University of Guilan Journal of Mathematical Modeling 2345-394X 14 3 2026 08 01 A stable and convergent fully discrete scheme for solving two-dimensional distributed-order fractional cable models 891 907 9421 10.22124/jmm.2026.32479.2945 EN Hamid Rezaei Department of Mathematics, College of Sciences, Yasouj University, Yasouj-, 75914-74831, Iran Meysam Asadipour Department of Mathematics, College of Sciences, Yasouj University, Yasouj-75914-74831, Iran Mohammad Hossein Derakhshan Department of Mathematics, College of Sciences, Yasouj University, Yasouj-75914-74831, Iran Journal Article 2025 12 08 This paper investigates a novel distributed-order time-fractional cable equation involving both Caputo and Riemann–<br />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 convergence<br />rates in both time and space. Numerical simulations confirm the theoretical predictions and demonstrate the accuracy<br />and efficiency of the scheme in capturing the underlying fractional dynamics. The proposed framework offers a powerful<br />and flexible tool for the numerical simulation of fractional-order systems with distributed memory, and can be extended<br />to a wide range of multi-term and distributed-order fractional partial differential equations. https://jmm.guilan.ac.ir/article_9421_1b45020e2f34cc4e3e7e8e1335df757d.pdf