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Volume 43, Issue 3
Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Zheng Ma, Chengming Huang & Anatoly A. Alikhanov

DOI: 10.4208/jcm.2401-m2023-0196

J. Comp. Math., 43 (2025), pp. 690-707.

Published online: 2024-11

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  • Abstract

This paper is concerned with the numerical solution of Volterra integro-differential equations with noncompact operators. The focus is on the problems with weakly singular solutions. To handle the initial weak singularity of the solution, a fractional collocation method is applied. A rigorous $hp$-version error analysis of the numerical method under a weighted $H^1$-norm is carried out. The result shows that the method can achieve high order convergence for such equations. Numerical experiments are also presented to confirm the effectiveness of the proposed method.

  • Keywords

Volterra integro-differential equation, Noncompact operator, Nonsmooth solution, Collocation method, Fractional polynomial, $hp$-version error analysis.

  • AMS Subject Headings

L05, L60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-690, author = {Ma , ZhengHuang , Chengming and Alikhanov , Anatoly A.}, title = {Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {3}, pages = {690--707}, abstract = {

This paper is concerned with the numerical solution of Volterra integro-differential equations with noncompact operators. The focus is on the problems with weakly singular solutions. To handle the initial weak singularity of the solution, a fractional collocation method is applied. A rigorous $hp$-version error analysis of the numerical method under a weighted $H^1$-norm is carried out. The result shows that the method can achieve high order convergence for such equations. Numerical experiments are also presented to confirm the effectiveness of the proposed method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2401-m2023-0196}, url = {http://global-sci.org/intro/article_detail/jcm/23555.html} }
TY - JOUR T1 - Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators AU - Ma , Zheng AU - Huang , Chengming AU - Alikhanov , Anatoly A. JO - Journal of Computational Mathematics VL - 3 SP - 690 EP - 707 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2401-m2023-0196 UR - https://global-sci.org/intro/article_detail/jcm/23555.html KW - Volterra integro-differential equation, Noncompact operator, Nonsmooth solution, Collocation method, Fractional polynomial, $hp$-version error analysis. AB -

This paper is concerned with the numerical solution of Volterra integro-differential equations with noncompact operators. The focus is on the problems with weakly singular solutions. To handle the initial weak singularity of the solution, a fractional collocation method is applied. A rigorous $hp$-version error analysis of the numerical method under a weighted $H^1$-norm is carried out. The result shows that the method can achieve high order convergence for such equations. Numerical experiments are also presented to confirm the effectiveness of the proposed method.

Ma , ZhengHuang , Chengming and Alikhanov , Anatoly A.. (2024). Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators. Journal of Computational Mathematics. 43 (3). 690-707. doi:10.4208/jcm.2401-m2023-0196
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