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2026, Volume 34, Issue 4: 2384-2401. doi: 10.3934/era.2026109
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An alternating direction implicit scheme with graded time steps for a fractional evolution equation with a weakly singular kernel

  • 1.
    College of Information Science and Engineering, Hunan Women's University, Changsha 410004, China
  • 2.
    Department of Mathematics and Statistics, Hunan Normal University, 410081, Changsha 410004, China
  • Published: 18 March 2026

  • This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $ O(k^2+h_x^2+h_y^2) $, where $ k $ is a parameter of the maximal time steps and $ h_x $, $ h_y $ are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.

    Citation: Man Luo, Shukun Liu, Da Xu. An alternating direction implicit scheme with graded time steps for a fractional evolution equation with a weakly singular kernel[J]. Electronic Research Archive, 2026, 34(4): 2384-2401. doi: 10.3934/era.2026109

    Related Papers:

  • This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $ O(k^2+h_x^2+h_y^2) $, where $ k $ is a parameter of the maximal time steps and $ h_x $, $ h_y $ are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.



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  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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