Overview

Many of the equations modeling complex dynamical systems in the natural and engineering sciences are partial differential equations. For these equations the independent variables are usually the time and one or more space variables. It is important to note that a vast number of scientific and technologically interesting phenomena can be modeled in terms of a relatively small set of linear and nonlinear partial differential equations. Of further significance is the fact that several techniques exist to calculate special solutions to these equations and these particular solutions are often the ones needed to analyze the phenomena of interest.

The major goals of this chapter are to introduce some special nonlinear partial differential equations and the techniques needed for determining particular solutions to them. The bibliography, at the end of the chapter, provides a list of books on both the theory and application of linear and nonlinear partial differential equations. Two excellent introductions to this subject are the books by Logan and Myint-U. Several books on specific application topics are also given:

• combustion (Bebernes and Eberly; Fickett),

• fluid flow (Courant and Friedrichs; Kreiss and Cole),

• mathematical biosciences (Britton; Murray; Okubo);

• perturbation methods (Kevorkian and Cole);

• reaction-diffusion processes (Britton; Fife; Smoller).

The book of Lin and Segel is well-written and contains many applications of both linear and nonlinear differential equations to interesting problems from several of the natural sciences. Another book giving the model based genesis of certain partial differential equations and various methods for determining solutions and/or their mathematical properties...