A positive and bounded convergent scheme for general space-fractional diffusion-reaction systems with inertial times

Abstract

ABSTRACT We consider a multidimensional system of hyperbolic equations with fractional diffusion, constant damping and nonlinear reactions. The system considers fractional Riesz derivatives, and generalizes many models from science. In particular, the system describes the dynamics of populations with temporal delays, whence the need to approximate nonnegative and bounded solutions is an important numerical task. Motivated by these facts, we propose a scheme to approximate the solutions. We prove the existence of the solutions under suitable regularity assumptions on the reaction functions. We prove that the scheme is capable of preserving positivity and boundedness. The technique has consistency of the second order in space and time. Using a discrete form of the energy method, we establish the stability and the convergence. As a corollary, we prove the uniqueness of the solutions. Some computer simulations in the two- and the three-dimensional scenarios are provided at the end of this work for illustration purposes.