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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(4 - \beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>-</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Mohadese Ramezani,
  • Reza Mokhtari

摘要

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 \(4 - \beta \) 4 - β , where \(\beta \) 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.