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

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.