SMALL RANDOM DISTURBANCE OF THE HEAT CONDUCTION EQUATION
Zoya Nagolkina, Yuri Filonov · APPLIED GEOMETRY AND ENGINEERING GRAPHICS · 2024
This work considers the heat conduction equation with a small random disturbance. Such an equation, under certain assumptions, can be interpreted as a mathematical model of the real process of heat transfer in a randomly heterogeneous medium. Let u(t,x) be the temperature of some body at the moment of time t at the point with the generalized coordinate x. That is, ut(x)-distribution of temperatures and is a continuous function of x. We will consider this function as an element of Hilbert's infinitely dimensional space H. This is possible if, for example, we introduce the appropriate norm. Moreover, if we consider the Sobolev space with the corresponding norm, then we can also take into account the boundary conditions. But in this work, we will limit ourselves to the problem with initial conditions, that is, the Cauchy problem in the corresponding space. Thus, the heat conduction equation can be considered as an equation with an unbounded operator A(t) in the Hilbert space H with the domain of definition densely embedded in this Hilbert space DA⊂H, and DA does not depend on t. The theory of deterministic levels of this type is developed in works There, the properties of evolutionary operator families depending on the location of the corresponding operator resolvent A(t) are considered. It is the estimate of in the Hilbert space norm that is key in proving the existence of a unique solution of a stochastic differential equation with an unbounded collapse operator in the Ito form. A small random perturbation of such an equation can be described in some cases as a stochastic Ito-type differential equation with unlimited decay and a small parameter in the diffusion term. A stochastic differential equation with a small perturbation in the diffusion term is considered using the standard methodology of series expansion in a small parameter. This was done in for a stochastic equation with a bounded nonlinear drift coefficient in Rn space. When the standard conditions for the existence of a unique solution of the Ito equation are met in the work, the functions are obtained in an explicit form, which are appropriate approximations of the expansion of the solution of the equation in a series by powers of a small parameter. The estimate of the residual term of this series is given. At the same time, all estimates are obtained in the mean square in the corresponding norms of the Hilbert space.