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 α, 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 variablez=xt−α/2is derived. The derivative then is an Erdelyi–Kober derivative depending on a parameter α. Its complete solution is given in terms of the Wright and the generalized Wright functions.