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<p>This study explores uniqueness result for a direct problem and a numerical technique for determining the time-varying source term and initial condition simultaneously in a non-local diffusion equation, utilizing additional measurements at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t = T_1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t = T_2\)</EquationSource> </InlineEquation>. The challenge is characterized by its ill-posedness. We establish the existence and uniqueness of the strong solution for the forward problem with homogeneous Dirichlet boundary conditions. Subsequently, applying integral equations theory, we transform the resulting integral equations into an ill-conditioned system of linear algebraic equations. The uniqueness of the inverse problem is further demonstrated. Tikhonov regularization method with the generalized cross-validation approach, is then utilized to solve these equations. Finally, we give the numerical procedures and provide several numerical experiments to illustrate the effectiveness of our proposed method.</p>

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Uniqueness results for forward and inverse problems and numerical method for simultaneous reconstruction of initial value and source term in a non-local diffusion equation

  • 期刊论文

  • 发表日期：

    2026年2月3日

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  • Jin Wen,
  • Meng-Yao Zhou,
  • Song-Shu Liu

摘要

This study explores uniqueness result for a direct problem and a numerical technique for determining the time-varying source term and initial condition simultaneously in a non-local diffusion equation, utilizing additional measurements at \(t = T_1\) and \(t = T_2\) . The challenge is characterized by its ill-posedness. We establish the existence and uniqueness of the strong solution for the forward problem with homogeneous Dirichlet boundary conditions. Subsequently, applying integral equations theory, we transform the resulting integral equations into an ill-conditioned system of linear algebraic equations. The uniqueness of the inverse problem is further demonstrated. Tikhonov regularization method with the generalized cross-validation approach, is then utilized to solve these equations. Finally, we give the numerical procedures and provide several numerical experiments to illustrate the effectiveness of our proposed method.