An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind

José Manuel Gutiérrez Jiménez ,M A Hernández‐Verón

doi:10.3390/sym13101957

José Manuel Gutiérrez Jiménez , M A Hernández‐Verón

Open Access

https://doi.org/10.3390/sym13101957

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Journal: Symmetry	Publication Date: Oct 17, 2021
License type: CC BY 4.0

Affiliation: University of La Rioja

Abstract

In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels.

Highlights

Scheme for Fredholm IntegralIn this paper, we are concerned with obtaining an approximate solution of Fredholm integral equations of a second kind given by Equations of Second Kind
The problem of solving integral equations is quite general, so in this work we focus on a particular type of integral equation called Fredholm integral equations [1]
We have considered the numerical solution of Fredholm integral equations of the second kind by means of iterative methods

Summary

Introduction

We are concerned with obtaining an approximate solution of Fredholm integral equations of a second kind given by Equations of Second Kind. We present a strategy based on the symmetry between the problem of numerically solving linear integral equations and the problem of approximating the inverse of a linear operator. With this idea, the use of iterative methods gives us an alternative way of approaching this inverse and the solution of the integral equation, instead of trying to calculate the exact solution of the problem (see [7,8,9,10]). One of the targets of this work is to use Newton’s method for solving G(z) = 0 In this point, we introduce the use of iterative methods for the calculus of the inverse. The local convergence of the method can be seen in [20]

Construction of an Iterative Scheme of Ulm-Type

Main Convergence Result

A Particular Case

Numerical Examples

Conclusions