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A corrected Crank–Nicolson scheme for the time fractional parabolic integro-differential equation with nonsmooth data

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Listed:
  • Chen, Ao
  • Chen, Xuejuan
  • Yan, Yubin
  • Guo, Wen

Abstract

This paper proposes a corrected Crank–Nicolson (CN) scheme for solving time fractional parabolic integro-differential equations which involve Caputo time fractional derivative and fractional Riemann–Liouville (R-L) integral. The weighted and shifted Grünwald–Letnikov (WSGL) formulae is adopted to approximate the time fractional Riemann–Liouville integral. The Crank–Nicolson scheme is applied to approximate the Caputo time fractional derivative. After appropriating corrections, the proposed scheme attains the optimal convergence order of O(τ2) with respect to the time step size τ for both smooth and nonsmooth data at any fixed time t. When combined with the Galerkin finite element method for spatial discretization, it forms a fully discrete scheme. The second-order error estimate for this scheme is rigorously established using the Laplace transform technique and verified by some numerical examples.

Suggested Citation

  • Chen, Ao & Chen, Xuejuan & Yan, Yubin & Guo, Wen, 2026. "A corrected Crank–Nicolson scheme for the time fractional parabolic integro-differential equation with nonsmooth data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 242(C), pages 279-296.
    • Handle: RePEc:eee:matcom:v:242:y:2026:i:c:p:279-296
    DOI: 10.1016/j.matcom.2025.12.001
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    References listed on IDEAS

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    1. Suganya, S. & Mallika Arjunan, M. & Trujillo, J.J., 2015. "Existence results for an impulsive fractional integro-differential equation with state-dependent delay," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 54-69.
    2. Luo, Ziyang & Zhang, Xingdong & Wang, Shuo & Yao, Lin, 2022. "Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
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