Table of Contents

1. Mathematical Preliminaries.- 1.1 Introduction.- 1.2 Characteristics and Classification.- 1.3 Orthogonal Functions.- 1.4 Sturm-Liouville Boundary Value Problems.- 1.5 Legendre Polynomials.- 1.6 Bessel Functions.- 1.7 Results from Complex Analysis.- 1.8 Generalised Functions and the Delta Function.- 2. Separation of the Variables.- 2.1 Introduction.- 2.2 The Wave Equation.- 2.3 The Heat Equation.- 2.4 Laplace’s Equation.- 2.5 Homogeneous and Non-homogeneous Boundary Conditions.- 2.6 Separation of variables in other coordinate systems.- 3. First-order Equations and Hyperbolic Second-order Equations.- 3.1 Introduction.- 3.2 First-order equations.- 3.3 Introduction to d’Alembert’s Method.- 3.4 d’Alembert’s General Solution.- 3.5 Characteristics.- 3.6 Semi-infinite Strings.- 4. Integral Transforms.- 4.1 Introduction.- 4.2 Fourier Integrals.- 4.3 Application to the Heat Equation.- 4.4 Fourier Sine and Cosine Transforms.- 4.5 General Fourier Transforms.- 4.6 Laplace transform.- 4.7 Inverting Laplace Transforms.- 4.8 Standard Transforms.- 4.9 Use of Laplace Transforms to Solve Partial Differential Equations.- 5. Green’s Functions.- 5.1 Introduction.- 5.2 Green’s Functions for the Time-independent Wave Equation.- 5.3 Green’s Function Solution to the Three-dimensional Inhomogeneous Wave Equation.- 5.4 Green’s Function Solutions to the Inhomogeneous Helmholtz and Schrödinger Equations: An Introduction to Scattering Theory.- 5.5 Green’s Function Solution to Maxwell’s Equations and Time-dependent Problems.- 5.6 Green’s Functions and Optics: Kirchhoff Diffraction Theory.- 5.7 Approximation Methods and the Born Series.- 5.8 Green’s Function Solution to the Diffusion Equation.- 5.9 Green’s Function Solution to the Laplace and Poisson Equations.- 5.10Discussion.- A. Solutions of Exercises.

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