Boundary Value Problems

This Book Has Been Designed For A One-year Graduate Course On Boundary Value Problems For Students Of Mathematics, Engineering, And The Physical Sciences. It Deals Mainly With The Three Fundamental Equations Of Mathematical Physics, Namely The Heat Equation, The Wave Equation, And Laplace's Equation. The Goal Of The Book Is To Obtain A Formal Solution To A Given Problem Either By The Method Of Separation Of Variables Or By The Method Of General Solutions And To Verify That The Formal Solution Possesses All The Required Properties. To Provide The Mathematical Justification For This Approach, The Theory Of Sturm–liouville Problems, The Fourier Series, And The Fourier Transform Are Fully Developed. The Book Assumes A Knowledge Of Advanced Calculus And Elementary Differential Equations. Contents: Linear Partial Differential Equationsthe Wave Equationgreen's Function And Sturm–liouville Problemsfourier Series And Fourier Transformsthe Heat Equationlaplace's Equation And Poisson's Equationproblems In Higher Dimensions Readership: Graduate Students In Applied Mathematics, Engineering And The Physical Sciences. Keywords:boundary Value Problems;green's Function;sturm-liouville Problems;symmetric Integral Operator;eigenvalues And Eigenfunctions;fourier Series And Fourier Transforms;heat Equation;wave Equation;laplace's Equation;bessel Functions;legendre Polynomials 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 This text has been designed for use on a one-year graduate course on boundary value problems. It deals mainly with the three fundamental equations of mathematical physics, namely the heat equation, the wave equation, and Laplace's equation.