Generalized Similarity Analysis of Partial Differential Equations

Abstract

The term similarity transformation of a partial differential equation is defined to be a transformation of independent and dependent variables occurring in the equation such that the number of independent variables appearing in the transformed equation is at least one less than in the original equation. If a partial differential equation has two independent variables, a similarity transformation would transform the equation into an ordinary differential equation. In fact, the major application of similarity transformations has been the reduction of certain classes of nonlinear partial differential equations to ordinarydifferential equations. This chapter describes the methods of similarity analysis, which include free parameter, separation of variables, group theoretic, and dimensional analysis. It discusses the relative merits of these methods and, in particular, advantages and disadvantages. The so-called free parameter analysis is a method of finding similarity solutions of partial differential equations by assuming that the dependent variables occurring in the partial differential equation can be expressed as similarity parameters, the number of which is one less than the total number of independent variables that occur in the original partial differential equation.