R\'esolution num\'erique du probl\`eme de Dirichlet $\Delta u = a\,u^3$ \`a l'aide du mouvement brownien

TL;DR

This paper develops stochastic algorithms to numerically solve Dirichlet boundary value problems involving a quasi-linear term $a u^3$, using probabilistic interpretations and Brownian motion simulations.

Contribution

It extends existing stochastic algorithms for boundary problems to handle nonlinear source terms like $a u^3$, providing new methods for such equations.

Findings

Numerical solutions obtained for positive and negative $a$.

Algorithms successfully extended to quasi-linear problems.

Results demonstrate the effectiveness of stochastic methods for nonlinear PDEs.

Abstract

In this paper, we are interested in numerical solution of some linear boundary value problems with Dirichlet boundary part, by the means of simulation of random walks. We use a probabilistic interpretation of solution $u$, assuming that the coefficient and the boundary data are sufficiently smooth, and applying It\^o's formula. From these stochastic representations of solution, we extend some algorithms obtained for standard boundary conditions to the quasi-linear source of the type $f(u)= a\,u^3$. For positive and negative parameter $a$, we then obtain numerical results by applying the stochastic methods based upon these generalized algorithms.