A computational scheme is proposed for the numerical solution of the stationary radiation transport equation ignoring scattering. It is proved that, at large optical thicknesses, the cosines and absorption coefficients averaged over the one-sided radiation intensities cannot be used in the zeroth approximation and it is necessary to use their first approximation.
The second term in the asymptotic expansion of the contribution from a nondegenerate stationary point is determined for the multidimensional case.
A one-dimensional model problem concerning the diffusion of a domain of turbulent perturbations in a homogeneous fluid is considered. It is assumed that the process takes place in a half-strip and is characterized by the equation of balance for the turbulent energy. A theorem on the existence of a solution is proved.
The problem of determining the field of permeability of an oil-bearing layer from the variations of the pressure and the yield of liquid at the boring wells can be stated as a problem for the minimum value of a functional and can be approximately solved by means of the methods of optimal control theory. The construction of numerical schemes is discussed. The results of a numerical solution of the problem enable one to draw conclusions regarding the uniqueness of the solution.
A numerical method for constructing external estimates of attainability sets for non-linear controlled systems of ordinary differential equations is proposed. In some cases the method enables the attainability set itself to be approximated.
Regularizing algorithms to determine approximations to L-pseudo-solutions are proposed on the basis of a generalized residual principle and a generalized residual method, when the initial data are specified only approximately. It is shown that the algorithms are equivalent.
An algorithm for finding the integral characteristics (optical thicknesses) of attenuation factors for illuminated regions containing inner zones of various forms is constructed and numerically implemented.
In the context of the numerical treatment of the two-dimensional Navier-Stokes equations, the boundary conditionsatasolid surface may be realized in different ways, some of which are examined. The approach considered here assumes that the equations of the system are solved separately. It is shown that then, irrespective of the specific formulation of the Navier-Stokes equations for an incompressible viscous liquid — in terms of velocity-pressure, velocity-vorticity or vorticity-stream function — the boundary conditions can be realized in an algorithmically universal way, based on a two-parameter formula previously proposed to approximate vorticity on a wall.
A study is made of numerical algorithms for canonical factorization, of two types: a) polar decomposition of an arbitrary bounded operator in H-space, and b) factorization of an operator-valued function of the unilateral shift operator in the space l2.
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