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<p>This article discusses a numerical method, a combination of L1-discretization for the time Caputo derivative and a non-symmetric interior penalty discontinuous Galerkin method for the spatial variable, for singularly perturbed time-fractional initial-boundary-value problems. Due to the availability of the Caputo derivative with respect to the time variable and a small perturbation parameter as the diffusion coefficient in this type of problem, the solution of the problem may produce some singularity near the initial time in the temporal direction and near the boundaries in the spatial direction. To capture these singularities properly in the numerical solution, we have used a temporal graded mesh and a spatial piecewise Shishkin mesh for the proposed numerical method. Finally, we establish stability and convergence analysis for the proposed method to evaluate its accuracy, which is further validated by showing some results using numerical experiments.</p>

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Stability and robust error analysis of an L1-NIPG method for time-fractional singularly perturbed differential equations

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • V. Santhanakumar,
  • Sandip Maji,
  • Srinivasan Natesan

摘要

This article discusses a numerical method, a combination of L1-discretization for the time Caputo derivative and a non-symmetric interior penalty discontinuous Galerkin method for the spatial variable, for singularly perturbed time-fractional initial-boundary-value problems. Due to the availability of the Caputo derivative with respect to the time variable and a small perturbation parameter as the diffusion coefficient in this type of problem, the solution of the problem may produce some singularity near the initial time in the temporal direction and near the boundaries in the spatial direction. To capture these singularities properly in the numerical solution, we have used a temporal graded mesh and a spatial piecewise Shishkin mesh for the proposed numerical method. Finally, we establish stability and convergence analysis for the proposed method to evaluate its accuracy, which is further validated by showing some results using numerical experiments.