Superconvergence Results for Weakly Singular Fredholm–Hammerstein Integral Equations. (4th April 2019)

• Record Type: Journal Article
• Authors: Mandal, Moumita; Nelakanti, Gnaneshwar
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 5(2019)
• Page Start: 548
• Page End: 570
• Publication Date: 2019-04-04
• Subjects: Fredholm–Hammerstein integral equations -- multi-Galerkin method -- piecewise polynomial -- superconvergence rates -- weakly singular kernels; 45B05 -- 45G10 -- 65R20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1561468
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this article, we consider the multi-Galerkin method for solving the Fredholm–Hammerstein integral equations with weakly singular kernels, using piecewise polynomial bases. We show that multi-Galerkin method has order of convergence O ( h r + α ) for the algebraic kernel, whereas for logarithmic kernel, it converges with the order O ( log h h r + 1 ) in uniform norm, where h is the norm of the partition and r is the smoothness of the solution. We also discuss the iterated multi-Galerkin method. We prove that iterated multi-Galerkin method has order of convergence O ( h r + 2 α ) for the algebraic kernel and for logarithmic kernel, it has order of convergence of order O ( ( log h ) 2 h r + 2 ) in uniform norm. Numerical examples are given to illustrate the theoretical results.