Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term

In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.