Количество страниц: 16 с.
We consider the queuing system (QS) with an infinite storage, one service device and exponential service. At the input of QS comes double stochastic Poisson flow whose intensity is a jump-like process with intervals of constancy distributed according to the exponential law. It is assumed that the input flow intensity values at the break points on the left and right are independent. In the earlier published works a sufficient condition of existence and uniqueness of the QS stationary regime was obtained. In this paper, the operator analysis of integral equations is performed with respect to the characteristics of the stationary SMO, the necessary condition of existence of the system of integral equations solution is obtained and the existence and uniqueness of the solution is proved. A stationary generating function of the solution in the form of a convergent series is found. A distinctive feature of this work is the construction of the 2nd model of QS and the use of the shift operator of coefficients of the generating function for stationary distribution of the customers number
Бондрова, О. В. Анализ уравнений СМО со скачкообразной интенсивностью входного потока / О. В. Бондрова, Т. А. Жук, Н. И. Головко // Математические заметки СВФУ. — 2018. — Т. 25, N 3 (99), июль-сентябрь. — С. 18-32.
DOI: 10.25587/svfu.2018.99.16948
Количество страниц: 24 с.
Shishkina, E. L. Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations / E. L. Shishkina, S. Abbas // Математические заметки СВФУ. — 2018. — Т. 25, N 3 (99), июль-сентябрь. — С. 68-91.
DOI: 10.25587/SVFU.2018.99.16952
Количество страниц: 12 с.
A mathematical model describing an equilibrium of cracked two-dimensional bodies with two mutually intersecting cracks is considered. One of these cracks is assumed to be straight, and the second one is described with the use of a smooth curve. Inequality type boundary conditions are imposed at the both cracks faces providing mutual non-penetration between crack faces. On the external boundary, homogeneous Dirichlet boundary conditions are imposed. We study a family of corresponding varia-tional problems which depends on the parameter describing the length of the straight crack and analyze the dependence of solutions on this parameter. Existence of the solution to the optimal control problem is proved. For this problem, the cost functional is defined by a Griffith-type functional, which characterizes a possibility of curvilinear crack propagation along the prescribed path. Meanwhile, the length parameter of the straight crack is chosen as a control parameter.
Лазарев, Н. П. Задача оптимального управления длиной поперечной трещины в модели равновесия двумерного тела с двумя пересекающимися трещинами / Н. П. Лазарев, Е. М. Рудой, Т. С. Попова // Математические заметки СВФУ. — 2018. — Т. 25, N 3 (99), июль-сентябрь. — С. 43-53.
DOI: 10.25587/SVFU.2018.99.16950
Количество страниц: 14 с.
We study existence of the left inverse, right inverse and inverse of Gaussian infinite matrices (those are the upper infinite triangular matrices with nonzero elements on the main diagonal). The existence of a unique inverse of the Gaussian matrix is proved. Also, an explicit expression for the inverse of the Gaussian matrix of any order is found, including the infinite case. Implementation of this expression is very convenient, since calculations are based on recurrence relations. Such approach can be extended to triangular infinite matrices (those are the lower infinite triangular matrices with nonzero elements on the main diagonal). Thus, there is the possibility of inversion of an infinite matrix of infinite rank, since such matrices decompose into the product of two matrices, a triangular and a Gaussian.
Об обращении бесконечных гауссовых матриц / Ф. М. Федоров. Н. Н. Павлов, С. В. Потапова, О. Ф. Иванова // Математические заметки СВФУ. — 2018. — Т. 25, N 3 (99), июль-сентябрь. — С. 54-67.
DOI: 10/25587/SVFU.2018.99.16951
Количество страниц: 2 с.
Егоров, В. А. Разностные схемы для уравнений типа "мелкой воды" при численном моделировании паводковых процессов / Егоров В. А. // Космо- и геофизические явления и их математические модели : тезисы докладов Всероссийской научной конференции, посвященной 80-летию Кузьмина А. И., г. Якутск, 23-24 октября 2002 г. – Якутск : Издательство ЯГУ, 2002. – С. 59.
Количество страниц: 14 с.
Егоров, В. А. Численные расчеты вязких течений в модельных руслах с поймой / В. А. Егоров // Математические заметки ЯГУ. – 2008. – Т. 15, N 2. – С. 92-105.
Количество страниц: 14 с.
We present a numerical simulation of thermo-electrochemical processes of a Li-ion battery. Mathematical model of thermo-electrochemical processes is described on a microscopic scale and contains nonlinear equations for concentration, potential and temperature. A Li-ion battery consists of three subdomains: two electrodes and the electrolyte. On the interface of electrodes and electrolyte there are Lithium ions intercalation and deintercalation processes which are described by the Butler—Volmer nonlinear equation. The main problem of numerical implementation is the discontinuity of concentration and potential at the interface of the subdomains. To take into account the discontinuity, we use mixed finite elements in spatial approximation of a coupled system: discontinuous Galerkin elements for concentration and potential and continuous Galerkin elements for temperature. The time approximation is performed using a fully implicit scheme. The nonlinear system of equations obtained by approximation is solved by the Newton method.
Захаров, П. Е. Численное моделирование термо-электрохимических процессов в LI-ION аккумуляторах / П. Е. Захаров, М. А. Никифорова // Математические заметки СВФУ. - 2018. - Т. 25, N 4 (100), октябрь-декабрь. - C. 102-114.
DOI: 10.25587/SVFU.2018.100.20557
Количество страниц: 18 с.
We consider filtration problems in the fractured media that are necessary when modeling the processes of extracting hydrocarbons from unconventional reservoirs, geothermal fields development, underground disposal of radioactive waste in aquifers, etc. Fracture networks in such oil pools can exist on different scales and differ in the nature of their occurrence. We discuss a mathematical model of fluid filtration in fractured porous media described by coupled equations of mixed dimension with assigning of a special flow function. The problem approximation is constructed through the finite difference method on structured grids using the embedded fracture model, which makes possible creating grids for the porous medium matrix independently of the fracture network grid. The construction of a conservative difference scheme is given for the matrix of porous medium with the use of an integro-interpolation method and generalized for coupled equations describing mathematical models of multicontinuum with hierarchical representation of fracture networks. The results of the numerical implementation of the two-dimensional model problem are presented.
Васильева, М. В. Консервативная разностная схема для задач фильтрации в трещиноватых средах // Математические заметки СВФУ. — 2018. — Т. 25, N 4 (100), октябрь-декабрь. — C. 84-101.
DOI: 10.25587/SVFU.2018.100.20556
Количество страниц: 10 с.
We study the solvability of the initial-boundary value problem for linear integro-differential equations with a lateral boundary condition correlating values of the solution or its conormal derivative with values of some integral operator on the solution. We prove existence and uniqueness theorems for regular solutions. Recently, nonlocal boundary value problems for parabolic and hyperbolic equations with integral conditions on the lateral boundary are intensively studied, primarily in the classical case of secondand fourth-order equations. The systematic study of nonlocal boundary value problems, the problems of finding periodic solutions to elliptic equations, began in the article by A. V. Bitsadze and A. A. Samarskii (1969). A great contribution to the development of the theory of nonlocal problems for differential equations of various classes was made by A. L. Skubachevsky (1997) and A. M. Nakhushev (2006, 2012).
Попов, Н. С. О разрешимости нелокальных краевых задач для интегродифференциальных уравнений / Н. С. Попов // Математические заметки СВФУ. — 2018. — Т. 25, N 4 (100), октябрь-декабрь. — C. 74-83.
DOI: 10.25587/SVFU.2018.100.20555
Количество страниц: 14 с.
We study the class of test functions constructed on the principle of Lizorkin spaces by means of mixed Fourier-Kipriyanov-Katrakhov transform. Initially, such classes of functions, constructed on the basis of a mixed Fourier-Bessel transform, were investigated by L. N. Lyakhov. The spaces introduced by him could not take into account “odd” orders of singular derivatives. But the latter appeared to be fundamentally necessary in the problems of determining the fundamental solutions of differential equations (ordinary and in partial derivatives). The integral Kipriyanov-Katrakhov transform (belonging to the class of Bessel transforms) is adapted to work with singular differential operators of the type where k takes values 0 or 1, is a singular differential Bessel operator and the order of differentiation is 2m. The spaces of the basic functions that represent the images of the mixed Fourier-Kipriyanov-Katrakhov transform of functions vanishing at the origin and infinity are considered in this paper. We study the possibility of approximating functions from weighted Lebesgue classes with power weight namely, the density theorem in the Lebesgue function space.
Половинкина, М. В. О плотности специального класса функций Лизоркина в весовом лебеговом пространстве L (gamma) p / М. В. Половинкина, С. А. Рощупкин // Математические заметки СВФУ. — 2018. — Т. 25, N 4 (100), октябрь-декабрь. — С. 60-73.
DOI: 10.25587/SVFU.2018.100.20554