This paper presents a novel numerical framework for solving a two-dimensional multi-term nonlinear convection-diffusion equation involving the Caputo-Fabrizio time-fractional derivative. The proposed method begins with a semi-discrete formulation based on a spatially continuous finite element method, and an optimal spatial error estimate of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2112_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$O(h^{2})$</EquationSource> </InlineEquation> for the semi-discrete scheme is rigorously established. To advance the solution in time, a fully discrete scheme is constructed by integrating a second-order finite difference discretization, ensuring compatibility with the nonlinear terms. The resulting algorithm is proven to be unconditionally stable in the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2112_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{2} $</EquationSource> </InlineEquation>-norm. Furthermore, under suitable regularity assumptions, the method attains a convergence rate of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2112_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mover accent="true"> <mi>τ</mi> <mo>‾</mo> </mover> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$O(h^{2} + \overline{\tau}^{2}) $</EquationSource> </InlineEquation>, where h and τ̅ represent the spatial and temporal discretization steps, respectively.

Finite element analysis of a multi-term nonlinear time-fractional convection-diffusion equation with Caputo-Fabrizio derivative

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2025年11月4日

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Allaoua Mehri,
Mohammed S. Abdo,
Hakima Bouhadjera,
Arafa Dawood,
Khaled Aldwoah,
Ria Egami

摘要

This paper presents a novel numerical framework for solving a two-dimensional multi-term nonlinear convection-diffusion equation involving the Caputo-Fabrizio time-fractional derivative. The proposed method begins with a semi-discrete formulation based on a spatially continuous finite element method, and an optimal spatial error estimate of order $O (h^{2})$ $O(h^{2})$ for the semi-discrete scheme is rigorously established. To advance the solution in time, a fully discrete scheme is constructed by integrating a second-order finite difference discretization, ensuring compatibility with the nonlinear terms. The resulting algorithm is proven to be unconditionally stable in the $L^{2}$ $L^{2} $ -norm. Furthermore, under suitable regularity assumptions, the method attains a convergence rate of $O (h^{2} + {\overline{τ}}^{2})$ $O(h^{2} + \overline{\tau}^{2}) $ , where h and τ̅ represent the spatial and temporal discretization steps, respectively.