One nonclassical higher order mathematical model with additive "white noise"'

A. A. Zamyshlyaeva

Journal of computational and engineering mathematics, Tome 1 (2014) no. 1, pp. 55-68

Résumé

Sobolev type equations theory experiences an epoch of blossoming. The majority of researches is devoted to the determined equations and systems. However in natural experiments there are the mathematical models containing accidental indignation, for example, white noise. Therefore recently even more often there arise the researches devoted to the stochastic differential equations. A new conception of "white noise", originally constructed for finite dimensional spaces, is spread here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher order Sobolev type equations theory and practical applications. The main idea is in construction of «noise» spaces using the Nelson–Gliklikh derivative. Abstract results are applied for the investigation of the Boussinesq–Lòve model with additive "white noise" within the Sobolev type equations theory. At studying the methods and results of theory of Sobolev type equations with relatively p-sectorial operators are very useful. We use already well proved at the investigation of Sobolev type equations the phase space method consisting in a reduction of singular equation to regular one, defined on some subspace of initial space. In the first part of article the spaces of noises are constructed. In the second — the Cauchy problem for the stochastic Sobolev type equation of higher order is investigated. As an example the stochastic Boussinesq–Lòve model is considered.

Détail

Keywords: Sobolev type equation, propagator, "white noise", K-Wiener process.