Adaptive numerical solutions of time-fractional advection–diffusion–reaction equations

In this paper, we propose an adaptive procedure, recently developed for fractional ordinary differential equations, for the solutions of time-fractional advection–diffusion–reaction equations involving the Caputo derivative. We focus on the adaptivity of the discretization in time direction defining a step size selection function that allows adapting the time step size according to the local behaviour of the solution. The new approach is easy to implement and reveals to have a low computational cost. Test problems are reported and comparisons with results found in literature confirm the accuracy and efficiency of the step-size selection procedure for the solutions of fractional partial differential equations.