Numerical Solution of Distributed-Order Fractional Diffusion Equations Using a High-Order Temporal Scheme

doi:10.1007/s42967-025-00501-6

Abstract

Abstract: Distributed-order fractional diffusion equations (DO-FDEs) are crucial for modeling complex processes in heterogeneous and anomalous systems. Unfortunately, they encounter significant analytical and computational challenges. This study introduces a novel numerical framework that extends high-order approximation formulas to accommodate the distributed-order fractional derivative. The framework achieves a temporal convergence order of 4-β, where β is the upper bound of the integral in the distributed-order derivative. The proposed scheme using the finite element method (FEM) in the spatial direction, offers an accurate approach for solving DO-FDEs. We examine stability and convergence analyses to validate the method’s applicability. Additionally, numerical experiments confirm theoretical results.

Key words: Distributed-order fractional diffusion equation (DO-FDE), Finite element method (FEM), Gr?nwall inequality, Stability analysis, Convergence analysis

Mohadese Ramezani, Reza Mokhtari. Numerical Solution of Distributed-Order Fractional Diffusion Equations Using a High-Order Temporal Scheme[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1351-1365.

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URL: https://www.camc.shu.edu.cn/EN/10.1007/s42967-025-00501-6

https://www.camc.shu.edu.cn/EN/Y2026/V8/I4/1351

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Numerical Solution of Distributed-Order Fractional Diffusion Equations Using a High-Order Temporal Scheme

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