The present study deals with the numerical solution of the G-heat equation. Since the G-heat equation is defined in an unbounded domain, we firstly state that the solution of the G-heat equation defined in a bounded domain converges to the solution of the G-heat equation when the measure of the domain tends to infinity. Moreover, after time discretisation by an implicit time marching scheme, we define a method of linearisation of each stationary problem, which leads to the solution of a large scale algebraic system. A unified approach analysis of the convergence of the sequential and parallel relaxation methods is given. Finally, we present the results of numerical experiments. . He teaches numerical analysis, optimisation and numerical solution of boundary value problems. His fields of interest are in numerical analysis, large scale nonlinear systems of evolution equations, optimal control, parallel computing and more particularly, domain decomposition methods for the solution of nonlinear boundary values problems; he is interested to apply the obtained theoretical results to applications concerning finance, image processing, mechanics, etc. He is also a scientific expert, advisor and referee for several international scientific committees and journals.

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In this work we deal with spectral gap and canonical measures related to a model called colored disordered lattice gas. We consider the approach used in the work of Dermoune and Heinrich [“A small step towards the hydrodynamic limit of a colored disordered lattice gas,” C. R. Math. Acad. Sci. 339, 507–511 (2004)]. We suggest a new computation for the canonical measures. Also, we propose the explicit form of the spectral gap for colored disordered lattice gas of exclusion processes which plays an important role in the study of hydrodynamic limit.

Let B be a continuous additive functional for a standard process (Xt)t∈ℝ+ and let (Yt)t∈ℝ be a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of (Xt)t∈ℝ+ conditioned on the strict past and future without duality hypothesis. We study excursions of a general regenerative system and of a regenerative system consisting of the closure of the set of times the regular points of B are visited. In both cases, those conditioned excursion laws depend only on two points Xg− and Xd, where ]g,d[ is an excursion interval of the regenerative set M. We use the (FDt)-predictable exit system to bring together the isolated points of M and its perfect part and replace the classical optional exit system. This has been a subject in literature before (e.g., Kaspi (1988)) under the classical duality hypothesis. We define an “additive functional” for (Yt)t∈ℝ with B, we generalize the laws cited before to (Yt)t∈ℝ, and we express laws of pairs of excursions

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