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Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations

Abstract 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 variable z  =  xt  − α/2 is 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.