THE SIMILARITY SOLUTION FOR SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Abstract

It has been seen that mathematical models of a diversity of real world phenomena consist of nonlinear differential equations subject to appropriate auxiliary conditions. In this chapter, invariance under group of Lie transformations is described in detail and the resulting qualitative properties and symmetry reductions.

A systematic investigation of continuous transformation groups was carried out by Lie (1882-1899). His original goal was the creation of a theory of integration for ode analogous to the Abelian theory for the solution of algebraic equations. He investigated the fundamental concept of the invariance group adimitted by a given system of differential equations. Today the mathematical approach whose object is the construction and analysis of the full invariance group admitted by a system of differential equations is called group analysis of differential equations. These groups, now, usually called Lie groups, and the associated Lie algebras have important real world applications.