A new approach of shifted Jacobi spectral Galerkin methods (SJSGM) for weakly singular Fredholm integral equation with non-smooth solution

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and u...

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Bibliographic Details
Published in:Numerical algorithms Vol. 96; no. 4; pp. 1553 - 1582
Main Authors: Kayal, Arnab, Mandal, Moumita
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2024
Springer Nature B.V
Subjects:
Algebra
Algorithms
Comparative studies
Computer Science
Exact solutions
Fredholm equations
Galerkin method
Integral equations
Mathematical analysis
Methods
Numeric Computing
Numerical Analysis
Operators (mathematics)
Original Paper
Polynomials
Regularity
Smoothing
Theory of Computation
Transformations (mathematics)
Weighting functions
Jacobi polynomials
Jacobi spectral Galerkin method
Fredholm integral equations
Superconvergence rate
Weakly singular kernel
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01677-9