Similarity solutions of nonlinear partial differential equations invariant to a family of affine groups

Abstract

This paper deals with the similarity solutions of second-order partial differential equations in one dependent and two independent variables that are invariant to a one-parameter family of one-parameter affine (stretching) groups. (Similarity solutions are solutions of the partial differential equation that are invariant to one group of the family.) The similarity solutions can be calculated merely by solving an ordinary differential equation of second order. It is shown that this ordinary differential equation is itself invariant to a group of affine transformations. This allows use of a theorem of Lie's, according to which the second-order ordinary differential equation can be reduced to one of first order. So the computational task for the class of problems considered here can be reduced to the solution of a first-order ordinary differential equation. An example, referring to heat transport in superfluid helium (HE-II), is worked in detail.