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<p>In this study, a novel distributed-order fractional partial differential equation model is proposed to describe complex dynamic processes with memory effects and non-local temporal behavior. The temporal derivative is defined in the sense of the Bessel fractional operator, which allows for an accurate representation of the underlying physical phenomena. A numerical solution is obtained by employing the Fractional Shifted Gegenbauer Polynomial (FSGP) method in time combined with a high-order finite difference scheme in space, providing an efficient and accurate approximation of the solution. Error analysis and convergence studies are conducted, and the method is shown to be both stable and convergent, with high accuracy in capturing temporal and spatial dynamics. The numerical results are presented in tables and figures, confirming the reliability and effectiveness of the proposed approach.</p>

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A numerical study of distributed-order fractional PDEs with Bessel time derivatives using fractional shifted Gegenbauer polynomials

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Ali Mankhi Jebur Alaayedi,
  • Safar Irandoust Pakchin,
  • Mohammad Hossein Derakhshan

摘要

In this study, a novel distributed-order fractional partial differential equation model is proposed to describe complex dynamic processes with memory effects and non-local temporal behavior. The temporal derivative is defined in the sense of the Bessel fractional operator, which allows for an accurate representation of the underlying physical phenomena. A numerical solution is obtained by employing the Fractional Shifted Gegenbauer Polynomial (FSGP) method in time combined with a high-order finite difference scheme in space, providing an efficient and accurate approximation of the solution. Error analysis and convergence studies are conducted, and the method is shown to be both stable and convergent, with high accuracy in capturing temporal and spatial dynamics. The numerical results are presented in tables and figures, confirming the reliability and effectiveness of the proposed approach.