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<p>This manuscript proposes an efficient hybrid numerical approach that combines high accuracy with low computational cost to approximate solutions of the time-space diffusion model governed by the Caputo and Riesz fractional derivatives. Addressing the fractional time derivative in the Caputo sense, we employ a combination of quadratic and linear interpolations, achieving an accuracy of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7287_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {O}\big \{\big (\Delta t\big )^{2-\gamma }\big )\big \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mrow> <mn>2</mn> <mo>-</mo> <mi>γ</mi> </mrow> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For the spatial discretisation of the Riesz space-fractional operator, we utilise a compact finite difference scheme with fourth-order accuracy. The stability and convergence properties of the proposed numerical method are rigorously analysed and verified. Finally, we provide with several numerical examples, supplemented by graphical and tabular illustrations, to demonstrate the accuracy, efficiency, and robustness of the proposed approach.</p>

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A hybrid numerical method with high accuracy to solve a time-space diffusion model in terms of the Caputo and Riesz fractional derivatives

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Mohammad Hossein Derakhshan,
  • Yadollah Ordokhani,
  • Pushpendra Kumar,
  • J. F. Gómez-Aguilar

摘要

This manuscript proposes an efficient hybrid numerical approach that combines high accuracy with low computational cost to approximate solutions of the time-space diffusion model governed by the Caputo and Riesz fractional derivatives. Addressing the fractional time derivative in the Caputo sense, we employ a combination of quadratic and linear interpolations, achieving an accuracy of \(\mathscr {O}\big \{\big (\Delta t\big )^{2-\gamma }\big )\big \}\) O { ( Δ t ) 2 - γ ) } . For the spatial discretisation of the Riesz space-fractional operator, we utilise a compact finite difference scheme with fourth-order accuracy. The stability and convergence properties of the proposed numerical method are rigorously analysed and verified. Finally, we provide with several numerical examples, supplemented by graphical and tabular illustrations, to demonstrate the accuracy, efficiency, and robustness of the proposed approach.