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<p>In this paper, a fully discrete scheme is presented for solving the time-fractional Allen–Cahn equation. The proposed scheme exhibits superlinear convergence in time. The space discretization is performed using the spectral Galerkin method. The time discretization is designed by combining the Mittag-Leffler function representation of the solution, the integrals of the Mittag-Leffler function and the piecewise polynomial interpolation method. Two methods have been developed to approximate the Mittag-Leffler function based on the Taylor series and the Wright function. The proposed time discretization is capable of achieving <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2942_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((1+\alpha )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>th order convergence, where the fractional order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2942_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> ranges from 0 to 1. Numerical experiments are presented to verify our theoretical results.</p>

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Mittag-Leffler Interpolation Integrator for the Time-Fractional Allen–Cahn Equation

  • 期刊论文

  • 发表日期：

    2025年5月21日

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  • Xing Liu,
  • Yumeng Yang

摘要

In this paper, a fully discrete scheme is presented for solving the time-fractional Allen–Cahn equation. The proposed scheme exhibits superlinear convergence in time. The space discretization is performed using the spectral Galerkin method. The time discretization is designed by combining the Mittag-Leffler function representation of the solution, the integrals of the Mittag-Leffler function and the piecewise polynomial interpolation method. Two methods have been developed to approximate the Mittag-Leffler function based on the Taylor series and the Wright function. The proposed time discretization is capable of achieving \((1+\alpha )\) ( 1 + α ) th order convergence, where the fractional order \(\alpha \) α ranges from 0 to 1. Numerical experiments are presented to verify our theoretical results.