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<p>This paper presents a novel and efficient numerical methodology for solving time-fractional partial integro-differential equations with time delay, employing a combination of finite difference, spectral, and finite block methods on both square and non-rectangular regions. The Caputo derivative is discretized employing the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\hbox {L2-1}_\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>L2-1</mtext> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> scheme, while a L–type approximation is applied to the Riemann–Liouville fractional integral at the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n+\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> time level. For square domains, a fully–discrete scheme is constructed using Chebyshev spectral methods via the differentiation matrix. For irregular geometries, the finite block method is developed to handle domain complexity effectively. Theoretical analyses of stability and convergence are investigated, and a series of numerical experiments demonstrate the accuracy and computational efficiency of the proposed method.</p>

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An efficient numerical scheme for time-fractional partial integro-differential equations with time delay on rectangular and non-rectangular domains

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Amin Ghoreyshi,
  • Mahmoud A. Zaky,
  • Mostafa Abbaszadeh,
  • Mehdi Dehghan

摘要

This paper presents a novel and efficient numerical methodology for solving time-fractional partial integro-differential equations with time delay, employing a combination of finite difference, spectral, and finite block methods on both square and non-rectangular regions. The Caputo derivative is discretized employing the \(\hbox {L2-1}_\sigma \) L2-1 σ scheme, while a L–type approximation is applied to the Riemann–Liouville fractional integral at the \(n+\sigma \) n + σ time level. For square domains, a fully–discrete scheme is constructed using Chebyshev spectral methods via the differentiation matrix. For irregular geometries, the finite block method is developed to handle domain complexity effectively. Theoretical analyses of stability and convergence are investigated, and a series of numerical experiments demonstrate the accuracy and computational efficiency of the proposed method.