Информационная система "Конференции"

Abstracts

Stochastic simulation and Monte-Carlo methods

The study of intense mass- and energy-transfer processes frequently requires the solution to the Cauchy problem for a three-dimensional hyperbolic diffusion equation: % begin{equation}label{eq:HypEqDiff} tau^* frac{partial^2 C}{partial t^2} + frac{partial C}{partial t}=% D left(frac{partial^2 C}{partial x^2} + frac{partial^2 C}{partial y^2} + frac{partial^2 C}{partial z^2}right) end{equation} % with the initial conditions of general type: % begin{equation}label{eq:InitCond} %C({mathbf{x}},0)=Phi({mathbf{x}}),% %partial C({mathbf{x}},0) / partial t= Psi({mathbf{x}}), C({mbx},0)=Phi({mbx}),% partial C({mbx},0) / partial t= Psi({mbx}), end{equation} % %where ${mathbf{x}} ={x, y, z}$, $C({mathbf{x}},t)$ is a concentration of the transferred where ${mbx} ={x, y, z}$, $C({mbx},t)$ is a concentration of the transferred mass, $tau^*$ is a relaxational time of the medium, $D$ is a diffusion coefficient. The probabilistic formula to the solution to the Cauchy problem (ref{eq:HypEqDiff}), (ref{eq:InitCond}) is % begin{equation}label{eq:solution} %C({mathbf{x}},t)= {mathbf{E}} { C^*({mathbf{x}},overline{t}) }, C({mbx},t)= {mbe} { C^*({mbx},overline{t}) }, end{equation} % %where ${mathbf{E}}{ldots}$ is an operation of mathematical expectation, $ %C^*({mathbf{x}},t)$ is the solution to the Cauchy problem for wave equation with the same where ${mbe}{ldots}$ is an operation of mathematical expectation, $ C^*({mbx},t)$ is the solution to the Cauchy problem for wave equation with the same initial conditions (ref{eq:InitCond}), $overline{t}$ is the "randomized" time: % begin{equation}label{eq:rndTime} overline{t} = int_0^t (-1)^{N_a(tau)} dtau, end{equation} % where $N_a(tau)$ is a Poisson process with the intensity $a=1/(2tau^*)$.

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