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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Matrix of Stationary Spectral Densities for the Solution of Two Linear Stochastic Parabolic Differential Equations with Several Delays

Author(s):

Igor Egorovich Poloskov

Perm State University, Department of Higher Mathematics, Head of the Department
Associate Professor, Doctor of Physics and Mathematics

polosk@psu.ru

Abstract:

The paper is devoted to the extension of the Guillouzic scheme proposed to calculate the spectral density for the solution of one linear ordinary stochastic differential equation of the first order with constant coefficients and delay, to a new class of models, namely, systems of evolutionary stochastic partial differential equations with several constant delays. In particular, the task of the study was to construct an explicit form of the matrix of spectral densities for the stationary random state vector field for a system of two linear parabolic equations with constant coefficients, three delays, and an additive input in the form of a vector of space-temporal stationary random fields with known characteristics. In addition to demonstrating the methodology and obtaining the sought relations for the matrix components, the paper determines sufficient conditions for the existence of these components in terms of the coefficients of the equations and delays. The obtained analytical formulas were exploited for drawing the level lines of the auto- and cross-spectral densities of the state field components for different values of the task parameters. The illustrative material was prepared in the environment of the Wolfram Mathematica package.

Keywords

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