Symmetries and Differential Equations

Description

Bluman & Cole (1969) was cited in the last chapter of Gilmore 2008 as a book on how to use Lie methods for solving differential equations

This is Bluman's most highly cited book on Lie ODE/PDE methods according to BookMetrix.

Bluman & Anco (2002) is a revision of the first 4 chapters.

A major portion of this book discusses work which has appeared since the publication of the book Similarity Methods for Differential Equations, Springer-Verlag, 1974, by the first author and J.D. Cole. The present book also includes a thorough and comprehensive treatment of Lie groups of tranformations and their various uses for solving ordinary and partial differential equations. No knowledge of group theory is assumed. Emphasis is placed on explicit computational algorithms to discover symmetries admitted by differential equations and to construct solutions resulting from symmetries. This book should be particularly suitable for physicists, applied mathematicians, and engineers. Almost all of the examples are taken from physical and engineering problems including those concerned with heat conduction, wave propagation, and fluid flows. A preliminary version was used as lecture notes for a two-semester course taught by the first author at the University of British Columbia in 1987-88 to graduate and senior undergraduate students in applied mathematics and physics. Chapters 1 to 4 encompass basic material. More specialized topics are covered in Chapters 5 to 7.

Since the early 1960s there has been a growing interest and considerable development in symmetry methods (group methods) for differential equations. Much of this is due to their utility in the analysis of nonlinear differential equations. Symmetry methods are very algorithmic. A number of symbolic packages exist for the calculation of the classical symmetry groups and their extensions. This eliminates the tedium in their calculation and makes their use more amenable.
The availability of the symmetry group permits unification and extension of existing ad hoc methods for the construction of explicit solutions for differential equations, in particular for nonlinear ones. Very often ingenious techniques for solving particular differential equations become obvious from the group point of view. It is surprising that symmetry methods are not more widely used. Hopefully, this will change as a result of this book and that of P. J. Olver [Applications of Lie groups to differential equations, Springer, New York, 1986; MR0836734] and C. Rogers and the reviewer [Nonlinear boundary value problems in science and engineering, Academic Press, Boston, MA, 1989; MR1017035]. These books contain many detailed examples and the latter has a large table of existing groups.
The present work includes a comprehensive treatment of Lie groups of transformations and a thorough discussion of basic symmetry methods for solving ordinary and partial differential equations. No knowledge of group theory is assumed. The emphasis is placed on the development of explicit computational algorithms to find symmetry groups and to apply them to construct solutions resulting from the symmetries.
The contents are as follows: 1. Dimensional analysis, modelling and invariance; 2. Lie groups of transformations and infinitesimal transformations; 3. Ordinary differential equations; 4. Partial differential equations; 5. Noether's theorem and Lie-Bäcklund symmetries; 6. Construction of mappings relating differential equations; 7. Potential symmetries; References; Author index; Subject index.
This is an excellent book, prepared with erudition and skill. It is suitable both as a reference and as a text. It has been used in courses at the University of British Columbia. Reviewed by W. F. Ames