Partial Differential Equations

Transcription

1 Partial Differential Equations

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: R. W. BROCKETT, Harvard University, Cambridge, Mass., U.S.A. J. CORONES, Iowa State University, U.S.A. and Ames Laboratory, U.S. Department of Energy, Iowa, U.S.A. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-c. ROTA, M.l T., Cambridge, Mass., U.S.A.

3 Richard Bellman Dept. of Electrical Engineering, University of Southern California, Los Angeles, U.S.A.; Center for Applied Mathematics, The University of Georgia, Athens, Georgia, U.S.A. and George Adomian Center for Applied Mathematics, The University of Georgia, Athens, Georgia, U.S.A. Partial Differential Equations New Methods/or Their Treatment and Solution D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP Dordrecht / Boston / Lancaster " ~.

4 Librvy of Congress Cataloging in Publication Dall BeUman, Richard Ernest, Partial differential equation s.. (Mathematics and its applications) Bibliography: p. Includes index. 1. Differential equlltions, Partial. I. Adomian, G. II. Title. III. Series: MathematiC$ and its applications (D. Reidel Publishing Company) QA374.B S IS.3'S ISBN-13: e-isbn- 13: DOl: / Published by D. Reidel Publishing Company P.O. Box 17, 3300 AA Dordrecht, HoUand Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers 190 Old Derby SUeet, Hingttam, MA 02043, U.S.A. In all other countries, sold and distributed by Kl uwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holand All Righu Reserved CI 1985 by D. Reidel Publishing Company. Softcover reprint of the hardcover It edition 1985 No part of the material protected by this copyright notice may be reproduced or utili:ted in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without wrilten permission from the oopyriglu owner

5 To Peter D. Lax

6 TABLE OF CONTENTS PREFACE PREFACE BY SECOND AUTHOR Xlll XVll CHAPTER I / MONOTONE CONVERGENCE AND POSITIVE OPERATORS 1. Introduction 2. Monotone Operators 3. Monotonicity 2 4. Convergence 2 5. Differential Equations with Initial Conditions 2 6. Two-Point Boundary Conditions 3 7. Nonlinear Heat Equation 3 8. The Nonlinear Potential Equation 3 Bibliography and Comments 4 CHAPTER II/CONSERVATION 5 1. Introduction 5 2. Analytic and Physical Preliminaries 5 3. The Defining Equations 9 4. Limiting Differential Equations Conservation for the Discrete Approximation Existence of Solutions for Discrete Approximation Conservation for Nonlinear Equations The Matrix Riccati Equation Steady-State Neutron Transport with Discrete Energy Levels Analytic Preliminaries Reflections, Transmission, and Loss Matrices Existence and Uniqueness of Solutions Proof of Conservation Relation Proof of Nonnegativ~ty Statement of Result 26 Bibliography and Comments 26 CHAPTER III/DYNAMIC PROGRAMMING AND PARTIAL DIFFERENTIAL EQUATIONS Introduction Calculus of Variations as a Multistage Decision Process 28 vii

7 viii TABLE OF CONTENTS 3. A New Formalism 4. Layered Functionals 5. Dynamic Programming Approach 6. Quadratic Case 7. Bounds Bibliography and Comments CHAPTER IV / THE EULER-LAGRANGE EQUATIONS AND CHARACTERISTICS Introduction Preliminaries The Fundamental Relations of the Calculus of Variations The Variational Equations The Eulerian Description The Lagrangian Description The Hamiltonian Description Characteristics 53 Bibliography and Comments 58 CHAPTER V / QUASILINEARIZATION AND A NEW METHOD OF SUCCESSIVE APPROXIMATIONS Introduction The Fundamental Variational Relation Successive Approximations Convergence 60 Bibliography and Comments 61 CHAPTER VI/THE VARIATION OF CHARACTERISTIC VALUES AND FUNCTIONS Introduction Variational Problem Dynamic Programming Approach Variation of the Green's Function Justification of Equating Coefficients Change of Variable Analytic Continuation Analytic Character of Green's Function Alternate Derivation of Expression for (x) Variation of Characteristic Values and Characteristic Functions Matrix Case Integral Equations 80 Bibliography and Comments 81

8 TABLE OF CONTENTS CHAPTER VII/THE HADAMARD VARIATIONAL FORMULA 1. Introduction 2. Preliminaries 3. A Minimum Problem 4. A Functional Equation 5. The Hadamard Variation 6. Laplace-Beltrami Operator 7. Inhomogeneous Operator Bibliography and Comments CHAPTER VIII/THE TWO-DIMENSIONAL POTENTIAL EQUATION 1. Introduct ion 2. The Euler-Lagrange Equation 3. Inhomogeneous and Nonlinear Cases 4. Green's Function 5. Two-Dimensional Case 6. Discretization 7. Rectangular Region 8. Associated Minimization Problem 9. Approximation from Above 10. Discussion 11. Semidiscretization 12. Solution of the Difference Equations 13. The Potential Equation 14. Discretization 15. Matrix-Vector Formulation 16. Dynamic Programming 17. Recurrence Equations 18. The Calculations 19. Irregular Regions Bibliography and Comments CHAPTER IX / THE T~EE-DIMENSIONAL POTENTIAL EQUATION 1. Introduction 2. Discrete Variational Problems 3. Dynamic Programming 4. Boundary Conditions 5. Recurrence Relations 6. General Regions 7. Discussion Bibliography and Comments CHAPTER X / THE HEAT EQUATION 1. Introduction 2. The One-Dimensional Heat Equation 3. The Transform Equation ix

9 x T ABLE OF CONTENTS 4. Some Numerical Results 5. Multidimensional Case Bibliography and Comments CHAPTER XI/NONLINEAR PARABOLIC EQUATIONS Introduction Linear Equation The Non-negativity of the Kernel Monotonicity of Mean Values Positivity of the Parabolic Operator Nonlinear Equations Asymptotic Behavior Extensions 126 Bibliography and Comments 127 CHAPTER XII/DIFFERENTIAL QUADRATURE Introduct ion Differential Quadrature Determination of Weighting Coefficients Numerical Results for First Order Problems Systems of Nonlinear Partial Differential Equation~ Higher Order Problems Error Representation Hodgkin-Huxley Equation Equations of the Mathematical Model Numerical Method Conclusion 146 Bibliography and Comments 147 CHAPTER XIII/ADAPTIVE GRIDS AND NONLINEAR EQUATIONS Introduction The Equation u = -uu 148 t x 3. An Example Discussion Extension Higher Order Approximations 151 Bibliography and Comments 152 CHAPTER XIV / INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS Introduction Burgers' Equation Some Numerical Examples ' Two-Dimensional Case Closure Techniques A Direct Method 163

10 TABLE OF CONTENTS 7. Extrapolation 8. Difference Approximations 9. An Approximating Algorithm 10. Numerical Results 11. Higher Order Approximation 12. Truncation 13. Associated Equation 14. Discussion of Convergence of u(n) 15. The Fejer Sum 16. The Modified Truncation Bibliography and Comments xi CHAPTER XV j GREEN'S FUNCTIONS Introduction The Concept of the Green's Function Sturm-Liouville Operator Properties of the Green's Function for the Sturm-Liouville Equation Properties of the 0 Function Distributions Symbolic Functions Derivative of Symbolic Functions What Space Are We Considering? Boundary Conditions Properties of Operator L Adjoint Operators n-th Order Operators Boundary Conditions for the Sturm-Liouville Equation Green's Function for Sturm-Liouville Operator Solution of the Inhomogeneous Equation Solving Non-Homogeneous Boundary Conditions Boundary Conditions Specified on Finite Interval [a, b] Scalar Products Use of Green's Function to Solve a Second-order Stochastic Differential Equation Use of Green's Function in Quantum Physics Use of Green's Functions in Transmission Lines Two-Point Green's Functions - Generalization to n-point Green's Functions 24. Evaluation of Arbitrary Functior,;:; for Nonhomogeneous Boundary Conditions by Matrix Equations 25. Mixed Boundary Conditions

11 xii TABLE OF CONTENTS 26. Some General Properties Nonnegativity of Green's Functions and Solutions Variation-Diminishing Properties of Green's Functions 233 Notes 235 Bibliography 235 CHAPTER XVI/APPROXIMATE CALCULATION OF GREEN'S FUNCTIONS 237 CHAPTER XVII/GREEN'S FUNCTIONS FOR PARTIAL DIFFERENTIAL EQUATIONS Introduction Green's Functions for Multidimensional Problems in Cartesian Coordinates Green's Functions in Curvilinear Coordinates Properties of 0 Functions for Multi-dimensional Case 246 CHAPTER XVIII/THE ITO EQUATION AND A GENERAL STOCHASTIC MODEL FOR DYNAMICAL SYSTEMS 248 Bibliography 252 XIX / NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THE DECOMPOSITION METHOD Parametrization and the A Polynomials n Inverses for Non-simple Differential Operators Multidimensional Green's Functions by Decomposition Method Relationships Between Green's Functions and the Decomposition Method for Partial Differential Equations Separable Systems The partitioning Method of Butkovsky Computation of the ~ 282 CHAPTE~ 8. The Question of Convergence 284 Bibliography 287 INDEX 289

12 PREFACE The purpose of this book is to present some new methods in the treatment of partial differential equations. Some of these methods lead to effective numerical algorithms when combined with the digital computer. Also presented is a useful chapter on Green's functions which generalizes, after an introduction, to new methods of obtaining Green's functions for partial differential operators. Finally some very new material is presented on solving partial differential equations by Adomian's decomposition methodology. This method can yield realistic computable solutions for linear or nonlinear cases even for strong nonlinearities, and also for deterministic or stochastic cases - again even if strong stochasticity is involved. Some interesting examples are discussed here and are to be followed by a book dealing with frontier applications in physics and engineering. In Chapter I, it is shown that a use of positive operators can lead to monotone convergence for various classes of nonlinear partial differential equations. In Chapter II, the utility of conservation technique is shown. These techniques are suggested by physical principles. In Chapter III, it is shown that dyn~mic programming applied to variational problems leads to interesting classes of nonlinear partial differential equations. In Chapter IV, this is investigated in greater detail. In Chapter V, we show. that the use of a transformation suggested by dynamic programming leads to a new method of successive approximations. In Chapter VI, we consider the variation of characteristic values and characteristic functions with the domain, restricting our attention to the one-dimensional case. We point out how the method may be applied to the multidimensional case. In Chapter VII, we apply this method to obtain the classic Hadamard variational formula. It is pointed out that the same method can treat multidimensional variational problems. In Chapter VIII, we consider the discretized form of the two-dimensional potential equation. This equation arises from the minimization of a quadratic form. The minimization is carried out by means of dynamic programming and the dimenxill

13 xiv PREFACE sionality difficulties are circumvented observing that the minimum itself is a quadratic form. In Chapter IX, we consider the discretized form of the three-dimensional potential equation. Dimensionality difficulties are now avoided by using a more general domain. In Chapter X, we consider the linear heat equation. Our procedure has two parts. First, we use the Laplace transform, obtaining an equation of potential type. Then, we employ a numerical inver3ion method. In Chapter XI, we consider nonlinear parabolic partial differential equations, obtaining an analogue of the classical Poincare-Lyapunov theorem for ordinary differential equations. The results required concerning the linear eq~ation are of interest in themselves. In Chapter XII, we give some examples of the use of differential quadrature. In Chapter XIII, we consider an adaptive grid to treat nonlinear partial differential equations. In Chapter XIV, we consider the expansion of the solution of a partial differential equation in orthogonal functions. This yields an infinite system of ordinary differential equations. To obtain numerical results from this system, some method of truncation must be employed. Many interesting stability questions arise in this way. The concluding chapters deal with Green's functions, methods of determining Green's functions for partial differential equations, and new and valuable methods of dealing with nonlinearity and/or randomness in multidimensional problems using the basic decomposition method first discussed by Adomian in Stochastic Systems (Academic Press, 1983); Stochastic Systems II and also Applications of Stochastic Systems Theory to Physics and Engineering will appear shortly. Chapter XV deals with Green's functions; Chapter XVI shows a method for approximate calculation of Green's functions which is very useful in computation; Chapter XVII deals with Green's functions for partial differential equations; Chapter XVIII deals with the Ito equation and a general stochastic model for dynamical systems. The last chapter is completely new and deals with solution of nonlinear partial differential equations by the decomposition method and provides examples as well. Randolph Rach of Raytheon Company has been very helpful here. A great deal of work has been done in the field of partial differential equations. We have made no attempt to cover it all. Rather, we have restricted our attention to areas where we have worked ourselves. Even here, as will be apparent from the text, there is still a great deal to be done.

14 RICHARD E. BELLMAN

15 PREFACE BY SECOND AUTHOR Before the final galley corrections to this book, Richard E. Bellman passed into history on March 19, 1984 at the age of 63. To the very end, his mind was filled with exciting ideas, and he planned further contributions. Having had the privilege of knowing him for over 20 years, and of close collaboration in recent years, it is clear that the loss to science, engineering, and mathematics is immeasurable and profound. Richard Bellman will be remembered around the world as one of the foremost mathematical scientists of this century. Richard Bellman's influence on present and future researchers and generations of students to come will be pervasive and deep. His incredibly productive pioneering work and creative thought in mathematics and its applications to science, engineering, medicine, and economics have made him a leader in his field. His helpfulness and compassion and courage endeared him to o~r hearts. G. ADOMIAN xvii