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<p>This investigation focuses on the design and analysis of computational simulations for the two-dimensional nonlinear multi-term variable-order time-fractional advection–reaction–diffusion equation. The linear B-spline function approximates the variable-order time derivative in the equation, while modified cubic B-spline functions are employed to discretize space derivatives. The convergence and stability of the methodology are thoroughly analyzed theoretically, provided by numerical examples that demonstrate its implementation and validity. It has been established that the approach is unconditionally stable and preserves an accuracy of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40435_2025_1840_IEq1_HTML.gif" Format="GIF" Height="30" Rendition="HTML" Resolution="120" Type="Linedraw" Width="192" /> </InlineMediaObject> </InlineEquation> , where <i>h</i> is the maximum of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1840_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(h_{\texttt{x}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi mathvariant="monospace">x</mi> </msub> </math></EquationSource> </InlineEquation> (step size in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1840_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(\texttt{x}-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">x</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> direction) and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1840_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(h_{\texttt{y}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi mathvariant="monospace">y</mi> </msub> </math></EquationSource> </InlineEquation> (step size in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1840_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(\texttt{y}-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">y</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> direction). The analysis of the theory is in good accordance with the computational results. Numerical examples are used for illustrative purposes. Numerical tests were performed to validate the theoretical findings and demonstrate that the suggested technique yields a more precise solution than other existing methods in the existing literature.</p>

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Design and analysis of computational simulations for the two-dimensional nonlinear multi-term variable-order time-fractional advection–reaction–diffusion equation

  • 期刊论文

  • 发表日期：

    2025年9月13日

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  • Ravi Kanth Asv,
  • Varela Pavankalyan

摘要

This investigation focuses on the design and analysis of computational simulations for the two-dimensional nonlinear multi-term variable-order time-fractional advection–reaction–diffusion equation. The linear B-spline function approximates the variable-order time derivative in the equation, while modified cubic B-spline functions are employed to discretize space derivatives. The convergence and stability of the methodology are thoroughly analyzed theoretically, provided by numerical examples that demonstrate its implementation and validity. It has been established that the approach is unconditionally stable and preserves an accuracy of order , where h is the maximum of \(h_{\texttt{x}}\) h x (step size in \(\texttt{x}-\) x - direction) and \(h_{\texttt{y}}\) h y (step size in \(\texttt{y}-\) y - direction). The analysis of the theory is in good accordance with the computational results. Numerical examples are used for illustrative purposes. Numerical tests were performed to validate the theoretical findings and demonstrate that the suggested technique yields a more precise solution than other existing methods in the existing literature.