A numerical method for solving some nonlinear problems

TL;DR

This paper introduces a global linearization numerical method for solving certain nonlinear equations in Banach spaces, offering an alternative to Newton-type methods with convergence guarantees and practical applications.

Contribution

It presents a novel global linearization approach that transforms nonlinear problems into linear ones, differing from traditional local linearization techniques.

Findings

The method converges under specific conditions.

Numerical examples demonstrate effectiveness.

Applicable to various nonlinear problems.

Abstract

A nonlinear equation in a Banach space is written as a linear equation with a linear operator depending on the unknown solution. This method, which we call a global linearization method, differs essentially from the local linearization methods of the Newton-type. Inverting the above linear operator by the methods known for linear operators one gets an equation which sometimes is much better for numerical solution than the original one. Some theorems about convergence of the proposed iterative process for solving the transformed equation are given. Examples of applications are considered.