Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation

In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection–diffusion–reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection–diffusion–reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.