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<p>In this paper, we propose an efficacious numerical scheme for time-fractional Burgers’ equation, which can precisely capture the swift transitions of its solutions featuring sharp boundary and interface layers. Specifically, the L1 formula on graded mesh is used to resolve the initial singularity in time direction, and a novel tailored finite point scheme that incorporates exponential basis functions is used in the spatial discretization. We rigorously carry out detailed theroretical analyis of the proposed scheme, including solvability, stability and convergence. With appropriate choice of the mesh grading parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2583_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation>, the convergence accuracy is <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2583_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">\(\min\{r\alpha,2-\alpha\}\)</EquationSource> </InlineEquation> order in time and second order in space. In addition, we provide several numerical experiments to verify the validity of the proposed scheme and to test its capability in accurately resolving sharp solution transitions encompassing boundaries and interfaces.</p>

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Optimized tailored finite point scheme for time-fractional burgers’ equations with sharp solution transitions

  • 期刊论文

  • 发表日期：

    2025年7月4日

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  • Jianxiong Cao,
  • Yihong Wang

摘要

In this paper, we propose an efficacious numerical scheme for time-fractional Burgers’ equation, which can precisely capture the swift transitions of its solutions featuring sharp boundary and interface layers. Specifically, the L1 formula on graded mesh is used to resolve the initial singularity in time direction, and a novel tailored finite point scheme that incorporates exponential basis functions is used in the spatial discretization. We rigorously carry out detailed theroretical analyis of the proposed scheme, including solvability, stability and convergence. With appropriate choice of the mesh grading parameter \(r\) , the convergence accuracy is \(\min\{r\alpha,2-\alpha\}\) order in time and second order in space. In addition, we provide several numerical experiments to verify the validity of the proposed scheme and to test its capability in accurately resolving sharp solution transitions encompassing boundaries and interfaces.