A note on the adaptive numerical solution of a Riemann–Liouville space-fractional Kawarada problem

Abstract

This paper concerns the approximation and numerical solution of a singular fractional reaction–diffusion problem. A Riemann–Liouville space-fractional derivative oriented Laplacian is incorporated. While our spatial discretization is fulfilled though combined standard and shifted Grünwald formulas, temporal integration is accomplished via an implicit adaptive Crank–Nicolson scheme. It is proven that under proper constraints of the spatial and temporal discretization parameters, the numerical procedure implemented is positive, monotone and numerically stable. Simulation experiments are given to validate correlations between the fractional derivative and critical values including critical lengths, quenching times and locations.