An iterative Monte Carlo method to solve nonlinear second-order differential equations

TL;DR

This paper introduces an iterative Monte Carlo algorithm designed to solve complex nonlinear second-order differential equations with Dirichlet boundary conditions, expanding the applicability of Monte Carlo methods in numerical analysis.

Contribution

The paper presents a novel iterative Monte Carlo approach specifically tailored for nonlinear second-order differential equations with boundary conditions, which was not previously available.

Findings

Successfully applied the method to a test example

Demonstrated the algorithm's effectiveness for complex equations

Provided insights into convergence and accuracy

Abstract

The Monte Carlo method is a thriving and mathematically beautiful numerical technique used extensively, nowadays, to deal with many demanding problems in diverse fields. Here, we present an iterative Monte Carlo algorithm to work out very general nonlinear second-order differential equations, with Dirichlet boundary conditions. An example of its usage is, also, reported.