An effective and simple scheme for solving nonlinear Fredholm integral equations

• Published in: Mathematical modelling and analysis Vol. 27; no. 2; pp. 215 - 231
• Main Authors: Shahsavaran, Ahmad, Fotros, Forough
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 27.04.2022
• Subjects: convergence and stability; Exact solutions; Fredholm equations; Fredholm integral equation; Gauss-Legendre integration; Integral equations; Interpolation; Lagrange polynomials; Laguerre polynomials; Mathematical analysis; Mathematical research; Newton methods; Numerical analysis; Polynomials; Quadratures; Iran; interpolation; convergence and stability; Fredholm integral equation; Gauss-Legendre integration; Lagrange polynomials
• ISSN: 1392-6292; 1648-3510; 1648-3510
• DOI: 10.3846/mma.2022.14194

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.