Group Analysis of Differential Equations

Description

Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the given differential equation with a known admitted group. The theory of differential invariants that is developed on an infinitesimal basis is elaborated in Chapter VII. The last chapter outlines the ways in which the methods of group analysis are used in special issues involving differential equations.This publication is a good source for students and specialists concerned with areas in which ordinary and partial differential equations play an important role.

Bluman & Anco (2002) cite this.

MR0511921

This book presents the classical theory of Lie transformation groups and its application to the study of systems of partial differential equations invariant under such groups. The reader will not find any general mechanism here since the author deals with neither the formal theory of systems of p.d.e. nor Lie pseudogroups. It is to be regretted that the few non-Russian references used throughout do not include the modern works of Spencer and others on these subjects. However, this book provides many nice examples coming from physics and mechanics.
We indicate below the main points of each chapter: (I) prolongations of one-parameter groups of transformations, (II) construction of the one-parameter groups of transformations preserving a system of p.d.e., relations between the bracket and prolongation; (III) dependence of the group of invariance on the form of arbitrary parametric functions involved in the system (e.g., Navier-Stokes p.d.e.), (IV) the fundamental theorems of Lie and the Maurer-Cartan structure equations, (V) invariants of involutive distributions of vector fields, (VI) partially invariant solutions of systems of p.d.e. admitting a group, (VII) differential invariants of extended Lie transformation groups and automorphic systems, (VIII) classification of second order linear p.d.e., Appendix: List of systems and their corresponding invariance groups. Reviewed by J.-F. Pommaret