Introduction to PARTIAL DIFFERENTIAL EQUATIONS THIRD EDITION

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2 Introduction to PARTIAL DIFFERENTIAL EQUATIONS THIRD EDITION K. SANKARA RAO Formerly Professor Department of Mathematics Anna University, Chennai New Delhi

3 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS, Third Edition K. Sankara Rao 2011 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Eleventh Printing (Third Edition) January, 2011 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Syndicate Binders, A-20, Hosiery Complex, Noida, Phase-II Extension, Noida (N.C.R. Delhi).

4 This book is dedicated with affection and gratitude to the memory of my respected Father (Late) KOMMURI VENKATESWARLU and to my respected Mother SHRIMATI VENKATARATNAMMA

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6 Contents Preface Preface to the First and Second Edition ix xi 0. Partial Differential Equations of First Order Introduction Surfaces and Normals Curves and Their Tangents Formation of Partial Differential Equation Solution of Partial Differential Equations of First Order Integral Surfaces Passing Through a Given Curve The Cauchy Problem for First Order Equations Surfaces Orthogonal to a Given System of Surfaces First Order Non-linear Equations Cauchy Method of Characteristics Compatible Systems of First Order Equations Charpit s Method Special Types of First Order Equations 42 Exercises Fundamental Concepts Introduction Classification of Second Order PDE Canonical Forms Canonical Form for Hyperbolic Equation Canonical Form for Parabolic Equation Canonical Form for Elliptic Equation 59 v

7 vi CONTENTS 1.4 Adjoint Operators Riemann s Method Linear Partial Differential Equations with Constant Coefficiants General Method for Finding CF of Reducible Non-homogeneous Linear PDE General Method to Find CF of Irreducible Non-homogeneous Linear PDE Methods for Finding the Particular Integral (PI) Homogeneous Linear PDE with Constant Coefficients Finding the Complementary Function Methods for Finding the PI 99 Exercises Elliptic Differential Equations Occurrence of the Laplace and Poisson Equations Derivation of Laplace Equation Derivation of Poisson Equation Boundary Value Problems (BVPs) Some Important Mathematical Tools Properties of Harmonic Functions The Spherical Mean Mean Value Theorem for Harmonic Functions Maximum-Minimum Principle and Consequences Separation of Variables Dirichlet Problem for a Rectangle The Neumann Problem for a Rectangle Interior Dirichlet Problem for a Circle Exterior Dirichlet Problem for a Circle Interior Neumann Problem for a Circle Solution of Laplace Equation in Cylindrical Coordinates Solution of Laplace Equation in Spherical Coordinates Miscellaneous Examples 154 Exercises Parabolic Differential Equations Occurrence of the Diffusion Equation Boundary Conditions Elementary Solutions of the Diffusion Equation Dirac Delta Function Separation of Variables Method Solution of Diffusion Equation in Cylindrical Coordinates Solution of Diffusion Equation in Spherical Coordinates Maximum-Minimum Principle and Consequences 215

8 CONTENTS vii 3.9 Non-linear Equations (Models) Semilinear Equations Quasi-linear Equations Burger s Equation Initial Value Problem for Burger s Equation Miscellaneous Examples 220 Exercises Hyperbolic Differential Equations Occurrence of the Wave Equation Derivation of One-dimensional Wave Equation Solution of One-dimensional Wave Equation by Canonical Reduction The Initial Value Problem; D Alembert s Solution Vibrating String Variables Separable Solution Forced Vibrations Solution of Non-homogeneous Equation Boundary and Initial Value Problem for Two-dimensional Wave Equations Method of Eigenfunction Periodic Solution of One-dimensional Wave Equation in Cylindrical Coordinates Periodic Solution of One-dimensional Wave Equation in Spherical Polar Coordinates Vibration of a Circular Membrane Uniqueness of the Solution for the Wave Equation Duhamel s Principle Miscellaneous Examples 270 Exercises Green s Function Introduction Green s Function for Laplace Equation The Methods of Images The Eigenfunction Method Green s Function for the Wave Equation Helmholtz Theorem Green s Function for the Diffusion Equation 310 Exercises Laplace Transform Methods Introduction Transform of Some Elementary Functions Properties of Laplace Transform Transform of a Periodic Function Transform of Error Function Transform of Bessel s Function Transform of Dirac Delta Function 337

9 viii CONTENTS 6.8 Inverse Transform Convolution Theorem (Faltung Theorem) Transform of Unit Step Function Complex Inversion Formula (Mellin-Fourier Integral) Solution of Ordinary Differential Equations Solution of Partial Differential Equations Solution of Diffusion Equation Solution of Wave Equation Miscellaneous Examples 375 Exercises Fourier Transform Methods Introduction Fourier Integral Representations Fourier Integral Theorem Sine and Cosine Integral Representations Fourier Transform Pairs Transform of Elementary Functions Properties of Fourier Trasnform Convolution Theorem (Faltung Theorem) Parseval s Relation Transform of Dirac Delta Function Multiple Fourier Transforms Finite Fourier Transforms Finite Sine Transform Finite Cosine Transform Solution of Diffusion Equation Solution of Wave Equation Solution of Laplace Equation Miscellaneous Examples 431 Exercises 443 Bibliography Answers and Keys to Exercises Index

10 Preface The objective of this third edition is the same as in previous two editions: to provide a broad coverage of various mathematical techniques that are widely used for solving and to get analytical solutions to Partial Differential Equations of first and second order, which occur in science and engineering. In fact, while writing this book, I have been guided by a simple teaching philosophy: An ideal textbook should teach the students to solve problems. This book contains hundreds of carefully chosen worked-out examples, which introduce and clarify every new concept. The core material presented in the second edition remains unchanged. I have updated the previous edition by adding new material as suggested by my active colleagues, friends and students. Chapter 1 has been updated by adding new sections on both homogeneous and nonhomogeneous linear PDEs, with constant coefficients, while Chapter 2 has been repeated as such with the only addition that a solution to Helmholtz equation using variables separable method is discussed in detail. In Chapter 3, few models of non-linear PDEs have been introduced. In particular, the exact solution of the IVP for non-linear Burger s equation is obtained using Cole Hopf function. Chapter 4 has been updated with additional comments and explanations, for better understanding of normal modes of vibrations of a stretched string. Chapters 5 7 remain unchanged. I wish to express my gratitude to various authors, whose works are referred to while writing this book, as listed in the Bibliography. Finally, I would like to thank all my old colleagues, friends and students, whose feedback has helped me to improve over previous two editions. It is also a pleasure to thank the publisher, PHI Learning, for their careful processing of the manuscript both at the editorial and production stages. Any suggestions, remarks and constructive comments for the improvement of text are always welcome. K. SANKARA RAO ix

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