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.

Ch. 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 -- ch. 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 -- ch. 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 -- ch. 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.…