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Virtual element approximation and BDF2 time-discrete scheme for a partial integro-differential equation with a singular Abel's kernel

Author

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  • Abbaszadeh, Mostafa
  • Zaky, Mahmoud A.
  • Dehghan, Mehdi

Abstract

This work focuses on developing a virtual element framework for analyzing nonlinear partial integro-differential equations with singular Abel-type kernels. To address the singularity in the integral term, we propose two strategies: one based on linear interpolation over a uniform temporal mesh and another employing a graded mesh. Notably, while the uniform mesh fails to retain second-order temporal accuracy, the graded mesh successfully achieves it. For spatial discretization, we adopt the virtual element method and derive rigorous error estimates for the semi-discrete scheme. Nonlinear terms are approximated via a second-order temporal discretization. Following this, we establish a fully-discrete scheme. We rigorously establish the unconditional stability and convergence rates of the proposed method in separate theorems. Finally, we validate the theoretical results through numerical experiments on various domains.

Suggested Citation

  • Abbaszadeh, Mostafa & Zaky, Mahmoud A. & Dehghan, Mehdi, 2025. "Virtual element approximation and BDF2 time-discrete scheme for a partial integro-differential equation with a singular Abel's kernel," Applied Mathematics and Computation, Elsevier, vol. 501(C).
    • Handle: RePEc:eee:apmaco:v:501:y:2025:i:c:s009630032500178x
    DOI: 10.1016/j.amc.2025.129451
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    References listed on IDEAS

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    1. Zhang, Hao & Liu, Mengmeng & Guo, Tao & Xu, Da, 2024. "Three finite difference schemes for generalized nonlinear integro-differential equations with tempered singular kernel," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 1199-1217.
    2. Hooshyarfarzin, Baharak & Abbaszadeh, Mostafa & Dehghan, Mehdi, 2025. "Numerical simulation and error estimation of the Davey-Stewartson equations with virtual element method," Applied Mathematics and Computation, Elsevier, vol. 489(C).
    3. Qiu, Wenlin & Xu, Da & Guo, Jing, 2021. "Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation," Applied Mathematics and Computation, Elsevier, vol. 392(C).
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    Cited by:

    1. Mahmoud A. Zaky, 2025. "Finite Difference/Fractional Pertrov–Galerkin Spectral Method for Linear Time-Space Fractional Reaction–Diffusion Equation," Mathematics, MDPI, vol. 13(11), pages 1-13, June.

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