Students’ Solutions Manual
                        PARTIAL DIFFERENTIAL
                               EQUATIONS
                            with FOURIER SERIES and
                          BOUNDARYVALUEPROBLEMS
                                 Third Edition
                                    ´
                              NAKHLEH.ASMAR
                               University of Missouri
                                         Contents
                           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   3
                           2       Fourier Series                                                                 9
                                          2.1 Periodic Functions    9
                                          2.2 Fourier Series   15
                                          2.3 Fourier Series of Functions with Arbitrary Periods   21
                                          2.4 Half-Range Expansions: The Cosine and Sine Series      29
                                          2.5 Mean Square Approximation and Parseval’s Identity       32
                                          2.6 Complex Form of Fourier Series     36
                                          2.7 Forced Oscillations    41
                                         Supplement on Convergence
                                          2.9 Uniform Convergence and Fourier Series     47
                                         2.10 Dirichlet Test and Convergence of Fourier Series   48
                           3       Partial Differential Equations in Rectangular Coordinates                      49
                                          3.1 Partial Differential Equations in Physics and Engineering    49
                                          3.3 Solution of the One Dimensional Wave Equation:
                                               The Method of Separation of Variables   52
                                          3.4 D’Alembert’s Method      60
                                          3.5 The One Dimensional Heat Equation       69
                                          3.6 Heat Conduction in Bars: Varying the Boundary Conditions       74
                                          3.7 The Two Dimensional Wave and Heat Equations         87
                                          3.8 Laplace’s Equation in Rectangular Coordinates     89
                                          3.9 Poisson’s Equation: The Method of Eigenfunction Expansions       92
                                         3.10 Neumann and Robin Conditions        94
                                                                                                                 Contents  iii
                              4       Partial Differential Equations in
                                      Polar and Cylindrical Coordinates                                                   97
                                              4.1 The Laplacian in Various Coordinate Systems          97
                                              4.2 Vibrations of a Circular Membrane: Symmetric Case           99
                                              4.3 Vibrations of a Circular Membrane: General Case          103
                                              4.4 Laplace’s Equation in Circular Regions       108
                                              4.5 Laplace’s Equation in a Cylinder       116
                                              4.6 The Helmholtz and Poisson Equations         119
                                                 Supplement on Bessel Functions
                                              4.7 Bessel’s Equation and Bessel Functions       124
                                              4.8 Bessel Series Expansions      131
                                              4.9 Integral Formulas and Asymptotics for Bessel Functions         141
                              5       Partial Differential Equations in Spherical Coordinates                             142
                                              5.1 Preview of Problems and Methods         142
                                              5.2 Dirichlet Problems with Symmetry         144
                                              5.3 Spherical Harmonics and the General Dirichlet Problem           147
                                              5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
                                                   and Wave Equations 153
                                             Supplement on Legendre Functions
                                              5.5 Legendre’s Differential Equation       156
                                              5.6 Legendre Polynomials and Legendre Series Expansions            162
                              6       Sturm–Liouville Theory with Engineering Applications                               167
                                              6.1 Orthogonal Functions       167
                                              6.2 Sturm–Liouville Theory        169
                                              6.3 The Hanging Chain        172
                                              6.4 Fourth Order Sturm–Liouville Theory          174
                                              6.6 The Biharmonic Operator         176
                                              6.7 Vibrations of Circular Plates      178
              iv  Contents
                              7       The Fourier Transform and Its Applications                                          179
                                              7.1 The Fourier Integral Representation        179
                                              7.2 The Fourier Transform        184
                                              7.3 The Fourier Transform Method          193
                                              7.4 The Heat Equation and Gauss’s Kernel           201
                                              7.5 A Dirichlet Problem and the Poisson Integral Formula           210
                                              7.6 The Fourier Cosine and Sine Transforms          213
                                              7.7 Problems Involving Semi-Infinite Intervals         217
                                              7.8 Generalized Functions       222
                                              7.9 The Nonhomogeneous Heat Equation             233
                                             7.10 Duhamel’s Principle       235
                              8       The Laplace and Hankel Transforms with Applications                                 238
                                              8.1 The Laplace Transform        238
                                              8.2 Further Properties of the Laplace transform        246
                                              8.3 The Laplace Transform Method           258
                                              8.4 The Hankel Transform with Applications          262
                              12 Green’s Functions and Conformal Mappings                                                 268
                                             12.1 Green’s Theorem and Identities        268
                                             12.2 Harmonic Functions and Green’s Identities          272
                                             12.3 Green’s Functions      274
                                             12.4 Green’s Functions for the Disk and the Upper Half-Plane           276
                                             12.5 Analytic Functions      277
                                             12.6 Solving Dirichlet Problems with Conformal Mappings            286
                                             12.7 Green’s Functions and Conformal Mappings            296
                              A OrdinaryDifferential Equations:
                                      Review of Concepts and Methods                                                    A298
                                              A.1 Linear Ordinary Differential Equations         A298
                                              A.2 Linear Ordinary Differential Equations
                                                   with Constant Coefficients       A308
                                              A.3 Linear Ordinary Differential Equations
                                                   with Nonconstant Coefficients         A322
                                              A.4 The Power Series Method, Part I        A333
                                              A.5 The Power Series Method, Part II        A340
                                              A.6 The Method of Frobenius        A348