A stable and convergent fully discrete scheme for solving two-dimensional distributed-order fractional cable models

Rezaei, Hamid; Asadipour, Meysam; Derakhshan, Mohammad Hossein

doi:10.22124/jmm.2026.32479.2945

A stable and convergent fully discrete scheme for solving two-dimensional distributed-order fractional cable models

Articles in Press

Document Type : Research Article

Authors

¹ Department of Mathematics, College of Sciences, Yasouj University, Yasouj-, 75914-74831, Iran

² Department of Mathematics, College of Sciences, Yasouj University, Yasouj-75914-74831, Iran

10.22124/jmm.2026.32479.2945

Abstract

This paper investigates a novel distributed-order time-fractional cable equation involving both Caputo and Riemann–
Liouville fractional derivatives, which models complex diffusion and memory effects in various physical and
biological systems. The proposed model incorporates a distributed-order fractional Laplacian term, a memory integral,
and a nonlinear source, capturing multiscale temporal dynamics and nonlocal behavior. A robust numerical scheme
is developed by applying a fractional Adams–Bashforth–Moulton predictor-corrector method for time discretization,
while central difference approximations are used for the spatial Laplacian. This results in a fully discrete scheme
that effectively combines the advantages of convolution quadrature with classical finite difference methods. A detailed
convergence and stability analysis of the numerical method is presented using an energy-based approach and a discrete
fractional Gr¨onwall inequality. The method is proven to be unconditionally stable and achieves optimal convergence
rates in both time and space. Numerical simulations confirm the theoretical predictions and demonstrate the accuracy
and efficiency of the scheme in capturing the underlying fractional dynamics. The proposed framework offers a powerful
and flexible tool for the numerical simulation of fractional-order systems with distributed memory, and can be extended
to a wide range of multi-term and distributed-order fractional partial differential equations.

Keywords