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 ␣ , 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 s xt y␣ r2 is derived. The derivative then is an Erdelyi᎐Kober derivative depending on a parameter ␣. Its complete solution is given in terms of the Wright and the generalized Wright functions.

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

“…In particular, it has been proved in [Buckwar and Luchko (1998)] that the only invariant of the symmetry group T λ of scaling transformations of the time-fractional diffusion-wave equation (1) has the form η(x, t, u) = x/t ν that explains the form of the scaling variable. Using the well known representation of the Wright function, which reads (in our notation) for z ∈ C…”

Lie group method and fractional differential equations

“…We complete this section in the light of references [4,6,18,25,26]. Lie theory allows us to assume that the nonlinear system of fractional differential equations (1.1) is invariant under the one parameter transformations (2.2).…”

Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

“…Note also, that in some important applications, we have a = 0. Apart from that, as can be seen from the self-similar analysis of the time-anomalous diffusion equation (see [9,12,42]), the particular version of the E-K operator that arises there requires c < 0. However, we defer the analysis of such case to our future work and in the present paper we assume that c > 0.…”

“…As both Gamma and Beta functions are readily and optimally implemented in many popular scientific software packages, we almost never need to compute the integral in (9). The important special case, a = 0, can be evaluated explicitly…”