Approximate Symmetries of Partial Differential Equations

Abstract

The theory of approximate symmetries was developed by Baikov, Gazizov, and Ibragimov [1989] in the 1980s. The idea behind this development was the extension of Lie’s theory to situations in which a small perturbation of the original equation is encountered. For such cases, the question arises of how the point symmetries or the group of the equations are altered if a small perturbation is added to the original equation. This question initiated the development of a group analysis method that is stable under small perturbations of the differential equation. The present chapter discusses the method of approximate symmetries. The method is based on the concept of an approximate group of transformations. Approximate symmetries are useful for partial differential equations depending on a small parameter ∈. This parameter is usually used in the standard theories to examine the differential equation in some limit. On the other hand, this parameter is also useful in the examination of Lie point symmetries.