Symmetry Methods for Differential Equations

Abstract

Symmetry is the key to solving differential equations. There are many well-known techniques for obtaining exact solutions, but most of them are special cases of a few powerful symmetry methods. Furthermore, these methods can be applied to differential equations of an unfamiliar type; they do not rely on special 'tricks'. Instead, a given differential equation is forced to reveal its symmetries, which are then used to construct exact solutions. This book is a straightforward introduction to the subject, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily. The book contains methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.

The evolution of space curves by curvature and torsion

“…We apply the standard Lie group method to determine the point symmetries of equations (6)-(9) (see [6], for example) by seeking infinitesimal transformations of the form…”

“…In this case, the invariants of the global transformation cannot easily be found, so we use the invariant surface condition to determine the form of similarity solution in the usual way (see, for example, [6]) from the infinitesimal. This gives solutions of the form…”

“…The methods we use to derive the Lie point symmetries (and the similarity reductions) of PDEs (6)- (9), and hence of the velocity law (1), are described in [2,3,6]. A more rigorous treatment of the subject can be found in Olver [12].…”

Solução do tipo peakon para uma equação evolutiva de terceira ordem

“…Em um trabalho recente, encontramos o grupo de simetrias da equação, veja [3]. Para maiores detalhes sobre simetrias, veja [2].…”

Simetrias e soluções de equações diferenciais evolutivas envolvendo o laplaciano infinito

“…e F 2,1 (a, b, c, z)é dada em (11). Exceto pelo fato de a constante k em (15) ser a constante de separação utilizada no método de separação de variáveis, a solução obtida aquié essencialmente a mesma solução obtida anteriormente, utilizando-se simetria radial na equação (1).…”