0 ) ∈ O ⊂ X is continuous and Tt ◦ Ts = Tt+s , t, s ≥ 0. Thus the domain of this semi-flow {Tt}t≥0 extends to O. Given u∞ ∈ ω(u(0)), we have tk ↑ +∞ such that u(tk ) → u∞ which implies u(t + tk ) → Tt(u∞ ) and u(s + tk ) → Ts (u∞ ), where s > t ≥ 0. Since we may assume t + tk < s + tk < t + tk+1 , it follows that J(Tt(u∞ )) = J(Ts (u∞ ))

(11)

similarly to (10), and, therefore, the ω-limit set ω(u(0)) is contained in E = {u∞ ∈ X | t ∈ [0, +∞) → J(Tt(u∞ )) is constant}. Now we turn to a more concrete problem where the above E is actually identified with the set of stationary solutions. In fact, the decreasing of t → J(u(t)) is actually the case if u = u(t) ∈ C([0, +∞), X) is subject to the gradient flow (7) provided with the properties that 1. J : X → R is continuously Fr´echet differentiable. 2. There arises a continuous embedding X → H with a Hilbert space H. 3. ut ∈ L2loc ([0, +∞), H).

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4. The right-hand side of (7) is regarded as an element in H by Riesz’s representation theorem. More precisely, the mapping t → J(u(t)) is absolutely continuous in this case and it holds that d J(u) = δJ(u)[ut ] = −ut 2H ≤ 0. dt Given u∞ ∈ ω(u(0)), we have tk ↑ +∞ such that u(tk ) → u∞ which implies v(t) = lim vk (t) k→∞

(12)

for v(t) = Tt (u∞ ) and vk (t) = Tt(u(tk )). Here we have vkt = −δJ(vk ).

(13)

If u ∈ H → δJ(u) ∈ H  is continuous with H  ∼ = H provided with the weak topology, we can take the limits k → ∞ in (13). In fact, using d J(vk ) = −vkt 2H , dt we have vt ∈ L2loc ([0, +∞), H),

v = −δJ(v),

t and Applications, Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis

318

Takashi Suzuki

and, consequently, d (14) J(v) = −vt 2H dt for v = v(t) defined by (12). Since the left-hand side of (14) is zero by u∞ ∈ E, it holds that vt = 0, v(t) = u∞ . The converse is obvious and hence E = {u∞ ∈ X | δJ(u∞ ) = 0} coincides with the set of (weak) stationary solutions. Here we mention that several alternative arguments are available to clarify global in time behavior of the solution to (7) using stationary solutions. We remind also that the compactness and the connectedness of the ωlimit set ω(u(0)) hold under the presence of the compactness of the orbit O = {u(t)}t≥0 , see [27, 59] for instance. The Sobolev exponent ps = n+2 n−2 , n ≥ 3, and ps = ∞, n = 1, 2, arises with the variational functional J(u) defined by (8). If this functional is restricted to X = H01 (Ω) with the bounded domain Ω ⊂ Rn , then it is well-defined and continuously Fr´echet differentiable there if and only if 1 < p ≤ ps . The limit case p = ps , n ≥ 3, arises with the Sobolev 2n embedding H01 (Ω) → L n−2 (Ω), v22n ≤ S∇v22,

v ∈ H01 (Ω)

n−2

(15)

with the Sobolev constant S = S(n) independent of Ω. Taking the problem

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−Δu = up , u ≥ 0

in Rn

(16)

and −Δu = up , u ≥ 0 u=0

in Rn+

on ∂Rn+ ,

(17)

we have the following theorem, see [8] for the proof using the moving plane. Theorem 2 ([18]) If 1 < p < ps , then (16) admits no non-trivial classical solution. Theorem 3 ([19]) If 1 < p ≤ ps , then (17) admits no non-trivial classical solution. Theorem 4 ([4]) If p = ps , any non-trivial solution to (17) is radially symmetric with respect to some point. In connection with the above theorems we note that radially symmetric solutions to (16) for p ≥ ps are classified. Also, Theorem 3 is extended to 1 < p < ps , where ps = n+1 n−3 ,  n ≥ 4, and ps = ∞, n = 1, 2, 3. Actually it is the (n − 1)-dimensional Sobolev exponent, see [10]. The variational structure clarifies also the dynamics of u − Δu = |u|p−1 u in Ω × (0, T ),

u = 0 on ∂Ω × (0, T ),

u = u (x) in Ω, (18)

t 0 2011. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated,

The Blowup Mechanism in Nonlinear Partial Differential Equations

319

where Ω is a bounded domain in Rn with smooth boundary ∂Ω. The fundamental relations are d J(u) = −ut 22 dt 1 1 J(u) = ∇u22 − up+1 p+1 2 p+1 1 d p−1 u22 = −2J(u) + up+1 p+1 . 2 dt p+1

(19)

Then the Poincar´e inequality implies that any global in time orbit to (18) is L2 -bounded and satisfies J(u) > 0, and, hence  +∞ 0

ut2 dt < +∞.

(20)

These facts are valid to any 1 < p < ∞, see [59] and the references therein.

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3.

Scaling Invariance

A lot of important properties concerning semilinear and degenerate parabolic equations such as (3) are products of the comparison principle, see [16, 68, 51], for instance. This comparison principle, unfortunately, does not work in many physical and biological models. Equation (3), however, is provided with the scaling invariance which is kept in some other cases, see [52, 53]. To understand the importance of this property, it may be useful to 2n = ps + 1 in (15) is selected from the scaling invariance observe that the exponent n−2 of this inequality. Any model is of course provided with the variational structure when it is derived from the variational principle. If the equation is provided with both scaling invariance and the varitional structure, this scaling property should have the origin in the underlying variational structure. In the case of (3), this property takes the form that if u = u(x, t) satisfies the equation, then so does 2 μ > 0. (21) uμ (x, t) = μ p−1 u(μx, μ2 t), This relation is efficient even in the study of the stationary problem. First, we pick up (16) for p = ps . Then its solution is radially symmetric with respect to some point by Theorem 2. Hence the scaling property (21) combined with the translation invariance classifies the multiplicity of the solution. More precisely, any solution to (16), p = ps , takes the form uμ,x0 (x) = μ

n−2 2

U (μx + x0 ),

μ > 0, x0 ∈ Rn ,

where U = U (r) > 0, r = |x|, satisfies Urr +

n−1 Ur + |U |p−1 U = 0 for r > 0, r

Ur (0) = 0, U (0) = α > 0

(22)

for α = 1 (and p = ps ). In particular, ∇u22 = E∗ is invariant for any non-trivial classical stationary solution to (16), p = ps , which is the origin of the energy quantization in elliptic and parabolic problems involved by this exponent, see [1, 60]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

320

Takashi Suzuki The scaling (21) can be used even in the study of (22). Trying 2

w(s) = r p−1 U (r),

s = log r

(23)

is quick to take the idea, see [51] for example, but originally this property is applied to (22) through the following process by [5, 34]. First, the inhomogeneous condition U (0) = α is normalized by u(r) = αv(s), r = βs with β > 0 determined later, which results in 1 n 2 , β = {(n − 2 − τ )τ α1−p } 2 , τ = n−2 p−1 1 n , β = (α1−p ) 2 p= n−2 1 n 2 1

(sn−1 v  ) + sn−1 |v|p−1v = 0, (sn−1 v  ) + (τ + 2 − n)sn−1 |v|p−1 v = 0,

and

v  (0) = 0.

v(0) = 1, Then, using

w(t) = sτ v(s),

s = et ,

we obtain the autonomous system w  + (n − 2τ − 2)w  + (n − τ − 2)τ (|w|p−1 − 1)w = 0 



w − (n − 2)w + |w| 

p−1

(24)

w=0

w + (n − 2τ − 2)w + (τ + 2 − n)τ (|w|p−1 + 1)w = 0 provided with



lim e−τ t w(t) = 1,

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t↓−∞

lim e−t {e−τ t w(t)} = 0

t↓−∞

(25)

according to those cases. Thus the features of radially symmetric non-trivial classical solutions to (16) for p ≥ ps are completely absorbed into positive solutions to (24)-(25). Here, n , n ≥ 3, arising above, is concerned with the oscillawe remind that the exponent po = n−2 tory property of (22). More precisely, any non-trivial solution to (22) takes infinitely many zeros if 1 < p ≤ po , finitely many zeros for po < p < ps , and no zero for p ≥ ps . We transform (24) into the system of (w, w ) which has two critical points (w,√w ) = √ n−1 n−2 √ n−1 (0, 0) and (w, w ) = (1, 0). The fourth exponent pjl = 1 + 4 n−4+2 (n−2)(n−10) = n−4−2 n−1 , n ≥ 11 and pjl = ∞, 1 ≤ n ≤ 10, arises with the linearized eigenvalues around the latter critical point. Thus it is spiral and nodal according to ps ≤ p < pjl and p ≥ pjl , respectively. This exponent is actually associated with the intersection property, see [25, 39]. More precisely, if p ≥ pjl , the radially symmetric stationary solutions are in order so that any distinct stationary solutions, including the singular solution U∗ (r) = cpr

2 − p−1

,

cp = [

1 2((n − 2)p − n) p−1 ] , 2 (p − 1)

do not interesect each other. If ps < p < pjl , any two of classical or the above singular solutions intersect infinitely many times. Finally, if p = ps , any two distinct classical solutions intersect once, while a classical solution and the singular solution intersect twice. Similar analysis is done for the other equations in [34] among which is −Δu = λeu , see also [17, 43, 44, 14]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Blowup Mechanism in Nonlinear Partial Differential Equations

321

The scaling property (21) is the origin of the blowup analysis recently developed. To understand this method, we shall take the stationary problem −Δu = up , u > 0

in Ω,

u = 0 on ∂Ω,

(26)

where Ω is a bounded domain in Rn with smooth boundary ∂Ω. Theorem 5 ([19]) If 1 < p < ps , there is C > 0 determined by Ω such that any classical solution u = u(x) to (26) satisfies u∞ ≤ C. A benefit of Theorem 5 is the development of the topological argument which guarantees the existence of the solution to (26), 1 < p < ps , see [11] for details. Thus we have a duality concerning the existence of the (non-trivial) solution between (16) and (26). (At this occasion we note that any non-trivial non-negative stationary solution to these equations is positive by the maximum principle.) This existence result of the solution to (26) is also derived from the variational method, while the non-existence result for p ≥ ps is obtained if Ω is star-shaped, see [49]. To prove Theorem 5, we assume the contrary, the existence of the solution sequence {uk } to (26) such that uk ∞ = uk (xk ) → +∞, 1

xk ∈ Ω.

(27)

1

Then, u ˜k (x) = μkp−1 uk (μk x + xk ), μk > 0, μkp−1 uk (xk ) = 1 satisfies

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−Δuk = upk , uk > 0

in Ωk ,

uk = 0 on ∂Ωk ,

where Ωk = μ−1 k (Ω − {xk }). It holds that μk → 0 by (27), while the elliptic regularity is applicable by uk ∞ = uk (0) = 1. Passing to a subsequence, we obtain the limit problem −Δu = up , 0 ≤ u ≤ u(0) = 1 in Rn or

−Δu = up , 0 ≤ u ≤ u(0) = 1

in Rn+ ,

u = 0 on ∂Rn+ ,

where Rn+ is a half space with the boundary ∂Rn+ containing the origin. This property is impossible by Theorems 2 and 3, and the proof of Theorem 5 is complete. A similar blowup analysis using (21) guarantees the following theorem, see [64] for the critical case p = ps . More precisely, inequality (20) derived from the variational structure reduces the problem to the elliptic theory, Theorems 2 and 3. Theorem 6 ([20]) Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω, 1 < p < ps , and u = u(·, t) be a global in time solution to (18). Then, there is a constant C > 0 determined by Ω and u0 ∞ such that supt≥0 u(t)∞ ≤ C.

The Fujita exponent pf is also detected by the scaling invariance (21). First, putting 1 μ = t− 2 , we obtain 1 1 t), y = xt− 2 , u(y, 1) = t p−1 uμ (x, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

322

Takashi Suzuki

and then use a close idea of taking (23). Thus the forward self-similar transformation arises with s = log t for (3). Translating t → t + 1, we obtain 1

v(y, s) = (t + 1) p+1 u(x, t) y = x/(t + 1)1/2, s = log(t + 1). Then this transformation implies vs − Δv −

y 1 · ∇v = v + |v|p−1 v. 2 p−1

1 Writing L = −Δ − y2 · ∇ = − K ∇ · (K∇), K = K(y) = e self-adjoint operator in 2

2



n

L (K) = {v ∈ L (R ) |

Rn

(28) |y|2 4

, we see that L is a

v(y)2K(y)dy < +∞}.

Its eigenvalues and eigenfunctions are obtained by the Fourier transformation [12]. The eigenvalues of L are λk = n+k−1 , k = 1, 2, · · ·, and the eigenspace of L at λ = λk has the 2 dimension   n+k−2 n−1 |y|2

which is actually generated by ∂ αe− 4 , |α| = k − 1. The first eigenvalue is in particular n n 1 2 and p = pf is the root of 2 = p−1 . More precisely, the trivial solution v = 0 to (28) becomes unstable if and only if 1 < p ≤ pf . We see more in this case by taking the inner

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|y|2

product between v(·, s) and the first eigenfunction of L, that is ϕ(y) = e− 4 . Then, we obtain the blowup of the non-trivial rescaled solution, see [35], and hence Theorem 1. A stationary solution to (28), denoted by f = f (y), can be a profile function to (3). 1 − 1 Thus, u(x, t) = t p−1 f (xt− 2 ) is called the forward self-similar solution. A non-trivial non-negative profile function f exists only if p > pf by Theorem 1. Actually, it takes the form f = f (|y|) if and only if 2

f (y) = o(|y|− p−1 ),

|y| → +∞.

(29)

Such a profile function exists if and only if pf < p < ps . In this case, furthermore, any non-negative non-trivial profile function satisifes (29) and is unique, see [46]. Then the associated self-similar solution approximates asymptotically the threshold solution to (3) concerning the global in time existence, see [36]. In the other case of p ≥ ps , there arises also infintely many profile functions not in the form of f = f (r), r = |y|, to (28) which, consequently, do not satisfy (29). A sharp feature of the solution set will be observed by reducing 1 y v + |v|p−1 v −Δv − · ∇v = 2 p−1 to the ODE n−1 r 1 + )fr + f + |f |p−1 f = 0, r > 0, fr (0) = 0, f (0) = α > 0. frr + ( r 2 p−1 (30) We have the following properties, where f = f (r, α) denotes the solution to (30). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Blowup Mechanism in Nonlinear Partial Differential Equations

323

2

1. The finite value L(α) = limr↑+∞ r p−1 f (r; α) exists for each α. 2. If L(α) = 0, it holds that f (r, α) = Ae−

r2 4

2

r p−1

−n

{1 + O(r −2 )},

r ↑ +∞

with some A > 0. 3. If p ≥ ps , then f (r, α) > 0, r > 0. 4. If pf < p < ps , there is a unique α = αp > 0 such that f (r, α) > 0, r > 0, and L(α) = 0. If 0 < α < αp , then f (r, α) > 0, r > 0, and it holds that L(α) > 0. If α > αp , then f (r, α), r > 0, takes zero. 5. If 1 < p ≤ pf , then f (r, α), r > 0, takes zero. The backward self-similar transformation to ut − Δu = |u|p−1 u, on the other hand, is obtained by 1

v(y, s) = (T − t) p−1 u(x, t) y = x/(T − t)1/2 , s = − log(T − t), that is

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vs − Δv +

y 1 · ∇v = − v + |v|p−1 v. 2 p−1

(31)

Taking the family {v(·, s + sk )} and the limit process sk ↑ +∞, this equation arises in the whole space Rn . Its linear part is associated with L2 (ρ), the function space composed of v = v(y) satisfying  Rn

v(y)2ρ(y)dy < +∞,

ρ(y) = e−|y|

2 /4

.

We have the variational structure comparable to (19). Actually E(v) =

 Rn

1 1 1 |∇v|2 + |v|2 − |v|p+1 dμ, 2 2 p+1

dμ = ρ(y)dy

acts as a Lyapunov function and hence the dynamics of (31) is controlled by the stationary solution. There are three constant (stationary) solutions to (31), v = 0 and 



1 p−1

1 . In the case of 1 < p ≤ ps , a Pohozaev type inequality implies v = ± p−1 that only these constants are admitted as the stationary solution to (31) contained in H 1 (ρ) = {v ∈ L2 (ρ) | ∇v ∈ L2 (ρ)n}, see [21]. n−4 , n ≥ 11 and pl = ∞, 1 ≤ n ≤ 10, arises in the context The fifth exponent pl = n−10 of radially symmetric stationary solutions to (31),

grr + (

n−1 r 1 − )gr − g + |g|p−1g = 0, r > 0, r 2 p−1

gr (0) = 0, g(0) = α > 0.

The expected properties are stated below some of which have been rigorously proven by now, see [66, 38]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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Takashi Suzuki

1. If p ≥ pl , then f (r, α), r > 0, takes zero. 2. If pc < p < pl , then f (r, α), r > 0, is positive for a finite number of α. 3. If ps < p ≤ pc , then f (r, α), r > 0, is positive for countably many α’s. 4. If 1 < p ≤ ps , then f (r, α), r > 0, takes zero. The solution (2) to (1) defines the blowup rate. Thus the rate of the blowup solution to (3) is classified into type I, 1

lim sup(T − t) p−1 u(t)∞ < +∞, t↑T

and type II,

1

lim sup(T − t) p−1 u(t)∞ = +∞. t↑T

The super-critical case p > pjl has been studied particularly in details for radially symmetric solutions, see [13]. As far as (3) or (18) is concerned, the type I blowup rate happens to be the rate of the backward self-similar solution

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u(x, t) = (T − t)

1 − p−1

v(y),

y = x/(T − t)1/2

with v = v(y) to be a non-trivial stationary solution to (31). The ODE solution (2) is regarded as a backward self-similar solution. Given tk ↑ T , the type I blowup rate assures a subsequence along which the solution converges to a self-similar solution in the scaled variable. The non-degeneracy of this convergence means that the limit is always non-trivial. The self-similar blowup, however, may not be consistent to the type I blowup rate. We have, actually, a type II blowup rate with self-similar profile for some p ≥ ps . Thus there is a case of 2

1

u(x, t) = r(t)− p−1 {A((x − x0 )r(t)− p−1 ) + o(1)}

locally uniformly in B(x0 , bR(t)),

where 0 < r(t)  R(t), A = A(y) is a non-trivial profile function, x0 is a blowup point, 1 b > 0 is a constant, and R(t) = (T − t) 2 is the standard (parabolic) backward scaling rate. The above A = A(y) can be a non-trivial stationary solution. It is obvious that the blowup rate of such a solution is type II. There may be, on the contrary, possible that a radially symmetric solution takes the type I blowup rate and is not provided with an asymptotically self-similar profile. Finally, there may be multiple blowup points with or without their collisions. The following result shows the type I blowup rate with a non-degenerate selfsimilar profile of the solution to (18) for the sub-critical nonlinearity 1 < p < ps which is obtained by a hierarchical argument. Theorem 7 ([22, 23]) If 1 < p < ps and Ω is convex, the blowup rate is type I in (18). It holds that 1

lim(T − t) p−1 u(x, t) = ± t↑T



1 p−1



1 p−1

1

locally uniformly in |x − x0 | ≤ C(T − t) 2 ,

(32) where C > 0 is arbitrary and x0 ∈ Ω is the blowup point: ∃ xk → x0 , ∃ tk ↑ T such that )| →Analysis +∞.and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. |u(xk , tkTheory, Partial Differential Equations:

The Blowup Mechanism in Nonlinear Partial Differential Equations The constant ± 1 p−1



1 p−1



1 p−1

325

in (2) also arises as a universal constant estimating (T −

t) u(t)∞ . Through the scaling (21), this property is reduced to the Liouville property, non-existence of the non-trivial full-orbit −∞ < t < +∞ to (3), see [51]. Theorem 8 ([50]) If 1 < p < ps , the equation ut − Δu = |u|p−1 u

in Rn × (−∞, +∞)

admits no positive, raidally symmetric, bounded classical solution. Here we remind the following form of the Liouville property which, combined with Theorem 7, implies a refined blowup estimate. Theorem 9 ([40]) If 1 < p < ps and u = u(x, t) is a classical solution of ut − Δu = |u|p−1 u satisfying

in Rn × (−∞, 0)

1

sup(−t) p−1 u(t)∞ < +∞, t 0.

(34)

The variational structure is the other factor and (33) is formulated by the model (B) gradient system ut = ∇ · (u∇δF (u)) derived from Helmholtz’s free energy F (u) =

 Rn

u(log u − 1) −

1 Γ ∗ u, u . 2

Then we obtain the free energy decreasing and the total mass conservation d F (u(t)) ≤ 0 dt d u(t)1 = 0, dt see [63].

Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

(35) (36)

The Blowup Mechanism in Nonlinear Partial Differential Equations

327

The scaling (34) is consistent to the mass conservation (36) if and only if n = 2. In this case, the threshold phenomenon arises in various contexts. First, the variational function has the scaling property F (uμ) = F (u) + (2λ −

λ2 ) log μ 4π

for λ = u1 and uμ (x) = μ2 u(μx). This property results in the critical mass λ = 8π and then the dual Trudinger-Moser inequality inf{F (u) | u ≥ 0, u1 = 8π} > −∞

(37)

arises, see [67, 41, 62]. Next, the stationary problem δF (u) = 0, u ≥ 0, u1 = λ, obeys the feature of mass quantization of the blowup of family of solutions [42, 56]. Here we just mention that the blowup unit is detected by the Liouville equation in the whole space which takes  ev 2 in R , ev < +∞ (38) −Δv = λ v 2 e 2 R R in the dual form, see [62, 63]. Thus the solution to (38) exists if and only if λ = 8π, see [8]. If n = 2 and λ = u0 1 < 8π, we can derive global in time existence of the solution to (33), using (35), (36), (37), and the parabolic regularity. The second moment relation

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d dt

 R2

|x|2 u(x, t)dx = 4λ −

λ2 , 2π

(39)

on the other hand, assures the blowup of the solution in the case of λ > 8π where the method of symmetrization is involved to derive (39). We remind here that the use of the second moment in the stationary state is regarded as a dual Pohozaev identity, see [63]. Thanks to the scaling property, this threshold λ = 8π concerning the total mass λ = u0 1 is localized and we obtain the following theorem. Theorem 10 ([57, 62, 63]) If n = 2, it holds that u(x, t)dx  8π

 x0 ∈S

R2

δx0 (dx)+f (x)dx

|x|2 u0 (x)dx < +∞, and T < +∞ in (33) then as t ↑ T in M(R2 ∪ {∞}) = C  (R2 ∪ {∞}),

(40) where T > 0 is the existence time of the solution, S is the blowup set, and 0 ≤ f = f (x) ∈ L1 (R2 ) ∩ C(R2 \ S).

The appearance of the delta functions in (40) is called the formation of collapse and is conjectured by [48]. Then the threshold of the total mass for the blowup of the solution was conjectured by [9]. First, n = 2 is detected by the dimension analysis which is just the scaling property compatible to the mass conservation stated above. Then the threshold mass λ = 8π is suspected by the study of stationary solutions and some numerical computations. The study of the stationary problem, called the mean field equation has been extensively promoted since [42, 61], see [7, 65], and based on this study, the above threshold value is reduced to 4π when the flux zero boundary condition is assumed on the bounded domain, Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

328

Takashi Suzuki

see [57]. Theorem 10 is regarded as an affirmative combination of the conjectures of the formation of collapse [48] and the threshold mass of the blowup of the solution [9]. The weak formulation is the starting point of the proof of Theorem 10, that is, d dt



1 ϕ(x)u(x, t)dx = 4λ− 2 4π R

  R2

(x − x ) ·(∇ϕ(x)−∇ϕ(x))u(x, t)u(x, t)dxdx , |x − x |2

where ϕ ∈ C02 (R2 ). This formulation arises as a generalization of (39) which results in the bounded variation in time of the local mass:  d dt



R2

ϕ(x)u(x, t)dx ≤ C∇ϕC 1 .

This property is combined with the standard ε-regularity, that is the existence of ε0 such that lim sup u(t)L1 (B(x0,r)) < ε0 t↑T

⇒

lim sup u(t)L∞ (B(x0,r/2) < +∞, t↑T

r > 0.

Then we obtain the finiteness of the blowup points and also the formation of collapse, u(x, t)dx 

 x0 ∈S

in M(R2 ∪ {∞}) = C(R2 ∪ {∞})

m(x0 )δx0 (dx) + f (x)dx 1

(41)

2

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with 0 ≤ f = f (x) ∈ L (R ) and m(x0 ) ≥ 8π, where S is the blowup set [57]. To derive m(x0 ) ≤ 8π, we use the local second moment using ϕ = ϕx0 ,R (x) supported near x0 ∈ S with the support radius R > 0. Under the assumption m(x0 ) > 8π, we have d dt

 R2

|x − x0 |2 ϕx0 ,R (x)u(x, t)dx ≤ −a

(42)

with a > 0 independent of 0 < T − t  1 and 0 < R  1 again by the method of symmetrization. Then we apply the backward self-similar variables z(y, s) = (T − t)u(x, t) y = (x − x0 )/(T − t)1/2 ,

s = − log(T − t).

Given sk ↑ +∞, we obtain {sk } ⊂ {sk } such that z(y, s + sk )dy  ζ(dy, s)

in C∗ (−∞, +∞; M(R2)).

We define tk ↑ T and t˜k < tk by sk = − log(T − tk ) and T − ˜tk = 2(T − tk ), respectively. We integrate (42) in (t˜k , tk ) and put R = bR(tk ) for b > 0 and R(t) = (T − t)1/2 . It follows that



|y|2 ϕ0,b(y), ζ(dy, s) ≤ −a + 2 |y|2 ϕ0,b(y), ζ(dy, s − log 2)



(43)

with k → ∞, where −∞ < s < +∞, see [63]. Since a hierarchical argument guarantees that the singular part of ζ(dy, s − log 2) is composed of a finite sum of delta functions, see [58], inequality (43) is a contradiction for 0 < b  1. Thus we obtain (41) with Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Blowup Mechanism in Nonlinear Partial Differential Equations

329

m(x0 ) = 8π. This fact may be regarded as the quantization of the collapse mass without their collisions. The method of symmetrization guarantees also d dt which implies

 R2

|x − x0 |2 ϕx0 ,R (x)u(x, t)dx ≥ −C



I(s) ≡ |y|2 , ζ(dy, s) ≤ C,

(44)

−∞ < s < +∞

(45)

by an integration of (44) on (t, T ) and the limiting process. More precisely, we put R = bR(t), sk + s = − log(T − t) and make k → ∞, b ↑ +∞. The rescaled limit ζ(dy, s) in (45), on the other hand, is a weak solution to zs − Δz = −∇ · z∇(Γ ∗ z +

|y|2 ) 4

in R2 × (−∞, +∞).

Having proven m(x0 ) = 8π, we have dI =I ds and hence I = 0 by (45). Then we can show ζ(dy, s) = 8πδ0 (dy), that is the formation of sub-collapse in M(R2 ) z(y, s)dy  8πδ0 (dy), Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

which guarantees the type II blowup rate for any blowup point x0 : ∀

lim u(t)L∞ (B(x0,b(T −t)1/2)) = +∞, t↑T

b > 0,

see [55, 47]. In this context, a solution constructed by [30] is to be noted satisfying u(x, t) ∼ r(t)−2 u(x/r(t)), |x| < bR(t) with some 0 < r(t)  R(t) = (T − t)1/2 , where u(y) = and b > 0. This profile results in lim Fx0 ,br(t)(u(t)) = +∞, t↑T

8 (1+|y|2 )2

∀

is the stationary solution

b>0

(46)

where 

1 u(log u − 1) − Fx0 ,R (u) = 2 B(x0,R)

  B(x0,R)×B(x0,R)

Γ(x − x )u ⊗ u

denotes the local free energy. Inequality (46) means the free energy transmission at the occasion of the blouwp of the solution. Thus we can summarize that ”mass and entropy are exchanged at the wedge of blowup envelope to create a clean quantized self”, see [62]. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

330

5.

Takashi Suzuki

Smoluchowski-Poisson Equation in Higher Dimension

In the case n ≥ 3, however, the scaling and the mass conservation are not consistent. There is an interesting solution provided with the blowup bulk traveling toward the origin. This blowup pattern has the type II blowup rate, slow concentration, and is stable. Other type II blowup patterns are also known, see [29, 28]. Unfortunately these results are restricted to the formal analysis but still efforts are made for rigorous arguments, see [3, 24, 54, 45]. A stable blowup pattern obtained by [28] to ut = ∇ · (∇u − u∇v), −Δv = u

in Rn × (0, T ), n ≥ 3

(47)

is not self-similar. This solution is radially symmetric, u = u(r, t), r = |x|, and hence ut = urr +

n−1 1 n−1 ur − n−1 (r n−1 uvr )r , vrr + vr = −u r r r

(48)

by (47). It is assumed that a bulk at r = R(t) with the width δ(t) and the hight h(t) is traveling toward the origin as t ↑ T satisfying δ(t)  R(t) ↓ 0, h(t) ↑ +∞. Then it follows that ωn−1 · R(t)n−1 · δ(t) · h(t) ∼ λ = u(t)1 (49) from the total mass conservation, where ωn−1 denotes the (n − 1)-dimensional volume of the unit shpere in Rn . The dimension analysis guarantees ur ∼ 1 · h(t)  |urr | ∼ h(t) r R(t) δ(t) δ(t)2

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

and therefore,

vr  |vrr | r

by the ansatz δ(t)  R(t). Thus (48) is reduced to ut = urr − (uvr )r , vrr = −u. Equality (50) implies

(50)

1 −vrrt = urr + (vr2 )rr , 2

and then we have

1 [vt + u + vr2 ]r = constant = 0 2 by putting r = 0. Hence it follows that 1 (vr )t + (u + vr2 )r = 0. 2

(51)

˙ Since r = R(t) is the wave front of u, its propagation speed is given by c = R(t). Then the Rankine-Hugoniot condition to (51) is described by 1 1 c[vr ]R(t) = [u + vr2 ]R(t) = [ vr2 ]R(t), Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and2 Applications, Nova Science 2 Publishers, Incorporated, 2011.

(52)

The Blowup Mechanism in Nonlinear Partial Differential Equations where

331

[ζ]R(t) = lim ζ(r) − lim ζ(r) = ζ(R(t)+ ) − ζ(R(t)− ). r↓R(t)

r↑R(t)

Regarding u∼

λ χr=R , vrr = −u, ωn−1 Rn−1

we notice vr (R(t)+ , t) = − and then (52) reads

λ , vr (R(t)− , t) = 0 ωn−1 R(t)n−1

˙ r (R(t)+ , t) = 1 vr (R(t)+ , t)2 . Rv 2

Thus we obtain R˙ = −

λ , 2ωn−1 R(t)n−1

and, hence R(t) ∼ c0 (T − t)1/n , c0 = Next, we take the replacement u ∼ h, follows that δv2 ∼ h, v ∼ δ 2 h, and hence

∂ ∂r

nλ . 2ωn−1

(53)

∼ δ −1 in (50). From the second equation, it

(uvr )r ∼ h2 .

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Then the first equation implies h h ∼ 2 − h2 . t δ Since t ∼ T , we have

1 1 − ∼ δ −2 . δ2 t Putting these relations to (49), we obtain h∼

1

(T − t)1− n · δ · δ −2 ∼ 1 and hence

1

2

δ ∼ (T − t)1− n , h ∼ (T − t)−2+ n .

The type II blowup rate, slow concentration, and non-self-similarity are thus observed in this blowup pattern. A self-similar blowup pattern is detected by using the volume function m(r, t) = r n−1 r u(r, t)dr in (47) which results in 0 mt = mrr −

n−1 1 mr + n−1 mmr . r r

Since (54) is scaling invariant under m (r, t) = μ−n+2 m(μr, μ2 t), μ > 0,

μ : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011. Partial Differential Equations: Theory, Analysis and Applications

(54)

332

Takashi Suzuki

we take the backward self-similar transformation m(r, t) = (T − t)

n−2 2

ϕ(y, s),

y = r/(T − t)1/2 , s = − log(T − t)

and then obtain ϕs = ϕyy − (

1 n−1 y n−2 + )ϕy + ϕ + n−1 ϕϕy . y 2 2 y

Writing ϕ = y n−2 g, we have gs = gyy + (

n−3 y g 1 − )gy − 2(n − 2) 2 + 2 ((n − 2)g + ygy )g. y 2 y y

(55)

We wish to construct a stationary to (55) satisfying g(y) ∼ y 2 , y ↓ 0. First, we take g(y) = V (η), η = ε−1/2 y, v  +

n − 3  2(n − 2) 1 1 v − v + 2 (ηvv  + (n − 2)v 2 ) − εηv  = 0. 2 η η η 2

Then, we assume ε = 0 and put v(η) = V (s), s = log η: Vss + (n − 4)Vs − 2(n − 2)V + V Vs + (n − 2)V 2 = 0.

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We write this autonomous equation as d dt



V H





=

H −(n − 4)H + 2(n − 2)V − (V H + (n − 2)V 2 )



(56)

and pick up the equilibria (V, H) = (0, 0) and (V, H) = (2, 0). The linearized equation around (V, H) = (0, 0) is d dt



V H





=

0 1 2(n − 2) −(n − 4)





V H

so that (0, 0) is a saddle because the linearized eigenvalues are 2, −n + 2. The linearized equation around (V, H) = (2, 0) is d dt



v H





=

0 1 −2(n − 2) −(n − 2)



v H



√

−(n−2)±

(n−2)(n−10)

using V = v + 2. Then we see that the linearized eigenvalues are . 2 The hetero-clinic orbit of (56) is constructed in this way and then creates an unstable selfsimilar blowup pattern to (47). Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

The Blowup Mechanism in Nonlinear Partial Differential Equations

333

References [1] A. Bahri, Critical Points at Infinity in Some Variational Problems, Longman, New York, 1980. [2] P. Biler and R. Sta´nczy, Mean field models for self-gravitating particles, Folia Matematica 13 (2006), 3-19. [3] M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel, and S. C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity 12 (1999) 1071-1098. [4] L.A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 271-297. [5] S. Chandrasekhar, An Introduction to the Theory of Stellar Structures, Dover, New York, 1957. [6] P.-H. Chavanis, Generalized kinetic equations and effective thermodynamics, Banach Center Publications 66 (2004) 79-101. [7] S.-Y.A. Chang, C.-C. Chen, and C.-S. Lin, Extremal functions for a mean field equations in two dimension, In; Lectures on Partial Differential Equations (ed. S.-Y.A. Chang, C.-S. Lin, and S.-T. Yau), International Press, New York, 2003.

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[8] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-622. [9] S. Childress and J.K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981) 217-237. [10] D.N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992) 425-434. [11] D.G. De Figueiredo, P.L. Lions, and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pure Appl. 61 (1982) 41-63. [12] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions for the heat equation, Nonlinear Anal. 33 (1987) 51-69. [13] M. Fila, Blow-up of solutions of supercritical parabolic equations, In; Handbook of Differential Equations, Evolutionary Equations (ed. C.M. Dafermos and E. Feireisl), vol. 2, pp. 105-158, Elsevier, Amsterdam. 2005. [14] M. Fila and P. Pol´acˇ ik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations 4 (1999) 163-196.

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[24] I.A. Guerra and M.A. Peletier, Self-similar blowu-up for a diffusion-attraction problem, Nonlinearity 17 (2004) 2137-2162. [25] C. Gui, W.-M. Ni, and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rn , Comm. Pure Appl. math. 45 (1992) 11531181. [26] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equation, Proc. Japan Acad. Ser. A 49 (1973) 503-505. [27] D. Henry, Geometry Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981. [28] M.A. Herrero, E. Medina, and J.J.L. Vel´azquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity 10 (1997) 1739-1754. [29] M.A. Herrero, E. Medina, and J.J.L. Vel´azquez, Self-similar blow-up for a reactiondiffusion system, J. Comp. Anal. Math. 97 (1998) 99-119. [30] M.A. Herrero and J.J.L. Vel´azquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583-623. [31] T. Hirono, T. Suzuki, and R. Takahashi, in preparation. [32] H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac. 34 (1991) 475-494. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[33] R. Ikehata and T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: local and asymptotic behavior of solutions, Differential and Integral Equations 13 (2000) 869-901. [34] D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by poisitive sources, Arch. Rational Mech. Anal. 49 (1973) 241-269. [35] O. Kavian, Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. Henri poincar´e, Analyse nonlin´eaire 4 (1987) 423-452. [36] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. Henri Poincar´e, Analyse non lin´eaire 13 (1996) 1-15. [37] K. Kobayashi, T. Sirao, and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977) 407-424. [38] L.A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, Differential Equations 24 (1989) 799-805. [39] Y. Li, Asymptotic behavior of positive solutions of equation Δu + K(x)up = 0 in Rn , J. Differential Equations 95 (1992) 304-330. [40] F. Merle and H. Zaag, Optimal esitmates for blowup rate and behavior nonlinear heat equation, Comm. Pure Appl Math. 51 (1998) 139-196.

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[41] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971) 1077-1092. [42] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities, Asymptotic Analysis 3 (1990) 173-188. [43] K. Nagasaki and T. Suzuki, Radial solutions for Δu + λeu = 0 on annului in higher dimensions, J. Differential Equations 100 (1992) 137-161. [44] K.Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation −Δu = λeu on circular domains, Math. Anal. 299 (1994) 1-15. [45] Y. Naito and T. Senba, Blow-up behavior of solutions to a chemotaxis system on higher dimensional domains, preprint. [46] Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations, J. Differential Equations 163 (2000) 407-428. [47] Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloquium Mathematics 111 (2008) 11-34.

[48] V. Nanjundiah, Chemotaxis, signal relaying, and aggregation morphology, J. Theor. Biol. 42 (1973) 63-105. Partial Differential Equations: Theory, Analysis and Applications : Theory, Analysis and Applications, Nova Science Publishers, Incorporated, 2011.

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[49] S.I. Pohozaev, Eigenfunctions of Δu + λf (u) = 0, Soviet Math. Dokl. 6 (1965) 1408-1411. [50] P. Pol´acˇ ik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Anal. 64 (2006) 1679-1689. [51] P. Quittner and P. Souplet, Superlinear Parabolic Equations, Blowup, Global Existence and Steady States, Birkh¨auser, Basel, 2007. [52] A.A. Samarski, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. [53] A.A. Samarski and A.P. Mikhailov, Principles of Mathematical Modeling, Taylor and Francis, London, 2002. [54] T. Senba, Blowup behavior of radial solutions to J¨ager-Luckhaus system in high dimensional domains, Funkcial Ekvac. 48 (2005) 247-271. [55] T. Senba, Type II blowup of solutions to a simplified Keller-Segel system in two dimensions, Nonlinear Analysis 66 (2007) 1817-1839. [56] T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl. 10 (2000) 191-224. [57] T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations 6 (2001) 21-50.

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[58] T. Senba and T. Suzuki, Blowup behavior of solutions to re-scaled L¨ager-Luckhaus system, Adv. Differential Equations 8 (2003) 787-820. [59] T. Senba and T. Suzuki, Applied Analysis, Imperial College Press, London, 2004. [60] M. Struwe, Variational Methods, Application to Partial Differential Equations and Hamiltonian Systems, third ed., Springer-Verlag, Berlin, 2000. [61] T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincar´e, Analyse nonlin´eaire, 9 (1992) 367-398. [62] T. Suzuki, Free Energy and Self-Interacting Particles, Birkh¨auser, Boston, 2005. [63] T. Suzuki, Mean Field Theories and Dual Variation, Atlantis Press, Amsterdam, 2008. [64] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J. 57 (2008) 3365-3396. [65] T. Suzuki and F. Takahashi, Nonlinear eigenvalue problem with quantization, Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 5 (ed. M. Chipot), pp. 277-370, Elsevier, Amsterdam, 2008.

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INDEX A

C

acquisitions, 274 activation energy, 105 actuation, 175, 191 actuators, 174 adaptation, 295 adjustment, 135, 209, 214 algorithm, 2, 19, 20, 36, 125, 196, 202, 206, 209, 213, 282, 285, 289, 290, 292, 295, 304, 310 alkalinity, 112, 134 amplitude, 28, 36, 144, 271, 273, 274, 278, 280, 286, 287, 289, 299, 301, 303, 304, 305, 308 anaerobic digestion, viii, 97, 100, 111, 116, 117, 125, 134, 135 applied mathematics, 224 atmosphere, x, 203, 265 atoms, 85, 90, 93, 271 avoidance, 58

calculus, viii, ix, 151, 152, 161, 171 candidates, 202, 268 case study, 211 Cauchy problem, 69, 70, 286, 287, 288, 296, 299, 300, 305, 307 causality, 3, 29 challenges, 198 chaos, 203, 265 Chaotic velocity fluctuations, x, 265 Chebyshev series, vii, 1, 8, 12, 13, 47, 171 chemical, viii, ix, 71, 72, 82, 83, 89, 90, 104, 134, 135, 173 chemical vapour deposition, viii, 71, 72, 82, 89 China, 141, 148 classes, ix, 223, 231, 233, 236, 237 classification, 236 closed string, 253, 254 clusters, 310 communication, 203 community, 201 compatibility, 226, 241, 244, 246 compensation, 180 competition, 25, 40 compilation, 11 complex numbers, 202 complexity, 89, 290 complications, 40, 284 composition, 82, 203 compressibility, 43 compression, 157, 158 computation, viii, ix, 9, 14, 18, 21, 22, 24, 27, 30, 36, 111, 141, 144, 151, 153, 157, 158, 170, 288

B Banach spaces, 56 base, 158, 268 benchmarking, 11, 25 bending, 224 bioinformatics, 277 biological processes, ix, 173 biomass, 111, 112, 114, 115, 116, 118, 124, 125, 134 bonds, viii, 71, 72, 85, 89 bounded linear operators, 175, 186 bounds, 211, 280 breakdown, 72 Brownian motion, 273

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340

Index

computer, viii, ix, 3, 11, 24, 31, 36, 46, 47, 151, 153, 154, 157, 170, 292 computing, 3, 10, 14, 18, 36, 170 conduction, x, 265, 267 conductivity, 267 configuration, 38, 176 confinement, 38 conflict, 12 congress, 199 conjugation, 228, 278 conservation, x, 53, 74, 75, 231, 242, 243, 250, 251, 260, 267, 278, 313 construction, 158, 242, 279, 287, 297 Continuous Wavelet, vi, x, 277, 286, 296 contour, 228, 251, 252, 254 contradiction, 315 controversies, 89 convergence, ix, 19, 20, 21, 22, 26, 27, 60, 63, 201, 218 cooling, 131, 132 correlation, 204, 273, 274 correlation function, 273 corruption, 289 cost, 54, 55, 56, 290 covering, 22 CRM, 263 crystallites, 89 crystallization, 92 cycles, 309 Czech Republic, 171

D damping, 152 data processing, 290 decay, 112, 315 decomposition, 2, 20, 99, 158, 229, 292 defect formation, viii, 71 defects, 85, 289, 295 deformation, 224 degenerate, 319 dependent variable, 39, 109, 140, 242, 243, 253 deposition, 82, 83, 89, 90, 91 depth, 77, 87, 88 derivatives, viii, ix, 8, 11, 12, 21, 67, 98, 109, 110, 124, 129, 146, 151, 152, 161, 162, 177, 182, 184, 226, 228, 230, 239, 241, 244, 247, 255, 283, 284, 286, 301, 303, 310 detection, viii, 71, 304, 306 deviation, 133, 268, 269, 270, 295

differential equations, vii, viii, ix, 3, 7, 18, 20, 21, 47, 111, 113, 151, 152, 153, 157, 170, 202, 208, 218, 220, 221, 223, 224, 225, 234, 235, 285 differential neural networks (DNN), ix, 201 diffusion, vii, viii, x, 1, 14, 15, 22, 25, 29, 71, 72, 73, 74, 77, 78, 79, 82, 83, 84, 85, 88, 90, 104, 152, 171, 205, 212, 265, 266, 270, 271, 272, 273, 274, 275, 277, 278, 282, 284, 285, 286, 287, 288, 296, 299, 304, 305, 309, 310, 311 diffusion process, 73, 90, 152, 282 diffusivity, viii, 71, 74, 85, 271 digestion, 115 dilation, 158, 234 Dirac equation, 254 Dirac-like partial differential equations, ix discontinuity, 104, 124, 134 discrete random variable, 59 discretization, 22, 120, 174, 180, 289, 290, 310 dispersion, 104, 142, 148, 205 displacement, 72, 306 distribution, x, 55, 242, 265, 266, 271, 275 doping, viii, 71, 82 duality, 321 dynamic viscosity, 266 dynamical systems, 2, 100, 203, 217, 310

E economics, 69 effusion, viii, 71 electric field, 40 electrical conductivity, 74 electricity, 73 electrochemistry, 152 electrodes, 82 electromagnetic, viii, ix, 151 electromagnetic fields, viii, ix, 151 electron, viii, 71, 89, 271 electron diffraction, 89 elementary particle, 253 employment, 54 energy, 38, 72, 118, 158, 178, 204, 224, 267, 278, 280, 299, 305, 319 engineering, viii, ix, 151, 199, 277 entropy, 266 environment, 11, 47, 78, 187, 191, 224 equality, 56, 62, 63, 67, 218, 278 equilibrium, 39, 40, 46, 224, 266, 271, 272 estimation problems, 99, 134 etching, 78, 83, 84, 85, 88, 89, 90, 91

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Index Euclidean space, ix, 223, 224, 225, 228, 229, 250, 252, 254, 257 evolution, viii, 2, 35, 36, 37, 38, 39, 46, 69, 84, 88, 89, 97, 123, 125, 129, 134, 142, 149, 187, 204, 224, 270, 271, 284, 309, 313, 314, 316 execution, 11 exposure, 84, 86, 93, 94 extraction, 89

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F families, x, 109, 277, 282, 300, 303, 310, 311 fatty acids, 134 FEM, 2, 5 FFT, 3, 290, 292, 295, 304 field theory, 224 filament, viii, 71, 93 film thickness, 83, 84, 92 films, 82 filters, 289 filtration, 289 finite element method, 2, 152, 221 First-Order Partial Differential Equations, viii, 97, 134, 135 flame, 3, 23 flame propagation, 3, 23 fluctuations, x, 2, 54, 180, 187, 196, 265, 272, 273, 274, 275, 309 fluid, vii, ix, 1, 2, 19, 25, 29, 33, 38, 40, 48, 103, 104, 106, 118, 120, 130, 131, 136, 173, 178, 179, 180, 183, 185, 187, 265, 266 Fokker-Planck equation, vii, 51, 52, 57, 58, 60, 63, 64 force, 38 formation, 83, 85, 89, 90, 91, 93 formula, 12, 66, 204, 229, 230, 237, 288, 289, 300 foundations, vii, 51 fractal structure, x, 265 fractional PDE, viii, ix, 151 France, 71, 173 free energy, x, 224, 313 freedom, 19, 248 friction, 267 fusion, 23, 26, 32, 38, 47, 48

G gauge theory, 258 Gauss and Morlet families, x, 277

341

generalized fully spectral weighted residual method (GWRM), vii, 1 genus, 225 geometry, 38, 40, 203, 204, 224, 225, 257, 267, 268, 272 Germany, 277 Ginzburg-Landau equations, ix global semi-analytical solutions, vii, 1 glow discharge, 82 graph, 155, 242, 277, 295, 301 gravitation, 253 gravity, 224, 252, 253, 266 grids, 311 grouping, 287 growth, viii, 23, 39, 44, 46, 69, 71, 78, 79, 89, 90, 91, 93, 293 growth rate, 44, 46

H Hamiltonian, ix, 223, 231, 256 Hamiltonian system, ix, 231 heat capacity, 105, 178 heat loss, 104, 284 heat transfer, 73, 74, 105, 178, 180, 183, 187, 190, 192, 267, 268 height, 267 Hilbert space, 101, 126, 205, 317 human, vii, viii, ix, 69, 151 human body, viii, ix, 151 Hungary, 199 hybrid, 128, 130 hydrogen, viii, 71, 72, 74, 77, 78, 82, 83, 84, 85, 88, 89, 90, 91, 92, 93, 94, 134 hydrogen atoms, 90 hydrogenated amorphous silicon, viii, 71, 82, 83 hydrogenation, 93 hydrolysis, 111, 112 hyperbolic systems, viii, 97, 99

I IAM, 220 ideal, vii, 1, 22, 38, 43, 44, 47, 105, 202, 303, 310 identification, ix, 153, 171, 201, 202, 203, 211, 213, 218 identification problem, ix, 201 identity, 18, 101, 103, 119, 156, 157, 229, 237 ideology, 278

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342

Index

image, x, 89, 275, 277, 278, 281, 282, 284, 285, 286, 299, 310, 311 images, x, 282, 284, 285, 288, 290, 296 immersion, 223, 229, 252, 254, 257 independent variable, 39, 146, 233, 234, 236, 241, 247, 248, 284, 286, 290 induction, 238, 273 inequality, 62, 64, 65, 132, 210, 219, 315, 316, 319, 321 infrared spectroscopy, viii, 71 ingredients, 224 inhibition, 112 inhibitor, 112 initial state, 9, 19 integration, 2, 4, 7, 8, 11, 17, 40, 48, 93, 98, 135, 142, 153, 154, 155, 156, 157, 160, 161, 162, 163, 164, 171, 205, 225, 252, 254, 260, 291, 292, 311 interface, 90, 93 invariants, 224, 233, 234 inversion, 20, 22, 30, 36, 47, 171, 202, 233 involution, 232, 247, 248, 250 iteration, 8, 11, 15, 19, 20, 21, 22, 34, 41, 44, 116 iterative solution, vii, 1, 8, 47

J

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Japan, 51, 68, 313

K KdV equations, viii, 141, 149 kinetics, 83, 90, 135 kinks, 148 Klein-Gordon equation, 203 Korea, 199

L Lagrange multipliers, 224 laminar, 267 languages, 11 laws, ix, x, 3, 38, 73, 99, 152, 175, 195, 201, 203, 231, 243, 250, 251, 260, 313 lead, 18, 19, 20, 74, 90, 282, 288, 304, 311 learning, 202, 212, 215 Lie algebra, 242, 244 light, 204, 282 linear dependence, 22, 270 linear function, 112

linear systems, 195, 306 liquid crystals, 204 liquid phase, 104 localization, 279, 284, 296, 299, 309 Lorenz system, x, 265, 269 Lyapunov function, 98, 131, 203, 210, 218, 316 Lyapunov matrix equation solving, viii, ix, 151 lying, 272

M machinery, 247 magnetic field, x, 38, 40, 48, 266, 271, 272, 273, 274 magnetic fusion, 26 magnetic moment, 271, 272 magnetic resonance, x, 265, 266, 270, 271 magnetic resonance images, x magnetic resonance imaging, 265, 271 magnetization, 271, 272, 273 magnetohydrodynamic (MHD), vii, 1, 22 magnitude, 140, 212, 213, 215, 216, 218, 272 mapping, 228, 229, 317 masking, 282 mass, viii, 38, 39, 71, 88, 104, 118, 120, 266 master equation, vii, 51, 52, 60, 61, 62, 63, 64, 65, 69, 70 materials, viii, 71 mathematics, 11, 17, 151, 201, 202, 224, 265 matrix, viii, ix, 2, 4, 19, 20, 22, 30, 36, 47, 101, 103, 108, 109, 111, 118, 119, 124, 140, 151, 153, 154, 155, 156, 157, 158, 160, 161, 162, 165, 166, 168, 171, 172, 196, 208, 209, 218, 225, 249, 250, 259, 261, 262, 291, 292 matter, 75, 108, 153, 288, 306 Maxwell equations, 38 measurement, viii, 97, 271, 273, 274 measurements, viii, x, 71, 83, 88, 99, 123, 124, 134, 192, 195, 196, 197, 265, 271 medical, 277 membranes, 224 memory, 3, 40, 291 Method of Characteristics (MC), viii, 97, 98, 99 methodology, ix, 3, 99, 100, 106, 116, 119, 120, 129, 176, 201, 202, 203, 206, 218 microbial community, 134 microcrystalline, viii, 71, 83, 89, 90, 91 migration, vii, 51, 52, 68, 69 Ministry of Education, 148 mixing, 104 model system, 232, 233, 238, 262

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Index modelling, 26, 32, 38, 152, 203 models, vii, viii, ix, 2, 25, 51, 52, 69, 70, 71, 89, 98, 99, 107, 119, 152, 173, 201, 202, 205, 216, 244, 319 modifications, x, 83, 84, 90 modulus, 243, 282, 293, 295, 296, 298, 299, 301, 306, 307, 308 molecules, 224 momentum, 257, 271 Monte Carlo method, 202 morphology, 285 MRI, 271 multidimensional, 2, 310 multiplication, 40 multiplier, 258

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N nanometer, 82, 84 natural science, 148, 277 neglect, 54, 91, 274, 295 Netherlands, 68, 198 neural network, ix, 141, 201, 202, 206, 209, 220 nodes, 36, 40, 106, 111, 152, 290, 291, 292, 310 nonlinear optics, 203 nonlinear systems, 209, 210, 218 no-slip boundary condition, x Nuclear Magnetic Resonance (NMR), v, 265, 267, 269, 271, 273, 274, 275, 276 nucleus, 271 null, 301

O one dimension, 18, 187 operations, 2, 20, 21, 22, 30, 36, 158 optical fiber, 203 optical properties, 82 optimization, 153 orbit, 316, 317, 318, 319 ordinary differential equations, 3, 22, 120, 122, 174, 203, 291 orthogonal functions, viii, ix, 151, 153, 154, 157, 161, 170 orthogonality, 5, 281 oscillation, 293, 295, 305, 309 overlap, 22 oxidation, 82 oxygen, 23

343

P parallel, 21, 196, 199, 292, 293, 310 partial differential equations (PDEs), vii, viii, ix, x, 1, 2, 3, 4, 8, 19, 22, 23, 26, 34, 39, 46, 47, 48, 63, 151, 152, 153, 170, 171, 173, 174, 199, 202, 203, 218, 220, 221, 223, 224, 225, 233, 241, 278, 285, 286, 310, 311 periodicity, 256, 295 permeability, 38 phase shifts, 273 phase transitions, 204 Philadelphia, 48, 220 physical laws, 47 physical phenomena, 152 physical properties, 224 physics, 2, 20, 25, 26, 33, 38, 66, 69, 73, 141, 148, 201, 202, 203, 204, 217, 223, 224, 265, 277, 311 plants, ix, 201, 203 platelets, 72 platform, 30 Poisson equation, x, 152, 313 polar, 233, 247, 249, 308 polymer, 111, 112 population, vii, 54, 58, 69 population density, 54, 58 portraits, 307 preparation, 43, 225 probability, 52, 55, 59, 60, 63, 64, 311 process control, 198 producers, 292 programming, 290 project, 2 propagation, vii, 152, 202, 203, 206, 224 propagators, 257, 260 proposition, 59, 61, 65, 123, 209, 210, 284 prototype, 202

Q QED, 237 quantization, x, 223, 253, 257, 258, 319 quantum devices, 141 quantum field theory, 224 quantum mechanics, 203, 204, 211 quantum state, 202, 203, 204 quantum theory, 252

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344

Index

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R radio, 82 radius, 23, 39, 84, 239, 308 ramp, 23, 24 reactions, 72, 111 reasoning, 131 recall, 46 recalling, 62, 66 rejection, 175, 181, 187, 188, 193, 196, 197 relativity, 204 relaxation, 271 reliability, 29 reproduction, 9 requirements, 281 researchers, 277 residual error, 9 residuals, 174 residues, 243 resolution, 12, 25, 30, 31, 33, 36, 153, 209, 304, 310 resources, 3 response, 99, 134 restrictions, 311 room temperature, 266 root, vii, 1, 9, 18, 19, 20, 47, 305 roots, 19, 109, 113, 229, 236 roughness, 83, 90 routines, 3, 25 rules, 81

S saturation, 89, 91, 112 scaling, x, 2, 52, 158, 236, 283, 287, 295, 313, 319, 320, 321 scattering, viii, 71, 148, 203 science, viii, ix, 151, 152 scope, ix, 151, 153 Secondary-ion mass spectroscopy (SIMS), viii, 71 self-organization, 69 semigroup, 194, 195 sensitivity, 135 sensors, 123 shape, 224, 225, 279, 281, 308 shock, 22, 26, 31, 224 shock waves, 224 showing, 28, 32, 47, 72, 218, 234, 315 signals, x, 196, 273, 277, 281, 283, 304, 306, 308, 309, 311

signs, 5, 15 silane, 72, 82 silicon, viii, 71, 72, 82, 83, 89 simulation, viii, 97, 98, 106, 107, 114, 116, 123, 132, 134, 175, 180, 187, 192, 196, 212, 213, 214 simulations, viii, ix, 100, 106, 125, 134, 151, 194, 212 Singapore, 262, 263, 264 Smoluchowski-Poisson equation, x smoothing, x, 277, 278, 282, 283, 284, 286, 287, 288, 309 sodium, 111, 112 software, viii, 141, 269, 292 solid phase, 89 solid waste, 111, 112, 114, 115, 116, 117, 124, 125, 134, 135 solitons, 148 solubility, 90, 93 solution space, 242 species, 98, 104 specific heat, 38, 266, 267 specifications, 206 spectral coefficients, vii, 1, 47 spectroscopy, viii, 71, 88, 276 spin, viii, x, 71, 253, 261, 262, 265, 266, 271, 272, 273, 274 spin dynamics, 265 stability, 4, 12, 19, 22, 23, 33, 38, 40, 44, 46, 47, 48, 99, 140, 174, 175, 177, 184, 192, 194, 195, 196, 203, 210, 313 standard deviation, 196 state, viii, ix, 3, 19, 21, 23, 52, 97, 98, 99, 100, 101, 106, 111, 114, 115, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 134, 140, 171, 174, 175, 177, 180, 182, 184, 187, 192, 194, 195, 196, 198, 201, 202, 204, 205, 211, 212, 214, 224 states, ix, 72, 119, 128, 140, 201, 203, 226, 253, 266 storage, 3, 292 stress, 224 string theory, x, 223, 224, 253 strong interaction, 252 structure, 22, 26, 54, 71, 89, 111, 112, 114, 119, 120, 122, 123, 125, 135, 137, 153, 202, 206, 208, 209, 211, 218, 233, 242, 244, 285, 290, 313, 316, 318, 319, 321 subdomains, vii, 1, 4, 19, 20, 21, 22, 24, 27, 28, 31, 38, 47, 48 subgroups, 233, 234, 240 substitution, 292 substrate, 82, 83, 84, 85, 89, 115, 116, 118 superconductivity, 202, 203, 204

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Index surface area, 112, 178 surface reactions, 89 surface region, 85 Sweden, 1, 48 symbolic computation software, viii, 141 symmetry, ix, 223, 233, 234, 237, 240, 241, 242, 244, 268 synchronization, 304, 306, 308, 309, 310 synthesis, 99, 198

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T target, 9 techniques, viii, 3, 71, 90, 97, 99, 195, 202, 264 temperature, x, 26, 74, 82, 84, 85, 88, 89, 90, 99, 103, 105, 123, 130, 132, 133, 134, 174, 175, 176, 178, 179, 180, 185, 187, 189, 191, 196, 265, 266, 267, 268, 269, 270, 271, 272, 275, 284 tension, 258 tertiary sector, 54 test procedure, 40 thermal expansion, 266 thermodynamics, 313 threedimensional Euclidean space, ix time resolution, 309 time step limitations, vii, 1, 47 time use, 31 time variables, 278 tokamak, 26, 48 topology, 310, 315, 317 torus, 53, 56 tracks, 180, 187 trade, 9 trade-off, 9 training, 214 trajectory, 106, 124, 134, 187, 210, 212, 221 transformation, 3, 6, 25, 43, 89, 90, 111, 120, 141, 142, 148, 204, 232, 233, 235, 236, 243, 244, 246 transformations, 148, 233, 234, 241, 313 translation, 158, 319 transmission, 89, 199 transmission electron microscopy (TEM), 89 transport, viii, ix, 71, 72, 79, 83, 89, 90, 134, 135, 173 treatment, 83, 87, 88, 89, 92, 135, 151, 221 trial, 2 turbulence, 203 Turkey, 199

345

U unemployment rate, 54 uniform, 85, 178, 210, 271, 272, 274, 275 united, 221 United Kingdom, 221 updating, 99 USA, 223

V vacancies, 85 vacuum, 38 vapor, 178 variables, vii, viii, ix, 4, 5, 6, 12, 14, 16, 40, 46, 47, 53, 54, 97, 98, 99, 100, 101, 106, 111, 114, 115, 118, 119, 120, 121, 122, 124, 163, 165, 173, 174, 175, 185, 187, 194, 195, 198, 231, 234, 239, 242, 243, 244, 247, 253, 254, 260, 271, 273 variations, 114, 115, 180, 187, 196, 274, 282 vector, 4, 18, 73, 98, 101, 102, 109, 118, 119, 124, 126, 127, 140, 160, 161, 175, 192, 198, 206, 225, 226, 228, 229, 234, 241, 242, 244, 247, 248, 249, 250, 251, 267, 308, 310 velocity, x, 29, 38, 39, 40, 99, 103, 104, 129, 144, 152, 178, 180, 183, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 306 viscosity, 25, 26, 27, 267

W wall temperature, 105 waste, viii, 97, 100, 111, 115, 134 wastewater, 135 water, 266, 270 wave propagation, 152 wavelet, vii, x, 158, 159, 160, 161, 164, 171, 220, 277, 278, 279, 280, 281, 282, 284, 285, 286, 287, 288, 289, 290, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 306, 307, 308, 309, 310, 311, 312 wavelet analysis, 286

Y yield, 7, 22, 112, 118, 128, 164, 169

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