Analytic Methods for Partial Differential Equations

Partial differential equations fall into several areas of mathematics: many of the greatest advances in modern science have been based on discovering the underlying PDE for the process in question. In this book, the emphasis is on the practical solution of problems rather than the theoretical background. An introductory chapter recaps the mathematical preliminaries required and exercises with solutions are provided. With its companion volume, Analytic Methods for Partial Differential Equations, it provides a complete introduction to the subject.

THIS BOOK IS THE COMPANION VOLUME TO ANALYTIC METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS. TOPICS ARE APPROACHED PRACTICALLY WITH THE EMPHASIS ON ACTUALLY SOLVING PROBLEMS CONTAINS NUMEROUS EXERCISES WITH WORKED SOLUTIONS. Includes supplementary material: sn.pub/extras

Contenu

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.