Design and numerical analysis of a logarithmic scheme for nonlinear fractional diffusion–reaction equations

Abstract

In this work, we consider a parabolic partial differential equation with fractional diffusion that generalizes the well-known Fisher’s and Hodgkin–Huxley equations. The spatial fractional derivatives are understood in the sense of Riesz, and initial–boundary conditions on a closed and bounded interval are considered here. The mathematical model is presented in an equivalent form, and a finite-difference discretization based on fractional-order centered differences is proposed. The scheme is the first explicit logarithmic model proposed in the literature to solve fractional diffusion–reaction equations. We establish rigorously the capability of the technique to preserve the positivity and the boundedness of the methodology. Moreover, we propose conditions under which the monotonicity of the numerical model is also preserved. The consistency, the stability and the convergence of the scheme are also proved mathematically, and some a priori bounds for the numerical solutions are proposed. We provide some numerical simulations in order to confirm that the method is capable of preserving the positivity and the boundedness of the approximations, and a numerical study of the convergence of the technique is carried out confirming, thus, the analytical results.