Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels. (18th November 2019)

• Record Type: Journal Article
• Authors: Kant, Kapil; Nelakanti, Gnaneshwar
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 15(2019)
• Page Start: 1787
• Page End: 1821
• Publication Date: 2019-11-18
• Subjects: Weakly singular Volterra integral equations -- Jacobi polynomials -- spectral Galerkin method -- spectral multi-Galerkin method -- superconvergence results; 45B05 -- 45G10 -- 65R20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1636278
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L 2 -norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L 2 -norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.