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<p>In this paper, a compact finite difference method is used for numerically solving the time-fractional diffusion equation with Caputo derivative. In the construction of the discrete scheme, the temporal and spatial terms are treated separately, where the time-fractional term is discretized by using a numerical approximation based on Lagrange interpolation polynomials, and the spatial term is discretized by using the compact finite difference scheme. The constructed numerical scheme achieves a convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(\tau^{4-\alpha}+h^6)\)</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(h\)</EquationSource> </InlineEquation> representing the temporal and spatial discretization parameters. The work contributes to a better knowledge of the dependability of the method by thoroughly examining convergence and error analysis. A rigorous stability analysis is conducted, further confirming the robustness of the proposed method. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research.</p>

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An asymptotic computational behavior of a time-fractional differential equation: a robust approach

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Yuelong Feng,
  • Xindong Zhang,
  • Leilei Wei,
  • Juan Liu

摘要

In this paper, a compact finite difference method is used for numerically solving the time-fractional diffusion equation with Caputo derivative. In the construction of the discrete scheme, the temporal and spatial terms are treated separately, where the time-fractional term is discretized by using a numerical approximation based on Lagrange interpolation polynomials, and the spatial term is discretized by using the compact finite difference scheme. The constructed numerical scheme achieves a convergence rate of \(O(\tau^{4-\alpha}+h^6)\) , with \(\tau\) and \(h\) representing the temporal and spatial discretization parameters. The work contributes to a better knowledge of the dependability of the method by thoroughly examining convergence and error analysis. A rigorous stability analysis is conducted, further confirming the robustness of the proposed method. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research.