

2017 year, number 2
N.B. Ayupova^{1,2}, V.P. Golubyatnikov^{1,2}, M.V. Kazantsev^{3}
^{1}Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090 ^{2}Novosibirsk State University, 1 Pirogova str., Novosibirsk, Russia, 630090 ^{3}Polzunov Altai State Technical University, Lenina avenue, 46, Barnaul, Altai region, Russian, 656038
Keywords: нелинейная динамическая система, модели генных сетей, дискретизация фазового портрета, гиперболические стационарные точки, циклы, теорема Брауэра о неподвижной точке, nonlinear dynamical systems, gene networks models, phase portrait's discretization, hyperbolic equilibrium points, cycles, Brower's fixed point theorem
Abstract >>
We consider a nonlinear 6dimensional dynamic system which is a model of functioning of one simple molecular repressilator and find sufficient conditions of existence of a cycle C in the phase portrait of this system. An invariant neighborhood of C which retracts to C has been constructed.

I.A. Blatov^{1}, A.I. Zadorin^{2}, E.V. Kitaeva^{3}
^{1}Volga region state university of telecommunications and informatics, Moskovskoe shosse, 77, Samara, Russia, 443090 ^{2}Sobolev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences, Omsk department, Pevtsova, 13, Omsk, Russia, 644043 ^{3}Samara national research University named after academician S.P. Korolyov, Moskovskoe shosse, 34, Samara, Russia, 443086
Keywords: сингулярное возмущение, пограничный слой, сетка Шишкина, параболический сплайн, модификация, оценка погрешности, singular perturbation, boundary layer, Shishkin mesh, parabolic spline, modification, estimation of error
Abstract >>
A problem of the Subbotin parabolic splineinterpolation of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it has been proved and in the case of the Shishkin grid it has been experimentally shown that with a parabolic splineinterpolation of functions with large gradients the error in the exponential boundary layer can unrestrictedly increase with a fixed number of grid nodes. A modified parabolic spline has been constructed. Estimates of the interpolation error of the constructed spline don't depend from a small parameter.

K.V. Voronin^{1,2}, A.V. Grigoriev^{1,3}, Yu.M. Laevsky^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, Russia, 630090 ^{2}Novosibirsk State University, 1 Pirogova str., Novosibirsk, Russia, 630090 ^{3}NorthEastern Federal University, 58 Belinsky str., Yakutsk, Republic of Sakha (Yakutia), Russia, 677027
Keywords: скважины, смешанная формулировка, смешанный метод конечных элементов, оценка погрешности, wells, mixed formulation, mixed finite element method, error estimate
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This paper deals with a numerical study of the diffusion problem in the presence of wells, at which integral boundary conditions are used. It is shown that the method proposed earlier is fully efficient and offers certain advantages as compared with the direct modeling of wells based on the finite element method. The results of calculations for the two wells are presented.

J.P. Jaiswal^{1,2,3}
^{1}Maulana Azad National Institute of Technology, Bhopal, M.P., 462051, India ^{2}Barkatullah University, Bhopal, M.P., 462026, India ^{3}Regional Institute of Education, Bhopal, M.P., 462013, India
Keywords: нелинейное уравнение, банахово пространство, слабое условие, полулокальная сходимость, граница ошибки, nonlinear equation, Banach space, weak condition, semilocal convergence, error bound
Abstract >>
The present paper is concerned with the study of semilocal convergence of a fifthorder method for solving nonlinear equations in Banach spaces under mild conditions. An existence and uniqueness theorem is proved and followed by error estimates. The computational superiority of the considered scheme over the identical order methods is also examined, which shows the efficiency of the present scheme from a computational point of view. Lastly, an application of the theoretical development is made in a nonlinear integral equation.

O.G. Monakhov, E.A. Monakhova
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: параллельный многовариантный эволюционный синтез, генетический алгоритм, генетическое программирование, декартово генетическое программирование, нелинейные модели, parallel multivariant evolutionary synthesis, genetic algorithm, genetic programming, Cartesian genetic programming, nonlinear models
Abstract >>
A parallel algorithm for solving the problem of constructing of nonlinear models (mathematical expressions, functions, algorithms, programs) based on given experimental data, a set of variables, basic functions and operations is proposed. The proposed algorithm of the multivariant evolutionary synthesis of nonlinear models has a linear representation of the chromosome, the modular operations in decoding the genotype to the phenotype for interpreting a chromosome as a sequence of instructions, the multivariant method for presenting a multiplicity of models (expressions) using a single chromosome. A comparison of the sequential version of the algorithm with a standard algorithm of genetic programming and the algorithm of the Cartesian Genetic Programming offers advantage of the algorithm proposed both in the time of obtaining a solution (by about an order of magnitude in most cases), and in the probability of finding a given function (model). In the experiments on the parallel supercomputer systems, estimates of the efficiency of the proposed parallel algorithm have been obtained showing linear acceleration and scalability.

K.K. Sabelfeld, E.G. Kablukova
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: нановискеры, адатомы, диффузия по поверхности, вероятность выживания, многократное рассеяние, устойчивое распределение по высотам, nanowires, adatoms, surface diffusion, survival probability, multiple scattering, selfpreserved height distribution
Abstract >>
In this paper a stochastic model of the nanowire growth by molecular beam epitaxy based on probability mechanisms of surface diffusion, mutual shading, adatoms rescattering and survival probability is proposed. A direct simulation algorithm based on this model is implemented, and a comprehensive study of the growth kinetics of a family of nanowires initially distributed at a height of about tens of nanometers to heights of about several thousands of nanometers is carried out. The time range corresponds to growing nanowires experimentally for up to 34 hours. In this paper we formulate a statement, which is numerically confirmed: under certain conditions, which can be implemented in real experiments, the nanowires height distribution becomes narrower with time, i.e. in the nanowires ensemble their heights are aligned in the course of time. For this to happen, it is necessary that the initial radius distribution of nanowires be narrow and the density of the nanowires on a substrate be not very high.

Swarn Singh^{1}, Suruchi Singh^{2}, R. Arora^{3}
^{1}University of Delhi, New Delhi, 110021, India ^{2}DM University of Delhi, New Delhi, 110007, India ^{3}AM University of Delhi, Delhi, 110039, India
Keywords: уравнение затухающей волны, SSPRK(2,2), метод экспоненциальных Всплайнов, телеграфное уравнение, трехдиагональный решатель, безусловно устойчивый метод, damped wave equation, exponential Bspline method, telegraphic equation, tridiagonal solver, unconditionally stable method
Abstract >>
In this paper, we propose a method based on collocation of exponential Bsplines to obtain numerical solution of nonlinear second order one dimensional hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The method is a combination of Bspline collocation method in space and two stage, second order strongstabilitypreserving RungeKutta method in time. The proposed method is shown to be unconditionally stable. The efficiency and accuracy of the method are successfully described by applying the method to a few test problems.

T. Hou, K. Wang, Y. Xiong, X. Xiao, Sh. Zhang
Beihua University, Jilin, 132013, China
Keywords: уравнение АлленаКана, конечноразностный метод, устойчивость дискретной ограниченности, максимумнорма, AllenCahn equation, finite difference method, discrete boundedness stability, maximum norm
Abstract >>
In this paper, we use finite difference methods for solving the AllenCahn equation which contains small perturbation parameters and strong nonlinearity. We consider a linearized secondorder three level scheme in time and a secondorder finite difference approach in space, and we establish discrete boundedness stability in maximum norm: if the initial data is bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. We will show that the main result can be obtained under certain.

