Scholarlink

Abstract <p>Linear Volterra equations of the first kind are considered. A class of problems with a unique solution is identified, for which collocation-variational solution methods are proposed. According to the proposed algorithms, an approximate solution is found at nodes of a uniform grid (collocation condition), which yields an underdetermined system of linear algebraic equations. The system thus obtained is supplemented with the minimization condition for the objective function, which approximates the squared norm of the approximate solution. As a result, we obtain a quadratic programming problem with a quadratic objective function (squared norm of the approximate solution) and equality constraints (collocation conditions). This problem is solved by applying the Lagrange multiplier method. Fairly simple third-order methods are considered in detail. Numerical results for test problems are presented. Further development of this approach for the numerical solution of other classes of integral equations is discussed.</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

Collocation-Variational Approaches to the Numerical Solution of Volterra Integral Equations of the First Kind

  • 期刊论文

  • 发表日期：

    2025年1月1日

查看全文
  • M. V. Bulatov

摘要

Abstract

Linear Volterra equations of the first kind are considered. A class of problems with a unique solution is identified, for which collocation-variational solution methods are proposed. According to the proposed algorithms, an approximate solution is found at nodes of a uniform grid (collocation condition), which yields an underdetermined system of linear algebraic equations. The system thus obtained is supplemented with the minimization condition for the objective function, which approximates the squared norm of the approximate solution. As a result, we obtain a quadratic programming problem with a quadratic objective function (squared norm of the approximate solution) and equality constraints (collocation conditions). This problem is solved by applying the Lagrange multiplier method. Fairly simple third-order methods are considered in detail. Numerical results for test problems are presented. Further development of this approach for the numerical solution of other classes of integral equations is discussed.