AND NONLINEAR SCIENCE SERIES. Partial Differential. Equations with MATLAB. Matthew P. Coleman. CRC Press J Taylor & Francis Croup
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1 CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES An Introduction to Partial Differential Equations with MATLAB Second Edition Matthew P Coleman Fairfield University Connecticut, USA»C) CRC Press J Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business A CHAPMAN & HALL BOOK
2 Contents Preface xi Prelude to Chapter Introduction 3 11 What are Partial Differential Equations? 3 12 PDEs We Can Already Solve 6 13 Initial and Boundary Conditions Linear PDEs Definitions Linear PDEs The Principle of Superposition Separation of Variables for Linear, Homogeneous PDEs Eigenvalue Problems 25 Prelude to Chapter The Big Three PDEs Second-Order, Linear, Homogeneous PDEs with Constant Co efficients The Heat Equation and Diffusion The Wave Equation and the Vibrating String Initial and Boundary Conditions for the Heat and Wave Equa tions Laplace's Equation The Potential Equation Using Separation of Variables to Solve the Big Three PDEs 71 Prelude to Chapter Fourier Series Introduction Properties of Sine and Cosine The Fourier Series The Fourier Series, Continued The Fourier Series Proof of Pointwise Convergence Fourier Sine and Cosine Series Completeness 124 vii
3 viii Contents Prelude to Chapter Solving the Big Three PDEs on Finite Domains Solving the Homogeneous Heat Equation for a Finite Rod 42 Solving the Homogeneous Wave Equation for a Finite String 43 Solving the Homogeneous Laplace's Equation on a Rectangular 138 Domain Nonhomogeneous Problems 153 Prelude to Chapter Characteristics First-Order PDEs with Constant Coefficients First-Order PDEs with Variable Coefficients The Infinite String Characteristics for Semi-Infinite and Finite String Problems General Second-Order Linear PDEs and Characteristics 201 Prelude to Chapter Integral Transforms The Laplace Transform for PDEs Fourier Sine and Cosine Transforms The Fourier Transform The Infinite and Semi-Infinite Heat Equations Distributions, the Dirac Delta Function and Generalized Fourier Transforms Proof of the Fourier Integral Formula 266 Prelude to Chapter Special Functions and Orthogonal Polynomials The Special Functions and Their Differential Equations Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials The Method of Frobenius; Laguerre Polynomials Interlude: The Gamma Function Bessel Functions Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials 317 Prelude to Chapter Sturm-Liouville Theory and Generalized Fourier Series Sturm-Liouville Problems Regular and Periodic Sturm-Liouville Problems 337
4 Contents ix 83 Singular Sturm-Liouville Problems; Self-Adjoint Problems The Mean-Square L2 or Norm and Convergence in the Mean Generalized Fourier Series; Parseval's Equality and Complete ness 361 Prelude to Chapter PDEs in Higher Dimensions PDEs in Higher Dimensions: Examples and Derivations The Heat and Wave Equations on a Rectangle; Multiple Fourier Series Laplace's Equation in Polar Coordinates: Poisson's Integral Formula The Wave and Heat Equations in Polar Coordinates Problems in Spherical Coordinates The Infinite Wave Equation and Multiple Fourier Transforms Postlude: Eigenvalues and Eigenfunctions of the Laplace Oper ator; Green's Identities for the Laplacian 456 Prelude to Chapter Nonhomogeneous Problems and Green's Functions Green's Functions for ODEs Green's Function and the Dirac Delta Function Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation Green's Functions for Equations of Evolution 525 Prelude to Chapter Numerical Methods Finite Difference Approximations for ODEs Finite Difference Approximations for PDEs Spectral Methods and the Finite Element Method 565 A Uniform Convergence; Differentiation and Integration of Fourier Series 579 B Other Important Theorems 585 C Existence and Uniqueness Theorems 591 D A Menagerie of PDEs 601 E MATLAB Code for Figures and Exercises 613
5 x Contents F Answers to Selected Exercises 627 References Index
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