International Conference «Mathematical and Informational Technologies, MIT-2013»
(X Conference «Computational and Informational Technologies for Science,
Engineering and Education»)

Vrnjacka Banja, Serbia, September, 5–8, 2013

Budva, Montenegro, September, 9-14, 2013

Prokhorov I.  

The numerical solution of the Cauchy problem for the radiative transfer equation with generalized matching conditions

Development and construction of the approximate solution methods for the unsteady transport equations is an urgent problem in the study of various mathematical models of physical processes. These are models of the atmospheric optics and propagation of gamma rays in matter, neutron diffusion and kinetic theory of gases, the growth population of cells and multicellular organisms.
The type of boundary conditions and the matching conditions at the boundaries of the media to describe various physical phenomena is an important aspect in the theory of boundary value problems for the kinetic equations.
For example, the effects of the influence of the environment in nuclear reactors, the reflection of the gas molecules on the solid walls, the refraction of the light flux at the boundary between media with different indices of refraction, the transition from one cell state to another.

It is proved that the initial-value problem for the unsteady transport equation in an inhomogeneous plane layer with generalized matching  conditions on the interfaces has the unique solution. It is shown that for continuous initial and boundary data and associated restrictions on the interface operator  the solution of the Cauchy problem is continuous in the domain of the continuity of coefficients. For the Fresnel conditions on the interface on the basis of the Monte Carlo method the algorithm for solving the problem is proposed.
Using estimates of the behavior of the solution for large time, the methods of reducing the statistical error are proposed. The computational experiments were carried out.

Abstracts file: Prokhorov_IV.doc

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