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


Creative Commons License

Amin A. Z. M. A., Zaky M. A., Hendy A. S., Hashim I., Aldraiweesh A.

Mathematics, vol.10, no.17, 2022 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 10 Issue: 17
  • Publication Date: 2022
  • Doi Number: 10.3390/math10173065
  • Journal Name: Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Keywords: memory kernels, multidimensional integral equations, nonsmooth solution, spectral algorithm
  • Istanbul Gelisim University Affiliated: No

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.