On the numerical Picard iterations method with collocations for the IVP

TL;DR

This paper introduces variants of the numerical Picard iteration method using Lagrange interpolation for solving IVPs, providing convergence analysis and numerical experiments to demonstrate effectiveness.

Contribution

It presents new variants of the numerical Picard iteration method with convergence results and numerical experiments for solving IVPs.

Findings

Convergence results established for fixed interpolation points.

Numerical experiments demonstrate the method's effectiveness.

Variants improve upon classical Picard iteration.

Abstract

Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term numerical emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed a convergence result is given. Finally some numerical experiments are reported.