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Abstract <p>In this study, we explore the numerical solutions of two nonlocal fractional problems using the finite element method (FEM). Many previous studies addressing elliptic problems have considered the simple form of the second member, where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2268_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(f = 1\)</EquationSource> <!--ComMat2570076Slimani-m1--> </InlineEquation>. However, in our current paper, we explore different formulas for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2268_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> <!--ComMat2570076Slimani-m2--> </InlineEquation>. In the case of the second parabolic problem, we utilize the matrices computed in the first problem to derive an ordinary differential equation.In order to calculate the exponential matrix in the analytical solution of the ordinary differential equation, we utilize the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2268_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\((2,2)\)</EquationSource> <!--ComMat2570076Slimani-m3--> </InlineEquation> Pade approximation. These methods are applied to three numerical examples to demonstrate the accuracy and efficiency of the proposed technique. Our findings suggest that this approach is highly effective and efficient.</p>

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Application of the Finite Element Method (FEM) to a Class of Nonlocal Elliptic and Parabolic Equations

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • A. Slimani,
  • Y. Bedrani,
  • T. Belhadi

摘要

Abstract

In this study, we explore the numerical solutions of two nonlocal fractional problems using the finite element method (FEM). Many previous studies addressing elliptic problems have considered the simple form of the second member, where \(f = 1\) . However, in our current paper, we explore different formulas for \(f\) . In the case of the second parabolic problem, we utilize the matrices computed in the first problem to derive an ordinary differential equation.In order to calculate the exponential matrix in the analytical solution of the ordinary differential equation, we utilize the \((2,2)\) Pade approximation. These methods are applied to three numerical examples to demonstrate the accuracy and efficiency of the proposed technique. Our findings suggest that this approach is highly effective and efficient.