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卷 58, 编号 2 (2018)

Article

On the 100th Anniversary of the Birthday of Academician Nikita Nikolaevich Moiseev

Kerimov M., Pospelov I.
Computational Mathematics and Mathematical Physics. 2018;58(2):153-154
pages 153-154 views

Geometric Theory of Reduction of Nonlinear Control Systems

Elkin V.

摘要

The foundations of a differential geometric theory of nonlinear control systems are described on the basis of categorical concepts (isomorphism, factorization, restrictions) by analogy with classical mathematical theories (of linear spaces, groups, etc.).

Computational Mathematics and Mathematical Physics. 2018;58(2):155-158
pages 155-158 views

Projective-Dual Method for Solving Systems of Linear Equations with Nonnegative Variables

Ganin B., Golikov A., Evtushenko Y.

摘要

In order to solve an underdetermined system of linear equations with nonnegative variables, the projection of a given point onto its solutions set is sought. The dual of this problem—the problem of unconstrained maximization of a piecewise-quadratic function—is solved by Newton’s method. The problem of unconstrained optimization dual of the regularized problem of finding the projection onto the solution set of the system is considered. A connection of duality theory and Newton’s method with some known algorithms of projecting onto a standard simplex is shown. On the example of taking into account the specifics of the constraints of the transport linear programming problem, the possibility to increase the efficiency of calculating the generalized Hessian matrix is demonstrated. Some examples of numerical calculations using MATLAB are presented.

Computational Mathematics and Mathematical Physics. 2018;58(2):159-169
pages 159-169 views

Inverse Problems in Economic Measurements

Shananin A.

摘要

The problem of economic measurements is discussed. The system of economic indices must reflect the economic relations and mechanisms existing in society. An achievement of the XX century is the development of a system of national accounts and the gross domestic product index. However, the gross domestic product index, which is related to the Hamilton–Pontryagin function in extensive economic growth models, turns out to be inadequate under the conditions of structural changes. New problems of integral geometry related to production models that take into account the substitution of production factors are considered.

Computational Mathematics and Mathematical Physics. 2018;58(2):170-179
pages 170-179 views

Scalarization Method in Multicriteria Games

Novikova N., Pospelova I.

摘要

Using a two-criteria two-person game as an example, the validity of the scalarization method applied for the parameterization of the set of game values and for estimating the players’ payoffs is investigated. It is shown that the use of linear scalarization by the players gives the results different from those obtained using Germeyer’s scalarization. Various formalizations of the concept of value of MC games are discussed.

Computational Mathematics and Mathematical Physics. 2018;58(2):180-189
pages 180-189 views

Some Continuous Methods for Solving Quasi-Variational Inequalities

Mijajlović N., Jaćimović M.

摘要

The continuous gradient projection method and the continuous gradient-type method in a space with a variable metric are studied for the numerical solution of quasi-variational inequalities, and conditions for the convergence of the methods proposed are established.

Computational Mathematics and Mathematical Physics. 2018;58(2):190-195
pages 190-195 views

New External Estimate for the Reachable Set of a Nonlinear Multistep Dynamic System

Lotov A.

摘要

An approach is proposed for estimating the reachable set of a nonlinear multistep dynamic system by approximating the effective hull of this set. The approximation of the effective hull relies on local optimization of specially chosen functions of the system’s state. Methods using global optimization of such functions are briefly described. Approximation methods based on local optimization are considered as applied to the effective hull of a reachable set, and a statistical estimate for the quality of the effective hull approximation is constructed.

Computational Mathematics and Mathematical Physics. 2018;58(2):196-206
pages 196-206 views

Primal Newton Method for the Linear Cone Programming Problem

Zhadan V.

摘要

A linear cone programming problem containing among the constraints a second-order cone is considered. For solving this problem, a primal Newton method which is constructed with the help of the optimality conditions is proposed. Local convergence of this method is proven.

Computational Mathematics and Mathematical Physics. 2018;58(2):207-214
pages 207-214 views

Computational Efficiency of the Simplex Embedding Method in Convex Nondifferentiable Optimization

Kolosnitsyn A.

摘要

The simplex embedding method for solving convex nondifferentiable optimization problems is considered. A description of modifications of this method based on a shift of the cutting plane intended for cutting off the maximum number of simplex vertices is given. These modification speed up the problem solution. A numerical comparison of the efficiency of the proposed modifications based on the numerical solution of benchmark convex nondifferentiable optimization problems is presented.

Computational Mathematics and Mathematical Physics. 2018;58(2):215-222
pages 215-222 views

Solution of Tikhonov’s Motion-Separation Problem Using the Modified Newton–Kantorovich Theorem

Belolipetskii A., Ter-Krikorov A.

摘要

The paper presents a new way to prove the existence of a solution of the well-known Tikhonov’s problem on systems of ordinary differential equations in which one part of the variables performs “fast” motions and the other part, “slow” motions. Tikhonov’s problem has been the subject of a large number of works in connection with its applications to a wide range of mathematical models in natural science and economics. Only a short list of publications, which present the proof of the existence of solutions in this problem, is cited. The aim of the paper is to demonstrate the possibility of applying the modified Newton–Kantorovich theorem to prove the existence of a solution in Tikhonov’s problem. The technique proposed can be used to prove the existence of solutions of other classes of problems with a small parameter.

Computational Mathematics and Mathematical Physics. 2018;58(2):223-229
pages 223-229 views

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

Pikulin S.

摘要

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.

Computational Mathematics and Mathematical Physics. 2018;58(2):230-237
pages 230-237 views

Some Fundamental Issues of Mathematical Simulation in Biology

Razzhevaikin V.

摘要

Some directions of simulation in biology leading to original formulations of mathematical problems are overviewed. Two of them are discussed in detail: the correct solvability of first-order linear equations with unbounded coefficients and the construction of a reaction–diffusion equation with nonlinear diffusion for a model of genetic wave propagation.

Computational Mathematics and Mathematical Physics. 2018;58(2):238-247
pages 238-247 views

Expansion of a Rarefied Gas Cloud in a Vacuum: Asymptotic Treatment

Zhuk V.

摘要

The unsteady expansion of a rarefied gas of finite mass in an unlimited space is studied. The long-time asymptotic behavior of the solution is examined at Knudsen numbers tending to zero. An asymptotic analysis shows that, in the limit of small Knudsen numbers, the behavior of the macroscopic parameters of the expanding gas cloud at long times (i.e., for small density values) has nothing to do with the free-molecular or continuum flow regimes. This conclusion is unexpected and not obvious, but follows from a uniformly suitable solution constructed by applying the method of outer and inner asymptotic expansions. In particular, the unusual temperature behavior is of interest as applied to remote sensing of rocket exhaust plumes.

Computational Mathematics and Mathematical Physics. 2018;58(2):248-253
pages 248-253 views

Nonclassical Transonic Boundary Layers: Toward Overcoming Dead-End Situations in High-Speed Aerodynamics

Bogdanov A., Diesperov V., Zhuk V.

摘要

Analytical models of unsteady free viscous-inviscid interaction of gas flows at transonic speeds, i.e., a transonic boundary layer with self-induced pressure (nonclassical boundary layer) are considered. It is shown that an adequate flow model can be constructed by applying methods of singular perturbations. The results of a comparative analysis of classical and regularized stability models for a boundary layer with self-induced pressure in the case of interaction at transonic speeds are overviewed.

Computational Mathematics and Mathematical Physics. 2018;58(2):254-263
pages 254-263 views

Absolute Instability of Incompressible Boundary Layer over a Compliant Plate

Savenkov I.

摘要

An incompressible boundary layer on a compliant plate is considered. The influence exerted by the tensile stress and bending stiffness of the plate on the stability of the boundary layer is investigated in the limit of high Reynolds numbers on the basis of the triple-deck theory. It is shown that upstream-propagating growing waves can be generated in a certain range of parameters characterizing the plate properties. As a result, the flow becomes absolutely unstable in the conventional sense.

Computational Mathematics and Mathematical Physics. 2018;58(2):264-273
pages 264-273 views

High-Order Multioperator Compact Schemes for Numerical Simulation of Unsteady Subsonic Airfoil Flow

Savel’ev A.

摘要

On the basis of high-order schemes, the viscous gas flow over the NACA2212 airfoil is numerically simulated at a free-stream Mach number of 0.3 and Reynolds numbers ranging from 103 to 107. Flow regimes sequentially varying due to variations in the free-stream viscosity are considered. Vortex structures developing on the airfoil surface are investigated, and a physical interpretation of this phenomenon is given.

Computational Mathematics and Mathematical Physics. 2018;58(2):274-285
pages 274-285 views

On a Heat Exchange Problem under Sharply Changing External Conditions

Khishchenko K., Charakhch’yan A., Shurshalov L.

摘要

The heat exchange problem between carbon particles and an external environment (water) is stated and investigated based on the equations of heat conducting compressible fluid. The environment parameters are supposed to undergo large and fast variations. In the time of about 100 μs, the temperature of the environment first increases from the normal one to 2400 K, is preserved at this level for about 60 μs, and then decreases to 300 K during approximately 50 μs. At the same periods of time, the pressure of the external environment increases from the normal one to 67 GPa, is preserved at this level, and then decreases to zero. Under such external conditions, the heating of graphite particles of various sizes, their phase transition to the diamond phase, and the subsequent unloading and cooling almost to the initial values of the pressure and temperature without the reverse transition from the diamond to the graphite phase are investigated. Conclusions about the maximal size of diamond particles that can be obtained in experiments on the shock compression of the mixture of graphite with water are drawn.

Computational Mathematics and Mathematical Physics. 2018;58(2):286-293
pages 286-293 views