Solution of Ordinary Differential Equations by Continuous Groups

Description

ch. 3 is on method of characteristics

cited on EqWorld

The book deals with the theory of Lie groups applied to solve ordinary differential equations. The emphasis is much more on the practical rather than the theoretical aspects of the subject. Accordingly, no excessive weight is given to mathematical proofs and derivations, which are nevertheless provided when necessary. Background material is required only up to a minimum level, and necessary tools (some group theory, the method of characteristics, etc.) are developed in the course of the text. The mathematical apparatus usually required in other books dedicated to continuous groups and differential equations is not required in this work. This mathematical apparatus is a consequence of the link between Lie algebra theory, one of the major research areas of mathematics, and Lie groups. This is certainly one of the main reasons why Lie groups for differential equations have not even more widespread use. However, not much abstraction is needed as long as one is interested only in solving differential equations by Lie group techniques. This work comes to present, in a self-contained way and in a clear style, the Lie groups symmetry approach for solving ordinary differential equations. No exhaustive material is presented. The topics covered include some theory of groups of continuous transformations, the method of characteristics, invariance concepts, the theory of extended groups, canonical coordinates for a Lie group, application of Lie groups to first- and second-order ordinary differential equations and to systems of first-order ordinary differential equations. Some new techniques are shown here for the first time, including the enlargement procedure created by the author. Also, the work includes tables of ordinary differential equations invariant under specified groups. Worked out examples and exercises for the reader are provided throughout the book.

Reviewed by Fernando Haas

Written by an engineer and sharply focused on practical matters, this text explores the application of Lie groups to solving ordinary differential equations (ODEs). Although the mathematical proofs and derivations in are de-emphasized in favor of problem solving, the author retains the conceptual basis of continuous groups and relates the theory to problems in engineering and the sciences.The author has developed a number of new techniques that are published here for the first time, including the important and useful enlargement procedure. The author also introduces a new way of organizing tables reminiscent of that used for integral tables. These new methods and the unique organizational scheme allow a significant increase in the number of ODEs amenable to group-theory solution.Solution of Ordinary Differential Equations by Continuous Groups offers a self-contained treatment that presumes only a rudimentary exposure to ordinary differential equations. Replete with fully worked examples, it is the ideal self-study vehicle for upper division and graduate students and professionals in applied mathematics, engineering, and the sciences.