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<p>This study focuses on identifying an unknown time-dependent source function in the heat equation under two distinct boundary conditions. The inverse problem is examined using an energy overspecification condition within the computational domain. The proposed approach combines the meshless local radial point interpolation method with the finite difference method. Through the energy method, it is demonstrated that the time-discrete scheme is unconditionally stable and convergent with an order of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3246_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal {O}}}(\delta t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the time variable. Despite the linear nature of the problem and its unique solution, it remains ill-posed due to the amplification of small input data errors in the output solution. To address this, the Tikhonov regularization method is employed, ensuring a stable solution when the input data is noisy. An effective approach for determining the regularization parameter is introduced, yielding significantly improved results compared to traditional methods such as the L-curve and the discrepancy principle. Numerical results illustrate the effectiveness of the proposed method, demonstrating accuracy for exact data and stability for noisy data.</p>

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Analysis of a time-dependent source function for the heat equation with nonlocal boundary conditions through a local meshless procedure

  • 期刊论文

  • 发表日期：

    2025年5月14日

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  • Elyas Shivanian,
  • Ahmad Jafarabadi,
  • Tannaz G. Chegini,
  • Abdollah Dinmohammadi

摘要

This study focuses on identifying an unknown time-dependent source function in the heat equation under two distinct boundary conditions. The inverse problem is examined using an energy overspecification condition within the computational domain. The proposed approach combines the meshless local radial point interpolation method with the finite difference method. Through the energy method, it is demonstrated that the time-discrete scheme is unconditionally stable and convergent with an order of \({{\mathcal {O}}}(\delta t)\) O ( δ t ) in the time variable. Despite the linear nature of the problem and its unique solution, it remains ill-posed due to the amplification of small input data errors in the output solution. To address this, the Tikhonov regularization method is employed, ensuring a stable solution when the input data is noisy. An effective approach for determining the regularization parameter is introduced, yielding significantly improved results compared to traditional methods such as the L-curve and the discrepancy principle. Numerical results illustrate the effectiveness of the proposed method, demonstrating accuracy for exact data and stability for noisy data.