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<p>In this article, we study a generating function of a grid that guarantees an optimal convergence rate of the L1 formula to the Caputo fractional derivative and propose new nonuniform meshes including sine graded one, linearly varying graded one, joined graded one, and joined linearly varying graded one. We also prove that the L1 finite element method for initial-boundary value problems of time-fractional reaction–diffusion equations is unconditionally stable and converges at an optimal convergence rate on our proposed temporal meshes. Finally, through numerical examples, we illustrate our main results by performing a comparative analysis with approximate solutions based on the graded and tanh meshes.</p>

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Error Estimation for the L1 Approximation of Caputo Fractional Derivative on Nonuniform Meshes and its Application based on a Generating Function

  • 期刊论文

  • 发表日期：

    2025年11月16日

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  • Kumsong Jong,
  • Huichol Choe,
  • Kyongjun Jang

摘要

In this article, we study a generating function of a grid that guarantees an optimal convergence rate of the L1 formula to the Caputo fractional derivative and propose new nonuniform meshes including sine graded one, linearly varying graded one, joined graded one, and joined linearly varying graded one. We also prove that the L1 finite element method for initial-boundary value problems of time-fractional reaction–diffusion equations is unconditionally stable and converges at an optimal convergence rate on our proposed temporal meshes. Finally, through numerical examples, we illustrate our main results by performing a comparative analysis with approximate solutions based on the graded and tanh meshes.