Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations

In this article a symmetry group of scaling transformations is determined for a partial differential equation of fractional order alpha, containing among particular cases the diffusion equation, the wave equation, and the fractional diffusion-wave equation. For its group-invariant solutions, an ordinary differential equation of fractional order with the new independent variable z = xt(-alpha/2) is derived. The derivative then is an Erdelyi-Kober derivative depending on a parameter cu. Its complete solution is given in terms of the Wright and the generalized Wright functions. (C) 1998 Academic Press.