Probabilistic numerical solution of wave equations with polynomial non-linearity

Abstract

We propose a Monte Carlo solution to the wave equations with polynomial non-linearity. Writing the probabilistic representation of the Monte Carlo solution, we are able to show its expected value retrieves the Duhamel’s solution of the wave equation. Based on a stochastically dominating branching process, we construct the proof of finding the probability generating function of the progeny problem, from which we are able to recover a quantitative estimate on the sufficient conditions for integrability.