On the numerical solutions of Fredholm--Volterra integral equation.
type: Item
instance of: scholarly article
title: On the numerical solutions of Fredholm--Volterra integral equation.
zbMATH Open document ID: 1033.65117
DOI: 10.1016/S0096-3003(02)00615-X
author: Khamis I. Mohamed
author: M. A. Abdou
author: Abdul Samad Bin Haji Ismail
published in: Applied Mathematics and Computation
publication date: 2003-11-25T00:00:00Z
review text: The authors describe the Toeplitz matrix method and the product Nystrom method for the mixed Fredholm-Volterra singular integral equation of the second kind: \[ \mu\phi(x,t)-\lambda\int_{-1}^1k(x,y)\phi(y,t)\,dy- \lambda\int_0^tF(t, \tau)\phi(x,\tau)\,d\tau= f(x,t),\quad 0\leqslant t\leqslant T,\;| x| \leqslant1,\tag{1} \] where \(k\), \(F\) and \(f\) are given functions and \(\mu\), \(\lambda\) are numerical parameters, while \(\phi(x, t)\) is the unknown function to be determined. The goal is to obtain, numerically, an approximate solution of (1), when the kernel of Fredholm integral term takes the form: \(k(x,y) =| y-x| ^{-\alpha}(\ln| y-x| )^q\), \(0\leqslant \alpha < 1, q\in\mathbb N\). Conditions are discussed to guarantee the existence of a unique solution of (1). A numerical technique is used to reduce (1) to a system of linear Fredholm integral equations of the second kind. The Toeplitz matrix method is applied to this system to get an approximate solution of (1). The product Nyström method is used to solve the same system to obtain another approximate solution of (1). The results are compared with the exact solution of the integral equation. The error of each method is calculated.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 74M15
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45F15
zbMATH DE Number: 2009068
zbMATH Keywords: mixed Fredholm-Volterra singular integral equation of the second kind
zbMATH Keywords: logarithmic kernel
zbMATH Keywords: Toeplitz matrix method
zbMATH Keywords: product Nystrom method
zbMATH Keywords: system of Fredholm integral equations
zbMATH Keywords: error bound
zbMATH Keywords: contact problem
reviewed by: Nikolay Yakovlevich Tikhonenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2009068