To the Memory of

My Parents

Prem K. Kythe

Fundamental Solutions for Differential Operators

and Applications

Birkhauser Boston • Basel • Berlin

Prem K. Kythe Department of Mathematics University of New Orleans New Orleans, LA 70148

Library of Congress Cataloging In-Publication Data

Kythe, Prem K. Fundamental solutions for differential operators and applications

/ Prem K. Kythe. p. cm.

Includes bibliographical references and index.

1. Theory of distributions (Functional analysis) 2. operators. 3. Boundary value problems. I. Title.

Differential

QA324.K97 1996 515'7272--dc20 96-24282

elP

Printed on acid-free paper © Birkhiiuser Boston 1996 Birkhiiuser

Softcover reprint of the hardcover 1 st edition 1996 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy­ing, recording, or otherwise, without prior permission of the copyright owner.

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ISBN-13: 978-1-4612-8655-4 e-ISBN-13: 978-1-4612-4106-5 DOl: 10.1007/978-1-4612-4106-5

Typeset by TechType Works, Gretna, LA. Printed and bound by The Maple Press Company, York, PA.

9 8 7 6 5 432 1

Contents

Preface, ix Notation, xv

Introduction, 1 1. Historical Background, 1 2. Modern Developments, 9

Chapter 1: Some Basic Concepts, 11 1.1. Definitions, 11 1.2. Green's Identities, 13 1.3. Distributions, 15 1.4. Fundamental Solutions, 29

Chapter 2: Linear Elliptic Operators, 37 2.1. Constant Coefficients, 37 2.2. Laplace Operator, 41 2.3. Helmholtz Operator, 44 2.4. Cauchy-Riemann Operator, 48 2.5. Nonhomogeneous Operator, 49 2.6. Maximum Principle, 51 2.7. Method of Images, 56

vi CONTENTS

Chapter 3: Linear Parabolic Operators, 60 3.1. Diffusion Operator, 60 3.2. Heat Potentials, 65 3.3. Cauchy Problem, 68 3.4. Maximum Principle, 75 3.5. Schr6dinger Operator, 79 3.6. Method of Images, 81

Chapter 4: Linear Hyperbolic Operators, 85 4.1. Wave Operator, 85 4.2. Harmonic Oscillators, 90 4.3. Wave Potentials, 95 4.4. Cauchy Problem, 100 4.5. Wave Propagation, 105 4.6. Maxwell's Equations, 112

Chapter 5: Nonlinear Operators, 117 5.1. Einstein-Kolmogorov Operator, 117 5.2. Fokker-Plank Operator, 119 5.3. Klein-Gordon Operator, 120 5.4. Dirac's Operator, 122 5.5. Transport Equation, 123 5.6. Transport Operator, 128 5.7. Biharmonic Operator, 129 5.8. Nonlinear Wave Equations, 133 5.9. The ~-function, 135 5.10. Quasihyperbolic Operator, 136

Chapter 6: Elastostatics, 138 6.1. Basic Relations, 138 6.2. Cauchy-Navier Operator, 142 6.3. Half-Space Solutions, 146 6.4. Axisymmetric Solutions, 150 6.5. Somigliana's Identity, 155

Chapter 7: Elastodynamics, 162 7.1. Elastodynamic Operator, 162 7.2. Wave Structures, 165 7.3. Bernoulli-Euler Operator, 170 7.4. Elastoplasticity, 174 7.5. Anisotropic Medium, 177

CONTENTS

Chapter 8: Fluid Dynamics, 180 8.1. Navier-Stokes Equations, 180 8.2. Aerodynamic Flows, 187 8.3. Non-Newtonian Flows, 191 8.4. Porous Media, 196 8.5. Underwater Acoustic Scattering, 202

Chapter 9: Piezoelectrics, 207 9.1. Green's Functions, 208 9.2. Dynamic Piezoelectric Operator, 211 9.3. Fundamental Solutions, 216 9.4. Steady-State Solutions, 220 9.5. Piezocrystal Waves, 225

Chapter 10: Boundary Element Methods, 231 10.1. Boundary Integral Equations, 232 10.2. Boundary Element Method, 234 10.3. Poisson Equation, 236 10.4. Transient Fourier Equation, 241 10.5. Laplace Transform BEM, 249 10.6. Elastostatic BEM, 254 10.7. Fracture Mechanics, 259

Chapter 11: Domain Integrals, 266 11.1. Dual Reciprocity Method, 267 11.2. Multiple Reciprocity Method, 274 11.3. Transient DRM, 277 11.4. Transient MRM, 279 11.5. Fourier Series Method, 280 11.6. Complex Variable BEM, 287

Chapter 12: Finite Deflection of Plates, 292 12.1. von Karman Equations, 292 12.2. Boundary Integral Equations, 295 12.3. Large Deflections, 298 12.4. Singularities in Biharmonic Problems, 306

Chapter 13: Miscellaneous Topics, 307 13.1. Poroelasticity, 307 13.2. Heat Conduction, 312 13.3. Thermoelasticity, 320 13.4. Neutron Diffusion, 327 13.5. Biomechanics, 331

vii

viii CONTENTS

Chapter 14: Quasilinear Elliptic Operators, 338 14.1. p-Laplacian, 339 14.2. Lane-Emden Equation, 344 14.3. Emden-Fowler Equation, 349 14.4. Black Hole Solutions, 353 14.5. Einstein-Yang-Mills Equation, 358

Appendix A: Transforms of Distributions, 364 AI. Fourier Transform, 364 A2. Laplace Transform, 372 A3. Inverse Laplace Transform, 374

Appendix B: Computational Aspects, 376

Appendix C: List of Differential Operators, 387

Bibliography, 391

Index,409

Preface

Overview

Many problems in mathematical physics and applied mathematics can be reduced to boundary value problems for differential, and in some cases, inte­grodifferential equations. These equations are solved by using methods from the theory of ordinary and partial differential equations, variational calculus, operational calculus, function theory, functional analysis, probability theory, numerical analysis and computational techniques. Mathematical models of quantum physics require new areas such as generalized functions, theory of distributions, functions of several complex variables, and topological and al­gebraic methods.

The main purpose of this book is to provide a self contained and system­atic introduction to just one aspect of analysis which deals with the theory of fundamental solutions for differential operators and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related applicable and computational features. The sub­ject matter of this book has its own deep rooted theoretical importance since it is related to Green's functions which are associated with most boundary value problems. The application of fundamental solutions to a recently devel­oped area of boundary element methods has provided a distinct advantage in that an integral equation representation of a boundary value problem is often

x PREFACE

more easily solved by numerical methods than a differential equation with specified boundary and initial conditions. This situation makes the subject more attractive to those whose interest is primarily in numerical methods. Re­cent advances in the area of boundary element methods, where the theory of fundamental solutions plays a pivotal role, has provided a prominent place in research in partial differential equations and related boundary value problems. With the current technological demand, no boundary element method can be advanced without further developments in the area of fundamental solutions.

In almost every good book on partial differential equations there has been a section devoted to fundamental solutions, but the subject in itself has never been explored in full detail. This is the first book of its kind devoted exclusively to this subject. The main motivation in writing this book is the desire to bring out a comprehensive and up-to-date text of theoretical developments and related applications.

Salient Features

The book provides a comprehensive and systematic coverage of the ba­sic theory and applications at a level that can readily be followed by some undergraduate seniors but definitely by graduate students and researchers in different fields of applied mathematics, engineering and mathematical physics where the theory of partial differential equations in itself or solutions of bound­ary value problems are investigated. The presentation, although motivated by current research trends in mathematical physics and engineering, has been carried out with rigorous mathematical methods and proofs. Except for the classical differential operators, the bulk ofthe subject matter and the examples are taken out of research papers published during the last two decades. The presentation in this sense is up to date as a glance at the bibliography at the end of the book will reveal. There are over 70 different differential operators, big or small, that are studied in this book and the fundamental solutions for them have been derived and recorded. A list of all these operators appears in Appendix C. This provides an encyclopedic feature to this book.

There are 14 chapters in the book, which deal with the fundamental solu­tions for linear and nonlinear differential operators from mathematical physics, theory of elasticity, fluid dynamics, piezoelectrics, to cosmology. Although Chapter 1 and Appendix A lay the basic foundation in the theory of distri­butions, Dirac delta function, and Fourier and Laplace transforms, the lin­ear elliptic, parabolic and hyperbolic operators are presented in Chapters 2

PREFACE xi

through 4. These chapters not only include the classical Laplace, Helmholtz, Cauchy-Riemann, diffusion, Schrodinger and wave operators, they also dis­cuss the nonhomogeneous operator, Cauchy problems, maximum principles, heat potentials, wave potentials, harmonic oscillators, wave propagation, and the method of images. Other linear and nonlinear operators, like the Einstein­Kolmogorov, Fokker-Plank, Klein-Gordon, Dirac's, transport, biharmonic, and quasihyperbolic are investigated in Chapter 5, with sections on the ~ func­tion and nonlinear wave equations. Chapter 6 deals with elastostatics where the basic relations of the linear theory of elasticity is presented, fundamen­tal solutions for the Cauchy-Navier operator are derived, and half-space and axisymmetric solutions are provided. The elastodynamic and the Bernoulli­Euler operators are studied in Chapter 7; wave structures and the problems of elastoplasticity and anisotropic media are presented. The Navier-Stokes equa­tions of fluid dynamics, with emphasis on aerodynamic and non-Newtonian fluid flows, especially those dealing with the viscoelastic and power-law flu­ids, are investigated in Chapter 8, which also shows applications in the areas of porous media flows and underwater acoustic scattering. Chapter 9 deals with both static and dynamic piezoelectric operators and presents piezocrystal wave theory. Chapters 10 and 11 cover the applications to the boundary element methods, and provide different methods like the dual reciprocity method, the multiple reciprocity method, the Fourier series method, and the complex vari­able boundary element method to solve the domain integral problem. Mostly potential flows, heat conduction, elastostatic, and ocean acoustic problems are presented. The finite deflection of plates the von Karman operator is considered in Chapter 12; boundary integrals equations are derived, and the problem of large deflections is investigated. Chapter 13 deals with miscel­laneous topics, which include operators in poroelasticity, heat conduction, thermoelasticity, neutron diffusion, and biomechanics. Quasilinear elliptic operators, and especially the p-Laplacian, are studied in Chapter 14, where applications to the dam problem as well as those of stellar dynamics and rela­tivistic physics are presented. The theory of radial solutions and, in particular those for the black hole solutions, are investigated. A discussion of the Lane­Emden, Emden-Fowler and Einstein-Yang-Mills equations completes this chapter.

An essay on the historical development of the subject since the time of George Green (1828) precedes the first chapter as Introduction. A quote from Courant and Hilbert's book Methoden der Mathematischen Physik (1924) that precedes this essay exhibits the first use of the term 'Grundlosung' (funda­mental solution). Another quote in the beginning from Marshall H. Stone is ornamental and provides the philosophical framework for the subject.

Although it is not possible to include every written word on the subject, the topics included in this book are carefully selected and meticulously presented,

xii PREFACE

thereby making it a very useful book to graduate students and researchers in mathematics, physics and engineering. The scope of the book can be judged from the table of contents.

Intended Readers

The book assumes a basic but thorough knowledge of advanced calculus, ordinary and partial differential equations, and complex analysis. It is in­tended to contribute to an effective study at the graduate level and to serve as a reference book for scientists, engineers, and mathematicians in industry. In most cases, through out the book the results are developed from the basics at a level consistent with the mathematical background of the intended readers. In a few instances the results are stated and the original references are cited. Each chapter also ends with a list of references that were used in presenting the subject matter of the chapter. The intended readers fall into one of the several categories. First, they are the students ready for a graduate course in the subject of the book. For them the book can be used as a textbook or a reference book. The second category is that of graduate students engaged in research in analytical and numerical solutions of boundary value problems, especially using the boundary element methods. For them the book should become a constant companion, filled with a vast amount of information on methodology and an up to date list of references. The third category com­prises of scientists and researchers in varied areas of applied mathematics, engineering and physics. For them the book is a vital source of information in some current trends in research where fundamental solutions for differential operators are important.

Computational Aspects

During the last decade mathematical modeling and computer simulation have been the two major activities in scientific research, technology and in­dustry. Some recent successful publications exhibit a careful presentation of text, graphics, and computational components. Keeping these aspects in mind, much effort has gone into the organization of the subject mat­ter in order to make this a useful book. The computational components have occupied a special feature of the book. A number of programs writ­ten in C that run on any PC or mainframe computer that uses MS-DOS,

PREFACE xiii

Z-DOS, MacOS, or UNIX operating system, with complete instructions are available in the form of a package which can be obtained by anonymous ftp at http://www.birkhauser.com/books/isbn/O-8167-3869-5. Details about this package can be found in Appendix B toward the end of the book. These programs solve a large number of interior and exterior boundary value problems in convex and nonconvex regions involving potential flows, heat conduction, porous medium flows, simple blood flows, and elastostatic dis­placements, stresses and strains. Nine benchmark problems described in Ap­pendix B are solved by using these programs. Their input and output files are available in the package. Since these programs are structured in various modules, they can easily be modified to solve other types of boundary value problems provided the fundamental solutions needed for the boundary integral equation formulation are known.

Acknowledgements

The help provided by my colleagues and friends is gratefully acknowledged. I thank the following persons for their constructive suggestions and valuable assistance: Mrs. Marshall H. Stone, Dr. Jan F. Andrus, Dr. Carroll F. Blake­more, Dr. Lew Lefton, Dr. Pratap Puri, Dr. Jairo Santanilla, Dr. Dongming Wei, Mr. M. R. Schaferkotter, Ms. Nancy Radonovich, and Ms. Gayle Bar­clay. I take this opportunity to thank my wife, Mrs. B. D. 'Kiran' Kulshrestha for continuous support and inspiration over the years, including the period of preparation of this book.

New Orleans, Louisiana May, 1996

... let me offer three poems - all scientific poems - which together occupy but a single line:

E = mc2 , E = hv, a(Aa) = (aA)a.

Like Chinese poems they must be taken in by the eye as well as by the ear. The first will be recognized at once as the famous equation between energy and mass, epitomizing a whole revolution in physics; it was discovered by Einstein early in this century as an essential principle of the theory of relativity. The second likewise states a revolutionary principle of modern physics, proposed by Planck shortly after the turn of the century and made the cornerstone ofthe quantum theory; it expresses a relation between energy and the frequency of vibration of a wave of light. An understanding of these two poems is impossible without a reasonably good grasp of the core of modern physics. Indeed, if one were to set out with the single aim of teaching their meaning and their implications, he would end by giving his students a very satisfactory introductory course in physics. In the same way, the last of these three poems is an epitome of a large part of what should be taught as an introduction to modern mathematics. Even for a mathematician this statement might remain obscure, unless he were given a hint to interpret the equation as symbolizing Stokes's theorem; and he would then understand at once just how much is needed from algebra, geometry, topology and analysis before the significance of this equation can be fully grasped. As it stands, the equation merely expresses a kind of abstract associative law, analogous to the one which is verified in case the three symbols a, A, and a designate cardinal numbers. It is when we give altogether different interpretations of these same symbols that our equation becomes an expression of Stokes's theorem.

- Marshall H. Stone, The Revolution in Mathematics,

Liberal Education, 47, 1961,304-327.

Notation

AxB A\B A Ac b B Bi Bij B(m,n) B(xo,r) B C

Cl,C2

cp

Cv

Ce

C Cijk1

CP(O) COO (0) CO'(Rn )

product of sets A and B complement of a set B with respect to a set A closure of a set A €-neighborhood of a set A body force diffusion coefficient (§5.1) domain integral Biot's criteria, j = 1,2 beta function open ball of radius r and center at Xo

magnetic induction field speed of sound, or light velocity of compression and transverse waves, respectively specific heat at constant pressure specific heat at constant volume effective wave velocity Cattaneo number elastic moduli class of functions continuous with derivatives on a domain 0 infinitely differentiable functions on 0 class of functions infinitely differentiable

on Rn and vanishing outside some bounded region set of functions in CP (Rn) that have compact support bending rigidity k-th partial derivative

xvi

Dij

D9

D V V(Rn) V' V+ D e

eijk

erf(x) E E1 Eijkl

E(m) E F ;: ;:-1

1 9 gij

G G(x, ~) H(x) Hn H~,2

h n H I 10 I Jo j k k Kn K(m)

NOTATION

rate of deformation tensor diffusion coefficient in a group 9

electric displacement field set of test functions set of all test functions from the class err (Rn) space of all distributions on err (Rn class of distributions in V' (R1) that vanish for t < 0 electric induction field specific internal energy piezoelectric material constants error function Young's modulus; dielectric constant flexural rigidity bending coefficients complete elliptic integral of second kind electric field external force (§ 13.5) Fourier transform inverse Fourier transform Laplace transform of f acceleration due to gravity potentials of the gravitational field Grashof number Green's function Heaviside unit step function Hermite functions Hankel functions of order n plate thickness (§9.1); thermal conductivity (§ 13.3) Planck's constant = 1.054 x 10-27 erg-sec (h = 27rn) magnetic field moment of inertia modified Bessel function of first kind identity matrix Bessel function of first kind current multi-index of dimension n (= (k 1,· .. ,kn» diffusivity (ch. 8); thermal conductivity (§ 13.2) Kirchhoff's effective shear force complete elliptic integral of first kind

K~,2

K K[,KII

L(D) L*

L (D, :t) £ £-1

m,mo m*(x,x') M M,Moo Mij

Mn Mns n,n n Nj(z) N(x, ~) p,q

P

P P

Pi

p

q q* r (r,e,z) (r,e,</J) Rn R+

Re Rf

Rij

8

NOTATION

modified Bessel functions of order n viscosity of non-Newtonian fluids (§8.3) stress intensity factor for mode I and I I cracks linear differential operator adjoint operator

transient operator

Laplace transform £{J(t)} = 1(8) inverse Laplace transform mass virtual moment (Chapter 3) Bending moment Mach numbers (Chapter 8) bending moments normal moment at the boundary twisting moment at the boundary outward normal vector unit normal vector nodal values at a point z global shape functions P = 8u/8x,q = 8u/8y, in variational calculus pressure; pore-water pressure; acoustic pressure traction vector (p = an) Prandtl number tractions

the function, p~ = dd In Ixl a x

flux (= 8u/8n); electric charge density per unit volume virtual flux = 8u* /8n radius polar cylindrical coordinates spherical coordinates Euclidean n-space set of nonnegative real numbers Reynolds number flow resistance Riemann-Christoffel tensor variable of the Laplace transform; storativity (§8.4) stress resultants

xvii

xviii

S S(XO, r) Sn(l) S S

S'(Rn )

S[f(xo)] tij

T Tij

U

Ui

U*(X,~)

U*(X, X') Vi

Vg

V(X, t) Vn(X, t) v

w X

a f3 f3k

r r± 8ij 8(x, ~) 8ij co e f

¢ 'IjJ

'IjJ(X,~n) 'l1 'l1(xo, a)

NOTATION

surface in Rn

boundary surface ofthe ball B(xo, r) surface area ofthe unit ball (= 27rn / 2 /r(n/2» boundary surface of a three-dimension region V complete countably-normed space with topology based on

convergent sequences (1) C S) set of distributions on S (Rn), (S' (Rn) c 1)' (Rn) ) discontinuity of function f at Xo

non-Newtonian stress tensor temperature; transmissivity (§8.4) stress energy tensor displacement inplane displacement fundamental solution virtual displacement (Chapter 3) velocity components velocity of neutrons in group 9

heat potential wave potentials velocity field (= (u, V, w)) deflection (= (Xl, ... ,Xn)) a point in Rn ; field point variable of the Fourier transform; coefficient of thermal expansion total contribution of delayed neutrons in group 9 individual contribution of delayed neutrons in group 9 parabolic boundary; boundary of a domain n future and past cones Kronecker delta Dirac delta function Kronecker delta small strain strain vector dielectric constant potential; velocity potential; piezometric head; electric potential stream function weights at the points ~n electric enthalpy spinor

*

NOTATION

permeability; coefficient of heat transfer (§ 13.3) effective wave number

material constants along orthotropic directions permeability tensor Lame's constants decay constant of delayed neutrons in group k

xix

Lame's constant; shear modulus (= G); magnetic permeability; dynamic viscosity of a fluid

kinematic viscosity of a fluid (= J.L/ p); elastic Poisson's ratio change in temperature

polytropic functions of index n class of all functions in Coo (Rn) which together with all

derivatives grow not faster than the polynomial normal modes density; charge density conductivity

normal stresses if i = j; shearing stresses if i :I j absorption cross section stress vector source point domain in Rn vorticity vector fraction of fission fraction of delayed neutrons characteristic or indicator function of a set A

~8 .8 ·8 gradient (= grad = i 8x + j 8y + k 8z)

82 82 82 Laplacian, = 8x2 + 8y2 + 8z2

biharmonic operator, = (V2)2 82

wave operator, = 8t2 - c2V2

wave operator for c = 1 linear deformation nonlinear deformation convolution tensor product Hodge star operator exterior or wedge product

... Diese heuristische UberJegung kehren wir nunmehr urn und machen sie so zu einer strengen mathematischen Theorie. Wir definieren von VOffi­

herein als Greensche Funktion des Differentialausdruckes L[u) bei gegebe­nen homogenen Randbedingungen eine Funktion K(x,~) von x und ~, welche folgende Bedingungen befriedigt:

1. K(x,~) ist bei festem ~ eine stetige Funktion von x und erftillt die vorgegebenen homogenen Randbedingungen.

2. Die Ableitungen erster und zweiter Ordnung von K nach x sind, abgesehen von der Stelle x = ~, tiberall in G stetig; an der Stelle x = ~ macht die erste Ableitung einen Sprung, gegeben durch

(95) dK(x, ~) IX=~+O = __ 1_

dx x=~-O p(~)'

3. Ausser an der Stelle x = ~ gentigt K als Funktion von x tiberall in G der Differentialgleichung L[K) = O.

Einige stetige Funktion, welch die Bedingungen 2, 3, abernicht notwendig die homogenen Randbedingungen erftillt, nennt man, nebenbei bemerkt, "Grundlosung" der Differentialgleichung L[u) = O.

- R. Courant and D. Hilbert,

Methoden der Mathematischen Physik,

Bd. I, 1924, 304.

Author

Prem Kishore Kythe (formerly Kulshrestha), b. India, 29 January 1930; U. S. citizen; Educ.: Ph. D. (Mathematics), Aligarh Muslim University, India, 1961; Prof. Exp.: Faculty member at Aligarh Muslim University 1958-60, at Indian Institute of Technology, Bombay, 1960-67; at University of New Orleans (UNO) 1967-; Professor of Mathematics at UNO since 1974; Invited speaker at the NATO Advanced Institute for Automatic Translation (from Russian) at Venice, Italy, July 1962; UNESCO Fellow in Linguistic Data Pro­cessing (Machine Translation) 1963; Consultant, Institute of Human Learning, University of California at Berkeley 1964; Guest participant in the Summer Linguistics Institute, University of Washington, Seattle, WA, June-Aug 1963; Participant in Summer School on Complex Function Theory, University of Cork, Ireland, 1971; Visitor at Mathematics Department, Imperial College, London, Fall 1973; Visiting Professor, Department of Computer Science, University of Illinois at Urbana-Champaign, Spring 1986; Reviewer for the Bulletin, Institute of Mathematics, Academia Sinica; Reviewer for the JEMT ASME; Reviewer for NSF for a research grant; Reviewer for Zentralblatt fUr Mathematik, and for Applied Mechanics Reviews; Over 15 books/monographs and some research papers translated from Russian into English; Over 40 re­search publications in the areas of univalent functions, boundary value prob­lems in continuum mechanics, differential equations, Laplace transform, wave theory, wave structure in rotating flows; Numerous citations in research arti­cles, and graduate text/monographs; Author of 'An Introduction to Boundary Element Methods'; Listed in American Men and Women of Science; has been teaching finite element analysis and boundary element methods at UNO.

Prem K. K ythe

Fundamental Solutions for Differential Operators and Applications