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High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

Author

Listed:
  • Ahmed Z. Amin

    (Department of Mathematical Sciences, Faculty of Science & Technology, University Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia)

  • Mahmoud A. Zaky

    (Department of Applied Mathematics, National Research Centre, Cairo 12622, Egypt
    Educational Technology Department, College of Education, King Saud University, Riyadh 11451, Saudi Arabia)

  • Ahmed S. Hendy

    (Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia
    Department of Mathematics, Faculty of Science Benha University, Benha 13511, Egypt)

  • Ishak Hashim

    (Department of Mathematical Sciences, Faculty of Science & Technology, University Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia)

  • Ahmed Aldraiweesh

    (Educational Technology Department, College of Education, King Saud University, Riyadh 11451, Saudi Arabia)

Abstract

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems.

Suggested Citation

  • Ahmed Z. Amin & Mahmoud A. Zaky & Ahmed S. Hendy & Ishak Hashim & Ahmed Aldraiweesh, 2022. "High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay," Mathematics, MDPI, vol. 10(17), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:17:p:3065-:d:897474
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    References listed on IDEAS

    as
    1. 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).
    2. Li, Dongfang & Zhang, Chengjian, 2020. "Long time numerical behaviors of fractional pantograph equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 244-257.
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