Open Access
Access granted
Subscription Access
Vol 63, No 8 (2023)
ОБЩИЕ ЧИСЛЕННЫЕ МЕТОДЫ
Generalization of the Fast Fourier Transform with a Constant Structure
Abstract
The widely popular famous fast Cooley–Tukey algorithms for the discrete Fourier transform of mixed radix are presented in two forms: classical and with a constant structure. A matrix representation of these algorithms is proposed in terms of two types of tensor product of matrices: the Kronecker product and the b-product. This matrix representation shows that the structure of these algorithms is identical to two fast Good algorithms for a Kronecker power of a matrix. A technique for constructing matrix-form fast algorithms for the discrete Fourier and Chrestenson transforms with mixed radix and for the discrete Vilenkin transform is demonstrated. It is shown that the constant-structured algorithm is preferable in the case of more sophisticated constructions
1241-1250
Approximation of Differential Operators with Boundary Conditions
Abstract
The use of spectral methods for solution of boundary value problems is very effective but involves great technical difficulties associated with the implementation of the boundary conditions. There exist several methods of such an implementation, but they are either very cumbersome or require a preliminary analysis of the problem and its reduction to an integral form. We propose a universal means of implementation of the boundary conditions for linear differential operators on a finite interval, which is very simple in its realization. The use of the rational arithmetic allows to assess the effectiveness of this method without interference of the round-off errors. We apply this approach for computation of rational approximations for some fundamental constants. We obtained approximations that in a number of cases are better than those that are given by convergents of regular continued fractions of these constants.
1251-1271
An Improved Difference Scheme for the Cauchy Problem in the Case of a Transport Equation
Abstract
The Cauchy problem for the regular transport equation is considered. The Richardson technique is used to construct an improved difference scheme that converges in the maximum norm with the second order of convergence.
1272-1278
On Simultaneous Determination of Thermal Conductivity and Volume Heat Capacity of Substance
Abstract
The study of nonlinear problems associated with heat transfer in substance is important for practice. Earlier, the authors proposed an efficient algorithm for determining the thermal conductivity from experimental observations of the dynamics of the temperature field in an object. In this work, we explore the possibility of extending the algorithm to the numerical solution of the problem of simultaneous identification of the temperature-dependent volume heat capacity and the thermal conductivity of the substance under study. The consideration is based on the Dirichlet boundary value problem for the one-dimensional nonstationary heat equation. The coefficient inverse problem in question is reduced to a variational problem, which is solved by applying gradient methods based on the fast automatic differentiation technique. The uniqueness of the solution to the inverse problem is analyzed.
1279-1295
ОБЫКНОВЕННЫЕ ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ
On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side
Abstract
A boundary value problem for a second-order ordinary differential equation with a parameter and discontinuous right-hand side is considered. Theorems on the number of solutions to the problem are established. The resulting solutions are illustrated by plots. The process of numerically solving the problem is described.
1296-1308
УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ
On Some Elliptic Boundary Value Problems in Conic Domains
Abstract
A model elliptic pseudodifferential equation in a polyhedral cone is considered, and the situation when some of the parameters of the cone tend to their limiting values is investigated. In Sobolev–Slobodetskii spaces, a solution of the equation in the cone is constructed in the case of a special wave factorization of the elliptic symbol. It is shown that a limit solution of the boundary value problem with an additional integral condition can exist only under additional constraints on the boundary function.
1309-1315
Application of IBSEF Method to Chaffee-Infante Equation in (1 + 1) and (2 + 1) Dimensions
Abstract
Применение улучшенного метода Бернулли к решению уравнения Чаффи–Инфанта для размерности (1 + 1) и (2 + 1)
. Рассмотрен модифицированный метод решения нестационарного уравнения Чаффи–Инфанта с помощью сведения к уравнению Бернулли в случае размерности (1 + 1) и (2 + 1). Используя представление решения в виде бегущей волны для некоторых частных случаев, удается получить явный вид решения.
1316
On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case
Abstract
The paper considers issues of unique solvability of systems of linear algebraic equations to which many inverse problems of geophysics are reduced as a result of discretization. Examples of degenerate and nondegenerate systems of different dimensions arising from the interpretation of gravity and magnetometric data are given.
1317-1331
Analogue of Kellogg’s Theorem for Piecewise Lyapunov Domains
Abstract
In weighted Hölder spaces, classes of smooth arcs and piecewise smooth contours are introduced that are invariant under power mappings. The boundary properties of conformal mappings are described in terms of these classes by analogy with Kellogg’s classical theorem.
1332-1342
МАТЕМАТИЧЕСКАЯ ФИЗИКА
Internal Gravity Waves from a Localized Source in Stratified Medium Flow with a Model Buoyancy Frequency Distribution
Abstract
The problem of calculating internal gravity wave fields generated by a localized source in a stratified flow of finite thickness with a model buoyancy frequency distribution is considered. By using analytical representations of the buoyancy frequency, an implicit form of the dispersion relation is obtained, which depends on the Bessel functions of real index. Numerical results for dispersion curves, lines of equal phase, and wave amplitudes for various wave modes and stratified flow velocities are presented. The thickness of the stratified layer, the vertical gradient of the buoyancy frequency, and the magnitude of the flow velocity are the main factors affecting the amplitude-phase spatial transformation of the wave fields excited downstream.
1343-1353
Simulation of Emission Processes in Strong Electromagnetic Fields
Abstract
The problem of calculating the processes of electron emission from metal surfaces in strong electromagnetic fields is considered with allowance for relativistic effects. One of the methods of simulation in these processes is the particle method combined with grid calculation of fields on the basis of Maxwell’s equations. Similar techniques have been developed since the 1960s to the present. However, existing approaches have certain limitations. In this work, for an axisymmetric geometry of the generating system, a new numerical technique simulating the processes of electron emission from metal cathode surfaces is presented. The technique uses the representation of large smoothed Gaussian particles and implements the calculation of electromagnetic fields on Cartesian spatial grids. The software implementation is oriented to parallel computing. The aim of numerical experiments was to determine the parameters of electron emission. Diode and triode cylindrical systems were chosen as test problems. In numerical calculations, the spatiotemporal characteristics of relativistic electron beams generated by emission processes are obtained, including the reproduction of the Child–Langmuir current. The numerical technique developed has confirmed its correctness and efficiency.
1354-1366
Оценка погрешности и оптимизация метода прямого статистического моделирования с учетом пространственной регуляризации
Abstract
Метод прямого статистического моделирования широко применяется для решения задач динамики разреженного газа. Представленная работа направлена на исследование погрешности, вносимой пространственной регуляризацией взаимодействия двух частиц. Рассматриваются два подхода к пространственной регуляризации и три алгоритма метода прямого статистического моделирования, реализующих эти подходы. Для этих алгоритмов построена верхняя граница погрешности в метрике пространства непрерывных функций и получены условно-оптимальные параметры, гарантирующие по вероятности заданный уровень погрешности. На примере классической задачи Фурье проведено численное исследование погрешности, вносимой регуляризацией, и тестирование построенных условно-оптимальных параметров. Библ. 28. Фиг. 4. Табл. 4.
1367-1379
Numerical Simulation of Phase Transfer During Cryosurgery for an Irregular Tumor Using Hybrid Approach
Abstract
Численное моделирование фазового перехода при криохирургии опухоли неправильной формы с использованием гибридного подхода
. Проведено численное решение уравнения теплопроводности в двумерной нестационарной модели Пеннеса, описывающей пораженные опухолевые клетки. Эллиптический криозонд различных размеров берется в центре вычислительной области таким образом, что местоположение зонда фиксируется на протяжении всего вычисления. Фазовый переход происходит за счет воздействия зонда с введением различных наночастиц золота Au, окиси алюминия Al2O3 и окиси железа Fe3O4. Эффективность охлаждения этих наночастиц, вводимых при очень низкой температуре, была изучена с помощью гибридного метода конечных элементов, в котором вся область разбивается на две подобласти. Результаты показаны в виде температурного профиля внутри расчетной области. Скорость охлаждения получена для различных наночастиц, и замечено, что введение наночастиц золота очень эффективно увеличивает скорость нагревания в области пораженных опухолевых клеток.
1380
ИНФОРМАТИКА
Методы построения оценок множеств достижимости в задаче моделирования потоков людей
Abstract
Работа посвящена математическому моделированию потоков людей в помещении. За основу взята модификация дискретной макромодели CTM, построенная на гарантированных оценках. Для описанной модели предложено два способа приближенного вычисления множества достижимости – количества людей в каждой комнате в последующий момент времени. Строятся интервальные оценки и оценки в форме совокупностей двумерных проекций. Предложенные алгоритмы проиллюстрированы численными примерами. Библ. 14. Фиг. 3.
1381-1394
On the Set of Stable Matchings in a Bipartite Graph
Abstract
The topic of stable matchings (marriages) in bipartite graphs gained popularity beginning from the appearance of the classical Gale and Shapley work. In this paper, a detailed review of selected and other related statements in this field that describe structured, polyhedral, and algorithmic properties of such objects and their sets accompanied by short proofs is given.
1395-1412