  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.
