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Siberian Journal of Numerical Mathematics

2017 year, number 2

On existence of a cycle in one asymmetric model of a molecular repressilator

N.B. Ayupova1,2, V.P. Golubyatnikov1,2, M.V. Kazantsev3
1Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
2Novosibirsk State University, 1 Pirogova str., Novosibirsk, Russia, 630090
3Polzunov 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 6-dimensional 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.

About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer

I.A. Blatov1, A.I. Zadorin2, E.V. Kitaeva3
1Volga region state university of telecommunications and informatics, Moskovskoe shosse, 77, Samara, Russia, 443090
2Sobolev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences, Omsk department, Pevtsova, 13, Omsk, Russia, 644043
3Samara 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 spline-interpolation 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 spline-interpolation 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.

On an approach to modeling wells

K.V. Voronin1,2, A.V. Grigoriev1,3, Yu.M. Laevsky1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, Russia, 630090
2Novosibirsk State University, 1 Pirogova str., Novosibirsk, Russia, 630090
3North-Eastern 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

Abstract >>
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.

Analysis of semilocal convergence in Banach spaces under relaxed condition and computational efficiency

J.P. Jaiswal1,2,3
1Maulana Azad National Institute of Technology, Bhopal, M.P., 462051, India
2Barkatullah University, Bhopal, M.P., 462026, India
3Regional 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 fifth-order 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.

A parallel algorithm of the multivariant evolutionary synthesis of nonlinear models

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.

A stochastic model of the nanowires growth by molecular beam epitaxy

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, self-preserved 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 3-4 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.

Numerical solution of second order one dimensional hyperbolic equation by exponential B-spline collocation method

Swarn Singh1, Suruchi Singh2, R. Arora3
1University of Delhi, New Delhi, 110021, India
2DM University of Delhi, New Delhi, 110007, India
3AM University of Delhi, Delhi, 110039, India
Keywords: уравнение затухающей волны, SSPRK(2,2), метод экспоненциальных В-сплайнов, телеграфное уравнение, трехдиагональный решатель, безусловно устойчивый метод, damped wave equation, exponential B-spline method, telegraphic equation, tri-diagonal solver, unconditionally stable method

Abstract >>
In this paper, we propose a method based on collocation of exponential B-splines 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 B-spline collocation method in space and two stage, second order strong-stability-preserving Runge-Kutta 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.

Discrete maximum-norm stability of a linearized second order finite difference scheme for Allen-Cahn equation

T. Hou, K. Wang, Y. Xiong, X. Xiao, Sh. Zhang
Beihua University, Jilin, 132013, China
Keywords: уравнение Аллена-Кана, конечно-разностный метод, устойчивость дискретной ограниченности, максимум-норма, Allen-Cahn equation, finite difference method, discrete boundedness stability, maximum norm

Abstract >>
In this paper, we use finite difference methods for solving the Allen-Cahn equation which contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three level scheme in time and a second-order 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.