Exact and Numerical Solutions of Time-Fractional type Advection--Diffusion Equation with a nonlinear source term by means of the Lie symmetries

In this paper, the authors analyze a time-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with a nonlinear source term. They determine the Lie symmetries and reduce the original fractional partial differential equation to a fractional ordinary differential equation (FODE). The authors solve the reduced fractional equation adopting the Caputo’s definition of derivatives of non-integer order in such a way the initial conditions have a physical meaning. The reduced fractional ordinary differential equation is approximated by the implicit second order backward differentiation formula. The analytical solutions, in terms of the Mittag-Leffler function for the linear fractional equation and numerical solutions, obtained by the finite difference method for the nonlinear fractional equation, are used to evaluate the solutions of the original advection-diffusion equation. Finally, comparisons between numerical and exact solutions and the error estimates show that the proposed procedure has a high convergence precision.