Analytic methods for partial differential equations / G. Evans, J. Blackledge and P. Yardley

The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. This book contain five chapters. In the first chapter some of the mathematical preliminaries are given (including orthogonal polynomials, special functions and a brief coverage of complex variables). The use of characteristics to classify partial differential equations leads to specific techniques in the following chapters. This is supported by brief derivations of the wave equation, the heat equation and Laplace's equation. The chapter is concluded with some background to generalized functions for use in the final chapter. The second chapter contains conventional coverage of separation of variables, applied to the heat equation and Laplace's equation in Cartesian, cylindrical polar and spherical polar coordinates. Chapter 3 is concerned with solutions involving characteristic curves, and seemed the natural place for first-order equations. Attention is paid to second-order equations and D'Alembert's solution of the wave equation, including the method of characteristics in an analytic setting. Integral transforms are covered in Chapter 4. The final chapter is on Green's functions. Topics here include Green's functions for the wave equation, the diffusion equation and Laplace's equation; Helmholtz and Schrödinger's equations with applications to scattering theory; Maxwell's equations; and Green's functions in optics with Kirchhoff diffraction theory. Approximation methods and Born series are also considered briefly. Most sections have a set of exercises, and fairly complete solutions can be found in the appendix. The exercises and solutions form an important part of the book and provide much insight to the ideas introduced in the text. Advanced undergraduate and non-specialist graduate students will find this book an invaluable and comprehensive introduction to the subject. This book can be recommended as a well written undergraduate text. (Zentralblatt)

Bibliogr. p. 293-295. Index