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.

在这项工作中，我们考虑具有分数扩散的抛物线偏微分方程，该方程推广了著名的Fisher方程和Hodgkin-Huxley方程。从Riesz的角度理解空间分数导数，并在封闭和有界区间上考虑初始边界条件。以等价形式表示数学模型，并提出了基于分数阶中心差的有限差分离散化。该方案是文献中提出的第一个显式对数模型，用于解决分数阶扩散反应方程。我们严格建立了该技术保持方法学的积极性和有界性的能力。此外，我们提出了在其中还保留数值模型的单调性的条件。一致性，提出了数值解的先验边界。我们提供一些数值模拟，以确认该方法能够保留逼近性的正值和有界性，并对该技术的收敛性进行了数值研究，从而确定了分析结果。