APPLIED PARTIAL DIFFERENTIAL EQUATIONS

APPLIED PARTIAL DIFFERENTIAL EQUATIONS

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DONALD W. TRIM THE UNIVERSITY OF MANITOBA

PWS-KENT PUBLISHING COMPANY

• • • • • • • • • • • • BOSTON

Contents

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Derivation of Partial Differential Equations of Mathematical Physics 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction 1 Heat Conduction 10 Transverse Vibrations of Strings; Longitudinal and Angular Vibrations of Bars 21 Transverse Vibrations of Membranes 33 Transverse Vibrations of Beams 40 Electrostatic Potential 45 General Solutions of Partial Differential Equations 47 Classification of Second-Order Partial Differential Equations

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Fourier Series 2.1 2.2 2.3

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Fourier Series 69 Fourier Sine and Cosine Series 84 Further Properties of Fourier Series

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T H R E E Separation of Variables

102

3.1 Linearity and Superposition 103 3.2 Separation of Variables 105 3.3 Nonhomogeneities and Eigenfunction Expansions

122

[F| fo] [ü] [R Sturm-Liouville Systems 4.1 4.2 4.3

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Eigenvalues and Eigenfunctions 141 Eigenfunction Expansions 150 Further Properties of Sturm-Liouville Systems

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Solutions of Homogeneous Problems by Separation of Variables 169 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Introduction 169 Homogeneous Initial Boundary Value Problems in Two Variables Homogeneous Boundary Value Problems in Two Variables 182 Homogeneous Problems in Three and Four Variables (Cartesian Coordinates Only) 189 The Multidimensional Eigenvalue Problem 197 Properties of Parabolic Partial Differential Equations 202 Properties of Elliptic Partial Differential Equations 210 Properties of Hyperbolic Partial Differential Equations 217

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Finite Fourier Transforms and Nonhomogeneous Problems 6.1 6.2 6.3

Finite Fourier Transforms 221 Nonhomogeneous Problems in Two Variables 225 Higher-Dimensional Problems in Cartesian Coordinates

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fs] [~E~| py] [ E ] [N" Problems on Infinite Spatial Domains 7.1 7.2 7.3 7.4

Introduction 252 The Fourier Integral Formulas 254 Fourier Transforms 264 Applications of Fourier Transforms to Initial Boundary Value Problems 276

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Special Functions 8.1 8.2 8.3 8.4 8.5 8.6

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Introduction 287 Gamma Function 289 Bessel Functions 291 Sturm-Liouville Systems and Bessel's Differential Equation 300 Legendre Functions 308 Sturm-Liouville Systems and Legendre's Differential Equation 315

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Problems in Polar, Cylindrical, and Spherical Coordinates 9.1 9.2 9.3

Homogeneous Problems in Polar, Cylindrical, and Spherical Coordinates 320 Nonhomogeneous Problems in Polar, Cylindrical, and Spherical Coordinates 335 Hankel Transforms 346

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Laplace Transforms 10.1 10.2 10.3 10.4 10.5

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Introduction 351 Laplace Transform Solutions for Problems on Unbounded Domains 360 The Complex Inversion Integral 365 Applications to Partial Differential Equations on Bounded Domains 376 Laplace Transform Solutions to Problems in Polar, Cylindrical, and Spherical Coordinates 390

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Green's Functions for O r d i n a r y Differential Equations 11.1 11.2 11.3 11.4 11.5 11.6

Generalized Functions 396 Introductory Example 403 Green's Functions 406 Solutions of Boundary Value Problems Using Green's Functions Modified Green's Functions 428 Green's Functions for Initial Value Problems 436

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12.4 12.5 12.6 12.7

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G r e e n ' s Functions for Partial Differential Equations 12.1 12.2 12.3

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Generalized Functions and Green's Identities 440 Green's Functions for Dirichlet Boundary Value Problems 443 Solutions of Dirichlet Boundary Value Problems on Finite Regions 455 Solutions of Neumann Boundary Value Problems on Finite Regions 463 Robin and Mixed Boundary Value Problems on Finite Regions 471 Green's Functions for Heat Conduction Problems 474 Green's Functions for the Wave Equation 478

Contents

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Convergence of Fourier Series

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Convergence of Fourier Integrals

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Vector Analysis

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Answers to Selected Exercises

Index

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