SOLUTIONS MANUAL
,Solutions Manual for Partial Differential Equations with FOURIER SERIE
and BOUNDARY VALUE PROBLEMS 2ND Edition
Contents
1 A Preview of Applications and Techniques 1
1.1 What Is a Partial Differential Equation? 1
1.2 Solving and Interpreting a Partial Differential Equation 4
2 Fourier Series 13
2.1 Periodic Functions 13
2.2 Fourier Series 21
2.3 Fourier Series of Functions with Arbitrary Periods 35
2.4 Half-Range Expansions: The Cosine and Sine Series 51
2.5 Mean Square Approximation and Parseval’s Identity 58
2.6 Complex Form of Fourier Series 63
2.7 Forced Oscillations 73
Supplement on Convergence
2.9 Uniform Convergence and Fourier Series 79
2.10 Dirichlet Test and Convergence of Fourier Series 81
3 Partial Differential Equations in Rectangular Coordinates 82
3.1 Partial Differential Equations in Physics and Engineering82
3.3 Solution of the One Dimensional Wave
Equation: The Method of Separation of
Variables 87
3.4 D’Alembert’s Method 104
3.5 The One Dimensional Heat Equation 118
3.6 Heat Conduction in Bars: Varying the Boundary Conditions 128
3.7 The Two Dimensional Wave and Heat Equations 144
3.8 Laplace’s Equation in Rectangular Coordinates 146
3.9 Poisson’s Equation: The Method of Eigenfunction Expansions 148
3.10 Neumann and Robin Conditions 151
4 Partial Differential Equations in
Polar and Cylindrical Coordinates 155
4.1 The Laplacian in Various Coordinate Systems 155
, Contents iii
4.2 Vibrations of a Circular Membrane: Symmetric Case 228
4.3 Vibrations of a Circular Membrane: General Case 166
4.4 Laplace’s Equation in Circular Regions 175
4.5 Laplace’s Equation in a Cylinder 191
4.6 The Helmholtz and Poisson Equations 197
Supplement on Bessel Functions
4.7 Bessel’s Equation and Bessel Functions 204
4.8 Bessel Series Expansions 213
4.9 Integral Formulas and Asymptotics for Bessel Functions 228
5 Partial Differential Equations in Spherical Coordinates 231
5.1 Preview of Problems and Methods 231
5.2 Dirichlet Problems with Symmetry 233
5.3 Spherical Harmonics and the General Dirichlet Problem 236
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
and Wave Equations 242
Supplement on Legendre Functions
5.5 Legendre’s Differential Equation 245
5.6 Legendre Polynomials and Legendre Series Expansions 251
6 Sturm–Liouville Theory with Engineering Applications 257
6.1 Orthogonal Functions 257
6.2 Sturm–Liouville Theory 259
6.3 The Hanging Chain 263
6.4 Fourth Order Sturm–Liouville Theory 265
6.6 The Biharmonic Operator 267
6.7 Vibrations of Circular Plates 269
, iv Contents
7 The Fourier Transform and Its Applications 271
7.1 The Fourier Integral Representation 271
7.2 The Fourier Transform 276
7.3 The Fourier Transform Method 286
7.4 The Heat Equation and Gauss’s Kernel 294
7.5 A Dirichlet Problem and the Poisson Integral Formula 303
7.6 The Fourier Cosine and Sine Transforms 306
7.7 Problems Involving Semi-Infinite Intervals 310
7.8 Generalized Functions 315
7.9 The Nonhomogeneous Heat Equation 325
7.10 Duhamel’s Principle 327
,Solutions Manual for Partial Differential Equations with FOURIER SERIE
and BOUNDARY VALUE PROBLEMS 2ND Edition
Contents
1 A Preview of Applications and Techniques 1
1.1 What Is a Partial Differential Equation? 1
1.2 Solving and Interpreting a Partial Differential Equation 4
2 Fourier Series 13
2.1 Periodic Functions 13
2.2 Fourier Series 21
2.3 Fourier Series of Functions with Arbitrary Periods 35
2.4 Half-Range Expansions: The Cosine and Sine Series 51
2.5 Mean Square Approximation and Parseval’s Identity 58
2.6 Complex Form of Fourier Series 63
2.7 Forced Oscillations 73
Supplement on Convergence
2.9 Uniform Convergence and Fourier Series 79
2.10 Dirichlet Test and Convergence of Fourier Series 81
3 Partial Differential Equations in Rectangular Coordinates 82
3.1 Partial Differential Equations in Physics and Engineering82
3.3 Solution of the One Dimensional Wave
Equation: The Method of Separation of
Variables 87
3.4 D’Alembert’s Method 104
3.5 The One Dimensional Heat Equation 118
3.6 Heat Conduction in Bars: Varying the Boundary Conditions 128
3.7 The Two Dimensional Wave and Heat Equations 144
3.8 Laplace’s Equation in Rectangular Coordinates 146
3.9 Poisson’s Equation: The Method of Eigenfunction Expansions 148
3.10 Neumann and Robin Conditions 151
4 Partial Differential Equations in
Polar and Cylindrical Coordinates 155
4.1 The Laplacian in Various Coordinate Systems 155
, Contents iii
4.2 Vibrations of a Circular Membrane: Symmetric Case 228
4.3 Vibrations of a Circular Membrane: General Case 166
4.4 Laplace’s Equation in Circular Regions 175
4.5 Laplace’s Equation in a Cylinder 191
4.6 The Helmholtz and Poisson Equations 197
Supplement on Bessel Functions
4.7 Bessel’s Equation and Bessel Functions 204
4.8 Bessel Series Expansions 213
4.9 Integral Formulas and Asymptotics for Bessel Functions 228
5 Partial Differential Equations in Spherical Coordinates 231
5.1 Preview of Problems and Methods 231
5.2 Dirichlet Problems with Symmetry 233
5.3 Spherical Harmonics and the General Dirichlet Problem 236
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
and Wave Equations 242
Supplement on Legendre Functions
5.5 Legendre’s Differential Equation 245
5.6 Legendre Polynomials and Legendre Series Expansions 251
6 Sturm–Liouville Theory with Engineering Applications 257
6.1 Orthogonal Functions 257
6.2 Sturm–Liouville Theory 259
6.3 The Hanging Chain 263
6.4 Fourth Order Sturm–Liouville Theory 265
6.6 The Biharmonic Operator 267
6.7 Vibrations of Circular Plates 269
, iv Contents
7 The Fourier Transform and Its Applications 271
7.1 The Fourier Integral Representation 271
7.2 The Fourier Transform 276
7.3 The Fourier Transform Method 286
7.4 The Heat Equation and Gauss’s Kernel 294
7.5 A Dirichlet Problem and the Poisson Integral Formula 303
7.6 The Fourier Cosine and Sine Transforms 306
7.7 Problems Involving Semi-Infinite Intervals 310
7.8 Generalized Functions 315
7.9 The Nonhomogeneous Heat Equation 325
7.10 Duhamel’s Principle 327
