ABSTRACT
The similarity method is one of the standard methods for obtaining exact solutions of partial differential equations (PDE). The number of independent variables in a PDE is reduced by one by making use of appropriate combinations of the original independent variables as new independent variables, called “similarity variables.” The similarity variables can themselves be identified by using the invariance properties of PDEs when subjected to finite or infinitesimal transformations. The first kind of self-similar solutions are those for which the exponent in the similarity variable is determined by dimensional arguments alone; a famous example of this kind of solution is the Taylor-Sedov self-similar solution describing the point explosion in a uniform medium. The method using the group of infinitesimal transformations to identify the similarity form of the solution has been much in vogue in the last several decades. However, the latter may give a clue to what kind of exact or asymptotic solutions one may look for.
