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卷 58, 编号 5 (2018)

Article

Trigonometric Tension B-Spline Method for the Solution of Problems in Calculus of Variations

Alinia N., Zarebnia M.

摘要

In this paper, the tension B-spline collocation method is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. The problems are reduced to an explicit system of algebraic equations by this approximation. We apply some numerical examples to illustrate the accuracy and implementation of the method.

Computational Mathematics and Mathematical Physics. 2018;58(5):631-641
pages 631-641 views

Optimal Strategy with One Closing Instant for a Linear Optimal Guaranteed Control Problem

Dmitruk N.

摘要

We consider an optimal guaranteed control problem for a linear time-varying system that is subject to unknown bounded disturbances. A control strategy is defined that guarantees steering the system to a given terminal set for any realization of disturbances and takes into account that at one future time instant the control loop will be closed. An efficient method for constructing the optimal control strategy and an algorithm for optimal feedback control based on this type of strategies are proposed.

Computational Mathematics and Mathematical Physics. 2018;58(5):642-658
pages 642-658 views

Quasi-Stable Structures in Circular Gene Networks

Glyzin S., Kolesov A., Rozov N.

摘要

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.

Computational Mathematics and Mathematical Physics. 2018;58(5):659-679
pages 659-679 views

Existence and Asymptotic Representation of the Autowave Solution of a System of Equations

Melnikova A., Chen M.

摘要

A singularly perturbed parabolic system of nonlinear reaction–diffusion equations is studied. Systems of this class are used to simulate autowave processes in chemical kinetics, biophysics, and ecology. A detailed algorithm for constructing an asymptotic approximation of a travelling front solution is proposed. In addition, methods for constructing an upper and a lower solution based on the asymptotics are described. According to the method of differential inequalities, the existence of an upper and a lower solution guarantees the existence of a solution to the problem under consideration. These methods can be used for asymptotic analysis of model systems in applications. The results can also be used to develop and justify difference schemes for solving problems with moving fronts.

Computational Mathematics and Mathematical Physics. 2018;58(5):680-690
pages 680-690 views

Kinetic Model and Magnetogasdynamics Equations

Chetverushkin B., D’Ascenzo N., Saveliev A., Saveliev V.

摘要

An original kinetic model for the molecular velocity distribution function is considered. Based on this model, the equations of ideal magnetogasdynamics (MGD) are derived and an original model for dissipative MGD is obtained. The latter model can be used to construct algorithms easily adaptable to high-performance computer architectures. As an example, results of high-performance computations of astrophysical phenomena are presented, namely, the formation of cosmic jets is modeled.

Computational Mathematics and Mathematical Physics. 2018;58(5):691-699
pages 691-699 views

Numerical Analysis of Spatial Hydrodynamic Stability of Shear Flows in Ducts of Constant Cross Section

Boiko A., Demyanko K., Nechepurenko Y.

摘要

A technique for analyzing the spatial stability of viscous incompressible shear flows in ducts of constant cross section, i.e., a technique for the numerical analysis of the stability of such flows with respect to small time-harmonic disturbances propagating downstream is described and justified. According to this technique, the linearized equations for the disturbance amplitudes are approximated in space in the plane of the duct cross section and are reduced to a system of first-order ordinary differential equations in the streamwise variable in a way independent of the approximation method. This system is further reduced to a lower dimension one satisfied only by physically significant solutions of the original system. Most of the computations are based on standard matrix algorithms. This technique makes it possible to efficiently compute various characteristics of spatial stability, including finding optimal disturbances that play a crucial role in the subcritical laminar–turbulent transition scenario. The performance of the technique is illustrated as applied to the Poiseuille flow in a duct of square cross section.

Computational Mathematics and Mathematical Physics. 2018;58(5):700-713
pages 700-713 views

Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

Elizarova T., Ivanov A.

摘要

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.

Computational Mathematics and Mathematical Physics. 2018;58(5):714-734
pages 714-734 views

Theoretical and Numerical Analysis of an Initial-Boundary Value Problem for the Radiative Transfer Equation with Fresnel Matching Conditions

Kim A., Prokhorov I.

摘要

A Cauchy problem for the time-dependent radiative transfer equation in a three-dimensional multicomponent medium with generalized matching conditions describing Fresnel reflection and refraction at the interface of the media is considered. The unique solvability of the problem is proven, a Monte Carlo method for solving the initial-boundary value problem is developed, and computational experiments for different implementations of the algorithm are conducted.

Computational Mathematics and Mathematical Physics. 2018;58(5):735-749
pages 735-749 views

On Determining Sources with Compact Supports in a Bounded Plane Domain for the Heat Equation

Solov’ev V.

摘要

The inverse problem of determining the source for the heat equation in a bounded domain on the plane is studied. The trace of the solution of the direct problem on two straight line segments inside the domain is given as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions whose derivatives satisfy the Hölder condition.

Computational Mathematics and Mathematical Physics. 2018;58(5):750-760
pages 750-760 views

Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion

Kazarinov N., Rudoy E., Slesarenko V., Shcherbakov V.

摘要

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.

Computational Mathematics and Mathematical Physics. 2018;58(5):761-774
pages 761-774 views

Computation of Traveling Waves in a Heterogeneous Medium with Two Pressures and a Gas Equation of State Depending on Phase Concentrations

Bedarev I., Fedorov A., Shul’gin A.

摘要

A fine structure theory of shock waves occurring in a gas–particle mixture was developed using an Anderson-type model with allowance for different phase pressures and with an equation of state for the gas component depending on the mean densities of both phases. The conditions for the formation of various types of shock waves based on the different speeds of sound in the phases were indicated. A high-order accurate TVD scheme was developed to prove the stability of some types of shock waves. The scheme was used to implement steadily propagating shock waves found in the stationary approximation, namely, shock waves of dispersive, frozen, and dispersive-frozen structures with one or two fronts.

Computational Mathematics and Mathematical Physics. 2018;58(5):775-789
pages 775-789 views

Numerical Methods for Computing Plausibility and Belief Distributions of Consequences of a Subjective Model of Object of Research

Balakin D.

摘要

Numerical methods for computing plausibility and belief distributions of consequences of a subjective model are considered. More precisely, related constrained optimization problems are studied. Error estimates of the proposed algorithms are obtained. Techniques for taking into account the information about the consequence available to the researcher for improving the accuracy of computations are discussed.

Computational Mathematics and Mathematical Physics. 2018;58(5):790-802
pages 790-802 views

Methods of the Convex Cone Theory in the Feasibility Problem of Multicommodity Flow

Grinberg Y.

摘要

The feasibility problem of multicommodity flow is reduced to finding out if a multidimensional vector determined by the network parameters belongs to a convex polyhedral cone determined by the set of paths in the network. It is shown that this representation of the feasibility problem makes it possible to formulate the feasibility criterion described in [1] in a different form. It is proved that this criterion is sufficient. The concepts of reference vectors and networks are defined, and they are used to describe a method for solving the feasibility problem for an arbitrary network represented by a complete graph.

Computational Mathematics and Mathematical Physics. 2018;58(5):803-812
pages 803-812 views

Octahedral Projections of a Point onto a Polyhedron

Zorkal’tsev V.

摘要

In computational methods and mathematical modeling, it is often required to find vectors of a linear manifold or a polyhedron that are closest to a given point. The “closeness” can be understood in different ways. In particular, the distances generated by octahedral, Euclidean, and Hölder norms can be used. In these norms, weight coefficients can also be introduced and varied. This paper presents the results on the properties of a set of octahedral projections of the origin of coordinates onto a polyhedron. In particular, it is established that any Euclidean and Hölder projection can be obtained as an octahedral projection due to the choice of weights in the octahedral norm. It is proven that the set of octahedral projections of the origin of coordinates onto a polyhedron coincides with the set of Pareto-optimal solutions of the multicriterion problem of minimizing the absolute values of all components.

Computational Mathematics and Mathematical Physics. 2018;58(5):813-821
pages 813-821 views

NP-Hardness of Some Euclidean Problems of Partitioning a Finite Set of Points

Kel’manov A., Pyatkin A.

摘要

Problems of partitioning a finite set of Euclidean points (vectors) into clusters are considered. The criterion is to minimize the sum, over all clusters, of (1) squared norms of the sums of cluster elements normalized by the cardinality, (2) squared norms of the sums of cluster elements, and (3) norms of the sum of cluster elements. It is proved that all these problems are strongly NP-hard if the number of clusters is a part of the input and are NP-hard in the ordinary sense if the number of clusters is not a part of the input (is fixed). Moreover, the problems are NP-hard even in the case of dimension 1 (on a line).

Computational Mathematics and Mathematical Physics. 2018;58(5):822-826
pages 822-826 views