Methods for Constructing Exact Solutions of Partial Differential Equations

Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations. Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Each solution of a system of differential equations corresponds to a particular process. Therefore, methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations. The emphasis is on the methods of differential constraints, degenerate hodograph and group analysis. These methods have become a necessary part of applied mathematics and mathematical physics. The book is primarily designed to present both fundamental theoretical and algorithmic aspects of these methods. The description of algorithms contains illustrative examples which are typically taken from continuum mechanics. Some sections of the book introduce new applications and extensions of these methods, such as integro-differential and functional differential equations, a new area of group analysis.§It should also be noted that the method of differential constraints is not well known outside Russia; there are only a few books in English where the idea behind this method (without analysis) is briefly mentioned. This book is a result of an effort to introduce, at a fairly elementary level, many methods for constructing exact solutions collected in one book. It is based on the author's research and various courses and lectures given to undergraduate and graduate students, as well as professional audiences over the past twenty-five years. The book is assembled, in a coherent and comprehensive way, from results that are scattered across many different articles and books published over the last thirty years.§The approach is analytical. Introductions to theories are followed by examples. The book is written for students, engineers and scientists with diverse backgrounds and interests. For a deeper coverage of a particular method or an application, the readers are referred to special-purpose books and/or scientific articles referenced in the book. The prerequisites for the study are standard courses in calculus, linear algebra, and ordinary and partial differential equations.