Boundary Value Problems

Chapter 1  Linear Partial Differential Equations
     1.1 Linear problems
     1.2 Classification
     1.3 Well-posed problems
     1.4 Method of general solutions
     1.5 Method of separation of variables
     1.6 Problems

Chapter 2  The Wave Equation
     2.1 The vibrating string
     2.2 The initial value problem
     2.3 The nonhomogeneous wave equation
     2.4 Uniqueness of the initial value problem
     2.5 Initial-boundary value problems
     2.6 Initial-boundary value problems for semi-infinite string
     2.7 Problems

Chapter 3  Green's Function and Sturm-Liouville Problems
     3.1 Solutions of second order linear equations
     3.2 Boundary value problems and Green's function
     3.3 Sturm-Liouville problems
     3.4 Convergence in the mean
     3.5 Integral operator with continuous, symmetric kernel
     3.6 Completeness of eigenfunctions of Sturm-Liouville problems
     3.7 Nonhomogeneous integral equation
     3.8 Further properties of eigenvalues and eigenfunctions
     3.9 Problems

Chapter 4  Fourier Series and Fourier Transforms
     4.1 Trigonometric Fourier series
     4.2 Uniform convergence and completeness
     4.3 Other types of Fourier series
     4.4 Application to the wave equation
     4.5 Fourier integrals
     4.6 Fourier transforms
     4.7 Contour integration
     4.8 Problems

Chapter 5  The Heat Equation
     5.1 Derivation of the heat equation
     5.2 Maximum Principle
     5.3 The initial-boundary value problem
     5.4 Nonhomongeneous problems and finite Fourier transform
     5.5 The initial value problem
     5.6 The initial value problem for the nonhomogeneous equation
     5.7 Nonhomogeneous boundary conditions for initial-boundary value problems
     5.8 Problems

Chapter 6  Laplace's Equation and Poisson's Equation
     6.1 Boundary value problems
     6.2 Green's identities and uniqueness theorems
     6.3 Maximum principle
     6.4 Laplace's equation in a rectangle
     6.5 Laplace's equation in a disc
     6.6 Poisson's integral formula
     6.7 Green's function for Laplace's equation
     6.8 Poisson's equation in a disc
     6.9 Finite Fourier transform for Poisson's equation
     6.10 Dirichlet problem in the upper-half plane
     6.11 Problems

Chapter 7  Problems in Higher Dimensions
     7.1 Classification
     7.2 Double Fourier series
     7.3 Laplace's equation in a cube
     7.4 The two-dimensional wave equation in a rectangular domain
     7.5 Bessel functions
     7.6 Singular Sturm-Liouville problem for Bessel's equation
     7.7 The two-dimensional wave equation in a circular domain
     7.8 Initial-boundary value problems for the heat equation
     7.9 Legendre's equation
     7.10 Properties of Legendre polynomials
     7.11 Legendre series and boundary value problems
     7.12 Laplace's equation in a sphere
     7.13 Poisson's integral formula in space
     7.14 Problems

Appendix A  Ascoli's Theorem

Appendix B  Answers for Selected Problems
     Chapter 1
     Chapter 2
     Chapter 3
     Chapter 4
     Chapter 5
     Chapter 6
     Chapter 7

Bibliography
Index