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

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 Gronwall inequality. The method is proven to be unconditionally stable and achieves optimal convergencerates in both time and space. Numerical simulations confirm the theoretical predictions and demonstrate the accuracyand efficiency of the scheme in capturing the underlying fractional dynamics. The proposed framework offers a powerfuland flexible tool for the numerical simulation of fractional-order systems with distributed memory, and can be extendedto a wide range of multi-term and distributed-order fractional partial differential equations.

کلیدواژه ها:

Distributed-order fractional differential equations ، cable equation ، fractional Adams--Bashforth--Moulton method ، Stability analysis ، Numerical simulation