scholarly journals Mathematical Model of a Surface Radiance Factor

2019 ◽  
Author(s):  
Александр Басов ◽  
Alexander Basov ◽  
Владимир Будак ◽  
Vladimir Budak
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Lower Boundary ◽  
Operator Method ◽  
Angular Spectrum ◽  
Discrete Ordinates ◽  
Optical Parameters ◽  
Anisotropic Part ◽  
Radiance Factor

The article is devoted to the creation of a surface radiance factor mathematical model. The basis of the model is the solution of the boundary value problem of the radiative transfer equation (RTE). The surface is considered as a structure consisting of several turbid layers, each of which is characterized by its optical parameters. The top of the structure is randomly rough, uncorrelated, Fresnel. The lower boundary reflects perfectly diffusely. The complexity of solving the RTE boundary value problem for real layers is due to the fact that the suspended particles in each layer are always much longer than the wavelength. This leads to a strong anisotropy of the radiance angular distribution according to Mie theory. The solution comes down to a system of equations by the discrete ordinates method that consists of several hundred of differential equations. Subtraction of the anisotropic part from the solution based on an approximate analytical solution of the RTE allows avoiding this problem. The approximation is based on a slight decrease in the anisotropic part of the angular spectrum. The matrix-operator method determines the general solution for a complex multilayer structure. The calculation speed can be increased without compromising the accuracy of the solution with the help of the synthetic iterations method. The method consists of two stages: the first one repeats the described one with a small number of ordinates; on the second one the iteration of it is performed. The model is realised in the Matlab software.

Mathematical model implementation in conditions of uncertainty and possible risks

2021 ◽  
pp. 137-145
Author(s):  
A. Kravtsov ◽  
◽  
D. Levkin ◽  
O. Makarov ◽  
◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Layer Structure ◽  
Boundary Value ◽  
Goal Function ◽  
Cauchy Problems ◽  
System Of Differential Equations

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

scholarly journals The first boundary value problem of the dynamics of an anisotropic elastic half-space under the action of subsonic transport loads

2020 ◽  
Vol 4 (78) ◽  
pp. 45-52
Author(s):  
G. K. ZAKIR’YANOVA ◽  
◽  
L. A. ALEXEYEVA ◽  
Keyword(s):  
Rock Mass ◽  
Boundary Value Problem ◽  
Half Space ◽  
Wide Class ◽  
Boundary Value ◽  
Operator Method ◽  
Elastic Half Space ◽  
Theory Of Elasticity ◽  
Wide Range ◽  
Anisotropic Elastic

The first boundary value problem of the theory of elasticity for an anisotropic elastic half-space is solved when a transport load moves along its surface. The subsonic Raleigh case is considered, when the velocity of motion is less than the velocity of propagation of bulk and surface elastic waves. The Green’s tensor of the transport boundary value problem is constructed and on its basis the solution of boundary value problems for a wide class of distributed traffic loads is given. To solve the problem, the methods of tensor and linear algebra, integral Fourier transform, and operator method for solving systems of differential equations were used. The obtained solution makes it possible to investigate the dynamics of the rock mass for a wide class of transport loads, in a wide range of velocities, both low velocities and high velocities, and to evaluate the strength properties of the rock mass under the influence of road transport. In particular, determine the permissible velocities of its movement and carrying capacity. In addition, a investigation on its basis of the movement of the day surface along the route will make it possible to establish criteria for the seismic resistance of ground structures and the permissible distances of their location from the route.

scholarly journals Interpolation of Boundary Condition at Time-Interval of Unknown Lenghth for the Problem of Convective Diffusion in a Three-Layered Water Filter

2019 ◽  
pp. 25-28
Author(s):  
Olha Chernukha ◽  
Yurii Bilushchak
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Mass Transfer ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Convective Diffusion ◽  
Lower Boundary ◽  
Time Interval ◽  
Water Filter ◽  
The Right

On the basis of mathematical model of convectivediffusion in a three-layered filter it is formulated a contactinitial-boundary value problem for description of mass transferof pollution accompanying the sorption processes. It is proposedthe algorithm for establishing the estimation of values of soughtfunction (concentration of pollution) at the lower boundary of thefilter on the basis of the interpolation of experimental data. It istaken into account that the right end of the interpolation segmentis unknown. It is determined the exact solutions of contact-initialboundaryvalue problems of mass transfer with provision forboth diffusive and convective mechanisms of transfer as well assorption processes, which is based on integral transformationsover space variables in the contacting regions. Is it designedsoftware and established regularities of convective diffusionprocess in the three-layered filter.

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

Mathematical model of magnetic field for a sectorial ferromagnetic cylinder

2021 ◽  
Vol 2021 (49) ◽  
pp. 19-25
Author(s):  
R. M. Dzhala ◽  
◽  
V. R. Dzhala ◽  
B. I. Horon ◽  
B. Ya. Verbenets ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Mathematical Model ◽  
Spatial Distribution ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Secondary Field

The solution of the boundary-value problem of magnetostatics for a circular ferromagnetic cylinder with a longitudinal sectorial cutout is described. The external primary magnetic field is orthogonal to the cylinder and directed at arbitrary azimuth relative to the cutout. A system of algebraic equations for finding the amplitudes of azimuthal expansions of the spatial distribution of the secondary field of the outer and sectorial partial regions of the cylinder is obtained by the method of rearrangement of functions.

scholarly journals The Mathematical Modeling of Metals Mass Transfer in Three Layer Peat Blocks

2015 ◽  
Vol 1 ◽  
pp. 87
Author(s):  
Ērika Teirumnieka ◽  
Ilmārs Kangro ◽  
Edmunds Teirumnieks ◽  
Harijs Kalis
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Mass Transfer ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Mathematical Models ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Difference Methods ◽  
The Mathematical Model

The mathematical model for calculation of concentration of metals for 3 layers peat blocks is developed due to solving the 3-D boundary-value problem in multilayered domain-averaging and finite difference methods are considered. As an example, mathematical models for calculation of Fe and Ca concentrations have been analyzed.

scholarly journals Simulation of the process of water purification in multilayered micro porous media

2019 ◽  
pp. 61-63
Author(s):  
Olena Prysiazhniuk ◽  
Andrii Safonyk ◽  
Anna Terebus
Keyword(s):  
Mathematical Model ◽  
Porous Media ◽  
Boundary Value Problem ◽  
Water Purification ◽  
Asymptotic Approximation ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
The Mathematical Model ◽  
Purification Of Water

The mathematical model of the process of adsorption purification of water from impurities in multilayer microporous filters is formulated. An algorithm for numerically-asymptotic approximation of solution of the corresponding nonlinear singularly perturbed boundary value problem is developed. The developed model allows to investigate the distribution of concentration of pollutant inside the filer.

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