Numerical solution of the Dzektser equation with “white noise” in the space of smooth differential forms defined on a torus

D. E. Shafranov

Journal of computational and engineering mathematics, Tome 7 (2020) no. 2, pp. 58-65

Résumé

In this paper, we present numerical solutions to the Cauchy problem for the Dzektser equation in the non-deterministic case in the space of differential forms on a two-dimensional torus. Based on the solutions constructed earlier by the author for the deterministic case and, in collaboration with other authors, abstract transition for the case of relatively sectorial operators from deterministic Sobolev type equations to non-deterministic ones in the space of differential forms defined on Riemannian manifolds without boundary, we construct trajectories of the solution to the Cauchy problem for the Dzektser equation in the spaces of differential forms with coefficients that are Wiener stochastic processes. Since these processes are non-differentiable in the usual sense, then the derivative is taken in the sense of Nelson–Glicklikh, and the Laplace–Beltrami operator is used instead of the Laplace operator on differential forms.

Détail

Keywords: relatively sectorial operator, differential forms, Nelson–Gliklikh derivative, Laplace–Beltrami operator.
Mots-clés : Sobolev type equations