| The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier＇s famous work on series expansions for the heat equation. |
| 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 1.8.1 Definition and Properties of a Generalised Function 1.8.2 Differentiation Across Discontinuities 1.8.3 The Fourier Transform of Generalised Functions 1.8.4 Convolution of Generalised Functions 1.8.5 The Discrete Representation of 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 Schr6dinger 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 5.10 Discussion A. Solutions of Exercises Bibliography Index |
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