Students Solutions Manual
                       PARTIAL DIFFERENTIAL
                              EQUATIONS
                           with FOURIER SERIES and
                         BOUNDARYVALUEPROBLEMS
                                 Second Edition
                                   ´
                             NAKHLEH.ASMAR
                              University of Missouri
                                         Contents
                                  Preface                                                                         v
                                  Errata                                                                         vi
                           1       APreview of Applications and Techniques                                        1
                                          1.1 What Is a Partial Differential Equation?     1
                                          1.2 Solving and Interpreting a Partial Differential Equation   2
                           2       Fourier Series                                                                 4
                                          2.1 Periodic Functions    4
                                          2.2 Fourier Series   6
                                          2.3 Fourier Series of Functions with Arbitrary Periods   10
                                          2.4 Half-Range Expansions: The Cosine and Sine Series      14
                                          2.5 Mean Square Approximation and Parseval’s Identity       16
                                          2.6 Complex Form of Fourier Series     18
                                          2.7 Forced Oscillations    21
                                         Supplement on Convergence
                                          2.9 Uniform Convergence and Fourier Series     27
                                         2.10 Dirichlet Test and Convergence of Fourier Series   28
                           3       Partial Differential Equations in Rectangular Coordinates                      29
                                          3.1 Partial Differential Equations in Physics and Engineering    29
                                          3.3 Solution of the One Dimensional Wave Equation:
                                               The Method of Separation of Variables   31
                                          3.4 D’Alembert’s Method      35
                                          3.5 The One Dimensional Heat Equation       41
                                          3.6 Heat Conduction in Bars: Varying the Boundary Conditions       43
                                          3.7 The Two Dimensional Wave and Heat Equations         48
                                          3.8 Laplace’s Equation in Rectangular Coordinates     49
                                          3.9 Poisson’s Equation: The Method of Eigenfunction Expansions       50
                                         3.10 Neumann and Robin Conditions        52
                                                                                                                 Contents  iii
                              4       Partial Differential Equations in
                                      Polar and Cylindrical Coordinates                                                   54
                                              4.1 The Laplacian in Various Coordinate Systems          54
                                              4.2 Vibrations of a Circular Membrane: Symmetric Case           79
                                              4.3 Vibrations of a Circular Membrane: General Case          56
                                              4.4 Laplace’s Equation in Circular Regions       59
                                              4.5 Laplace’s Equation in a Cylinder       63
                                              4.6 The Helmholtz and Poisson Equations         65
                                                 Supplement on Bessel Functions
                                              4.7 Bessel’s Equation and Bessel Functions       68
                                              4.8 Bessel Series Expansions      74
                                              4.9 Integral Formulas and Asymptotics for Bessel Functions         79
                              5       Partial Differential Equations in Spherical Coordinates                              80
                                              5.1 Preview of Problems and Methods         80
                                              5.2 Dirichlet Problems with Symmetry         81
                                              5.3 Spherical Harmonics and the General Dirichlet Problem           83
                                              5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
                                                   and Wave Equations 86
                                             Supplement on Legendre Functions
                                              5.5 Legendre’s Differential Equation       88
                                              5.6 Legendre Polynomials and Legendre Series Expansions            91
                              6       Sturm–Liouville Theory with Engineering Applications                                94
                                              6.1 Orthogonal Functions       94
                                              6.2 Sturm–Liouville Theory        96
                                              6.3 The Hanging Chain        99
                                              6.4 Fourth Order Sturm–Liouville Theory          101
                                              6.6 The Biharmonic Operator         103
                                              6.7 Vibrations of Circular Plates      104
              iv  Contents
                              7       The Fourier Transform and Its Applications                                          105
                                              7.1 The Fourier Integral Representation        105
                                              7.2 The Fourier Transform        107
                                              7.3 The Fourier Transform Method          112
                                              7.4 The Heat Equation and Gauss’s Kernel           116
                                              7.5 A Dirichlet Problem and the Poisson Integral Formula           122
                                              7.6 The Fourier Cosine and Sine Transforms          124
                                              7.7 Problems Involving Semi-Infinite Intervals         126
                                              7.8 Generalized Functions       128
                                              7.9 The Nonhomogeneous Heat Equation             133
                                             7.10 Duhamel’s Principle       134
                              8       The Laplace and Hankel Transforms with Applications                                 136
                                              8.1 The Laplace Transform        136
                                              8.2 Further Properties of the Laplace transform        140
                                              8.3 The Laplace Transform Method           146
                                              8.4 The Hankel Transform with Applications          148
                              12 Green’sFunctions and Conformal Mappings                                                  150
                                             12.1 Green’s Theorem and Identities        150
                                             12.2 Harmonic Functions and Green’s Identities          152
                                             12.3 Green’s Functions      153
                                             12.4 Green’s Functions for the Disk and the Upper Half-Plane          154
                                             12.5 Analytic Functions      155
                                             12.6 Solving Dirichlet Problems with Conformal Mappings            160
                                             12.7 Green’s Functions and Conformal Mappings            165
                              A OrdinaryDifferential Equations:
                                      Review of Concepts and Methods                                                    A167
                                              A.1 Linear Ordinary Differential Equations         A167
                                              A.2 Linear Ordinary Differential Equations
                                                   with Constant Coefficients       A174
                                              A.3 Linear Ordinary Differential Equations
                                                   with Nonconstant Coefficients         A181
                                              A.4 The Power Series Method, Part I        A187
                                              A.5 The Power Series Method, Part II        A191
                                              A.6 The Method of Frobenius        A197