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<p>This article explores the approximate solution of a class of semilinear parabolic singularly perturbed reaction–diffusion problems in a single spatial dimension. The approach is based on the application of an extrapolated Crank-Nicolson orthogonal spline collocation method. The proposed schemes integrate extrapolated Crank-Nicolson method for the time stepping and an orthogonal spline collocation method for the spatial discretization on a Shishkin mesh. The study derives parameter uniform error estimates in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_802_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^m_\varepsilon , m=1,2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>ε</mi> <mi>m</mi> </msubsup> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> norms. The numerical experiments are performed to compute error and order of convergence which supports the theoretical findings.</p>

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Extrapolated Crank-Nicolson Orthogonal Spline Collocation Methods for the Singularly Perturbed Semilinear Parabolic Reaction-Diffusion Problems

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Pankaj Mishra,
  • Jewel Howlader,
  • Kapil K. Sharma

摘要

This article explores the approximate solution of a class of semilinear parabolic singularly perturbed reaction–diffusion problems in a single spatial dimension. The approach is based on the application of an extrapolated Crank-Nicolson orthogonal spline collocation method. The proposed schemes integrate extrapolated Crank-Nicolson method for the time stepping and an orthogonal spline collocation method for the spatial discretization on a Shishkin mesh. The study derives parameter uniform error estimates in \(H^m_\varepsilon , m=1,2\) H ε m , m = 1 , 2 norms. The numerical experiments are performed to compute error and order of convergence which supports the theoretical findings.