This book is a significant update of the first four chapters of Symmetries and Differential Equations (1989; reprinted with corrections, 1996), by George W. Bluman and Sukeyuki Kumei. Since 1989 there have been considerable developments in symmetry methods (group methods) for differential equations as evidenced by the number of research papers, books, and new symbolic manipulation software devoted to the subject. This is, no doubt, due to the inherent applicability of the methods to nonlinear differential equations. Symmetry methods for differential equations, originally developed by Sophus Lie in the latter half of the nineteenth century, are highly algorithmic and hence amenable to symbolic computation. These methods systematically unify and extend well-known ad hoc techniques to construct explicit solutions for differential equations, especially for nonlinear differential equations. Often ingenious tricks for solving particular differential equations arise transparently from the symmetry point of view, and thus it remains somewhat surprising that symmetry methods are not more widely known. Nowadays it is essential to learn the methods presented in this book to understand existing symbolic manipulation software for obtaining analytical results for differential equations. For ordinary differential equations (ODEs), these include reduction of order through group invariance or integrating factors. For partial differential equations (PDEs), these include the construction of special solutions such as similarity solutions or nonclassical solutions, finding conservation laws, equivalence mappings, and linearizations.
From the reviews of the second edition:
"The book by Bluman and Anco is an introduction to symmetry-based methods ... . The book is written in a clear and straightforward style, using many excellent worked examples ... . As someone who teaches an undergraduate course on symmetry methods, I would recommend Bluman and Anco’s book as a good source of examples and supplementary reading. ... for anyone wishing to master techniques for obtaining first integrals of ODEs, this book is outstanding." (Peter Hydon, UK Nonlinear News, June, 2003)
"The book under review devoted to symmetry and integration methods for differential equations is a significant update ... . The book has a Preface and Introduction well presenting its aim. ... Moreover, it has a final section Discussion which puts its contents into perspective by summarizing major results, by referring to related works and by introducing related material. ... the book should be particularly suitable for applied mathematicians, physicists and engineers. ... the publication of the present book should be highly welcome." (J. Synnatzschke, Zeitschrift für Analysis und ihre Anwendungen, Vol. 22 (1), 2003)
