A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

José M. Gutiérrez,Miguel Á. Hernández-Verón

doi:10.3390/math9010083

José M. Gutiérrez, Miguel Á. Hernández-Verón

Open Access

https://doi.org/10.3390/math9010083

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Journal: Mathematics	Publication Date: Jan 1, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: University of La Rioja

Abstract

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

Highlights

Both linear and nonlinear integral equations appear in numerous fields of science and engineering [4,5,6], because many physical processes and mathematical models can be described by them, so that these equations provide an important tool for modeling processes [7]
The definition of an integral equation given previously is very general, so in this work, we focus on some particular integral equations that are widely applied, such as Fredholm integral equations [8]
We propose the use of iterative methods to approach this inverse and the solution of the integral Equation (1)

Summary

Introduction

Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind. Mathematics 2021, 9, 83. If the operator F defined in (2) is a contraction, the Banach fixed point theorem [10], guarantees the existence of a unique fixed point of F in C[ a, b] This fixed point can be approximated by the iterative scheme: y0 given in C[ a, b], y n +1 = F ( y n ), n ≥ 0. The iterative scheme (3) is known as the method of successive approximations for operator F It converges to the fixed point y∗ for any function y0 ∈ C[ a, b]. It is clear that the Banach fixed point theorem does not allow us to locate a fixed point in a domain that is not all the space C[ a, b] Both the successive approximation method and Picard’s method do not need either inverse operators or derivative operators.

Newton’s Method for the Calculus of Inverse Operators

Separable Kernels

Non-Separable Kernels

A Picard-Type Iterative Scheme from the Newton Method

Conclusions