Approximate symmetry of time-fractional partial differential equations with a small parameter

Abstract

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.