Numerical solution of the hoff equation with additive "white noise"' in spaces of differential forms on a torus

D. E. Shafranov

Journal of computational and engineering mathematics, Tome 8 (2021) no. 2, pp. 46-55.

Résumé

The paper is devoted to the search for numerical solutions to the Cauchy problem for the linear stochastic equation in space of smooth differential forms on a torus. Based on the previously obtained results on the type of analytical solution to the stochastic version of the Hoff equation in spaces of smooth differential forms on smooth compact Riemannian manifolds without boundary, we choose several terms from the analytical solution in order to construct graphs of the numerical solution for various values of the coefficients and the inhomogeneous term. Since these equations are Sobolev type equations with a degenerate operator at the derivative, we can solve various initial-boundary value problems using the theory of degenerate analytic groups of resolving operators. In the deterministic case, the solution is based on the phase subspace of the original space. In spaces of differential forms, we use the invariant form of the Laplacian, i.e. the Laplace – Beltrami operator. The phase space method is also used in non-deterministic case, but we use the Nelson – Gliklikh derivative due to the non-differentiability of "white noise" in the usual sense. In this paper, a two-dimensional torus plays the role of a smooth compact oriented Riemannian manifold without boundary. Numerical solutions are found using the Galerkin–Petrov method and are presented for several fixed time points as graphs of the coefficients of differential forms obtained in Maple.

Détail

Keywords: Sobolev type equation, Nelson–Gliklikh derivative, Laplace–Beltrami operator, differential forms, Riemannian manifold.