Scholarlink

Abstract <p>The conventional processes to tackle solutions of linear Fredholm integral equations over real intervals of substantial but finite lengths is based on two steps: First, we discretize the equation which gives a huge linear algebraic system. Second, we need to use some iterative schemes to approach the solution of this algebraic system. In this paper, we propose a new process denoted by iterate-discretize, which leads to the development of the iterative theory tackling this type of Fredholm integral equations over such intervals. This new process is dependent on two parts: We transform the Fredholm equation into a matrix of bounded linear operators defined over small subintervals, whence we construct a new functional variant of the iterative SOR scheme adapted to this operators’ matrix. As a second part, we don’t need to discretize the whole operators’ matrix but only its diagonal part. Within the context of study, some properties are provided to make sure the convergence of the generalized SOR scheme. Ultimately, a regular and weakly singular equation are studied and we obtain positive results that ground the new process. Moreover, we provide graphical approximations to the optimal relaxation parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega \)</EquationSource> <!--ComMat2570211Mahcene-m1--> </InlineEquation> for which convergence is optimized.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Approximating Solutions of Integral Equations Over Large Intervals: A Generalized SOR Scheme for Matrices of Bounded Linear Operators

  • 期刊论文

  • 发表日期：

    2026年3月1日

查看全文
  • M. G. Mahcene,
  • S. Lemita

摘要

Abstract

The conventional processes to tackle solutions of linear Fredholm integral equations over real intervals of substantial but finite lengths is based on two steps: First, we discretize the equation which gives a huge linear algebraic system. Second, we need to use some iterative schemes to approach the solution of this algebraic system. In this paper, we propose a new process denoted by iterate-discretize, which leads to the development of the iterative theory tackling this type of Fredholm integral equations over such intervals. This new process is dependent on two parts: We transform the Fredholm equation into a matrix of bounded linear operators defined over small subintervals, whence we construct a new functional variant of the iterative SOR scheme adapted to this operators’ matrix. As a second part, we don’t need to discretize the whole operators’ matrix but only its diagonal part. Within the context of study, some properties are provided to make sure the convergence of the generalized SOR scheme. Ultimately, a regular and weakly singular equation are studied and we obtain positive results that ground the new process. Moreover, we provide graphical approximations to the optimal relaxation parameter \(\omega \) for which convergence is optimized.