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Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints

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  • Lu, Jiashu
  • Yang, Mengna
  • Nie, Yufeng

Abstract

This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the L∞ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

Suggested Citation

  • Lu, Jiashu & Yang, Mengna & Nie, Yufeng, 2022. "Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints," Applied Mathematics and Computation, Elsevier, vol. 431(C).
    • Handle: RePEc:eee:apmaco:v:431:y:2022:i:c:s0096300322004192
    DOI: 10.1016/j.amc.2022.127345
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    References listed on IDEAS

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    1. Sohrabi, S. & Ranjbar, H. & Saei, M., 2017. "Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 299(C), pages 141-152.
    2. Yao, Guoqing & Tao, DongYa & Zhang, Chao, 2022. "A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    3. Tian, Hao & Zhang, Jing & Ju, Lili, 2020. "A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 370(C).
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