A probabilistic algorithm approximating solutions of a singular PDE of porous media type

Nadia Belaribi 1, 2 François Cuvelier 1 Francesco Russo 2, 3, *
* Corresponding author
3 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.
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Submitted on : Friday, November 12, 2010 - 6:59:05 PM
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  • HAL Id : inria-00535806, version 1
  • ARXIV : 1011.3107

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Nadia Belaribi, François Cuvelier, Francesco Russo. A probabilistic algorithm approximating solutions of a singular PDE of porous media type. 2010. ⟨inria-00535806⟩

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