Rubric: "01.01.00 Mathematics"
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Sorokin V. G. (Ishlinsky Institute for Problems in Mechanics), Vyazmin A. V. (Moscow State University of Mechanical Engineering)
doi: 10.18698/2309-3684-2014-4-5373
In the article we explored nonlinear hyperbolic delay reaction-diffusion equations with varying transfer coefficients. A number of generalized separable solutions were obtained. Most of the equations considered contain arbitrary functions. Global nonlinear instability conditions of solutions of hyperbolic delay reaction-diffusion systems were determined. The generalized Stokes problem for a linear delay diffusion equation with periodic boundary conditions was solved.
Polyanin A., Sorokin V., Vyazmin A. Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 53-73
Dubrovin V. M. (Bauman Moscow State Technical University), Butina T. A. (Bauman Moscow State Technical University), Polyakova N. S. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2015-4-3852
The objective of this research is to examine the shock wave with cylindrical shell and to propose a method for calculating its dynamic stability under axial compressive timevarying load. For weak shock waves we conducted comparative analysis of the exact solution and the existing approximate solutions. We evaluated the wave radiation effect after the shell deformation. The case of linearly varying load was considered as an example.
Dubrovin V., Butina T., Polyakova N. Modeling of the process of interaction of the shock wave with cylindrical shell considering wave radiation effect. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 38-52
519.63 Development and testing for methods of solving stiff ordinary differential equations
Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Khodzhaeva S. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-95119
The paper is aimed at research of the (m,k)-method, CROS, finite superelement method and 4-stage explicit Runge–Kutta method for solving stiff systems of ordinary differential equations. Analysis of tests results showed that the best choice for systems with high stiffness is CROS. The finite superelement method is the «precise» method for solving linear systems of ordinary differential equations, the best supporting method for its implementation is (4,2)-method. The variation of the finite superelement method has been built and tested for solving nonlinear problems, this method proved to be unsuitable for problems with high stiffness.
Galanin M., Khodzhaeva S. Development and testing for methods of solving stiff ordinary differential equations. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 95-119
517.9:519.6 Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system
Malinetsky G. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Faller D. S. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)
doi: 10.18698/2309-3684-2014-3-111125
The article discusses emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of reaction–diffusion models. We studied the dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors. We have shown that chaos is emerging in an unconventional pattern with chaotic regimes emerging and disappearing repeatedly. We had already studied this unconventional pattern for one-dimensional maps with a sharp apex and a quadratic minimum. We applied numerical analysis to study characteristic properties of the system, such as bistability and hyperbolicity zones, crisis of chaotic attractors.
Malinetsky G., Faller D. Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 111-125
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics)
doi: 10.18698/2309-3684-2015-4-337
We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N
n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.
Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37
Frolov A. V. (Water Problems Institute of the Russian Academy of Sciences)
doi: 10.18698/2309-3684-2016-3-7992
The paper considers long-term fluctuations of the Caspian Sea level as a nonlinear system output with positive and negative feedbacks. The Caspian Sea model with due consideration of an outflow into the Kara-Bogaz-Gol Bay is designed. Density distribution of the sea level is obtained as a solution to the corresponding Fokker — Planck — Kolmogorov equation. The bimodal probability density of the sea level distribution, which meets the endorheic Caspian Sea (if you cut off the Kara-Bogaz-Gol Bay), is shown to turn into the single-mode probability density in case of simultaneous influence of evaporation and seawater outflow into the Kara-Bogaz-Gol Bay on the sea level.
Frolov A. Modelling influence of outflow into the Kara-Bogaz-Gol Bay on probability density of the Caspian Sea level fluctuations. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 79-92
519.237.07 Factorial modeling using neural network
Chauvigny V. A. (Sobolev Institute of Mathematics, Omsk branch, Siberian Branch of the Russian Academy of Sciences), Goltiapin V. V. (Sobolev Institute of Mathematics, Omsk branch, Siberian Branch of the Russian Academy of Sciences)
doi: 10.18698/2309-3684-2016-2-85103
The paper deals with the factorial modeling of the initial stage arterial hypertension. The modeling was carried out by the factorization method based on the neural network and the back propagation of error algorithm. This factorization method is an alternative to the classical factor analysis. We implemented an algorithm for constructing the factorial structure based on the neural network in software. This method has been improved for the factor rotation and obtaining an interpretable solution. The hypertension factorial structure obtained by this factorization method is in accordance with the results of the factorial modeling by other methods.
Chauvigny V., Goltiapin V. Factorial modeling using neural network. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 85-103
519.8 Stochastic models of the two unit duel fight
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-2-6984
On the basis of the theory of continuous Markov processes we developed models of the two unit duel fight. We obtained computing formulas for calculating the basic fight indicators. Moreover, we found that the pre-emptive strike of one of the units participating in the fight has a significant impact on the fight outcome of the units which are similar in forces. The strike has a negligible impact, if one of the units has a significant advantage. The findings of the research show that the use of model with constant effective firing rates can lead to significant errors in the evaluation of its results. Finally, we found that the pre-emptive strike, coupled with a high degree of effective firing rate growth, can sometimes compensate for more than the double initial superiority of the opponent. We show the possibility of using approximations of the effective firing rate of the fighting units by the different functions of the fight time.
Chuev V., Dubogray I. Stochastic models of the two unit duel fight. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 69-84
519.8 “Mixed” probabilistic models of bilateral military operations of numerous groups
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Dyakova L. N. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2017-1-91101
The purpose of this work was to develop "mixed" probabilistic models of bilateral military operations according to the theory of continuous Markov processes. In our research we obtained calculation formulas for estimating the main combat indices of groups small in number. Moreover, we developed a numerical algorithm to calculate the main combat indices of numerous groupings and made a comparison with the results of combat simulation using a deterministic model of two-way combat operations, the model being developed according to themean-value method dynamics. Findings of the research show that the correlation of the forces of the opposing sides, rather than their initial numbers, affects the errors in the mean-value method dynamics.
Chuev V., Dubogray I., Dyakova L. “Mixed” probabilistic models of bilateral military operations of numerous groups. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 91-101