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1
Content available Asymptotic analysis of a closed G-network of unreliable nodes
Rusilko Tatiana
Journal of Applied Mathematics and Computational Mechanics
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2022
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Vol. 21, nr 2
91--102
EN
A closed exponential queueing G-network of unreliable multi-server nodes was studied under the asymptotic assumption of a large number of customers. The process of changing the number of functional servers in network nodes was considered as the birth-death process. The process of changing the number of customers at the nodes was considered as a continuous-state Markov process. It was proved that its probability density function satisfies the Fokker-Planck-Kolmogorov equation. The system of differential equations for the first-order and second-order moments of this process was derived. This allows us to predict the expectation, the variance and the pairwise correlation of the number of customers in the G-network nodes both in the transient and steady state.
2
Content available Diffusion approximation of the network with limited number of same type customers and time dependent service parameters
Matalytski M., Kopats D.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 2
77--84
EN
The article presents research of an open queueing network (QN) with the same types of customers, in which the total number of customers is limited. Service parameters are dependent on time, and the route of customers is determined by an arbitrary stochastic transition probability matrix, which is also dependent on time. Service times of customers in each line of the system is exponentially distributed. Customers are selected on the service according to FIFO discipline. It is assumed that the number of customers in one of the systems is determined by the process of birth and death. It generates and destroys customers with certain service times of rates. The network state is described by the random vector, which is a Markov random process. The purpose of the research is an asymptotic analysis of its process with a big number of customers, obtaining a system of differential equations (DE) to find the mean relative number of customers in the network systems at any time. A specific model example was calculated using the computer. The results can be used for modelling processes of customer service in the insurance companies, banks, logistics companies and other organizations.
3
Content available The island model as a Markov dynamic system
Schaefer R., Byrski A., Smołka M.
International Journal of Applied Mathematics and Computer Science
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2012
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Vol. 22, no. 4
971-984
EN
Parallel multi-deme genetic algorithms are especially advantageous because they allow reducing the time of computations and can perform a much broader search than single-population ones. However, their formal analysis does not seem to have been studied exhaustively enough. In this paper we propose a mathematical framework describing a wide class of island-like strategies as a stationary Markov chain. Our approach uses extensively the modeling principles introduced by Vose, Rudolph and their collaborators. An original and crucial feature of the framework we propose is the mechanism of inter-deme agent operation synchronization. It is important from both a practical and a theoretical point of view. We show that under a mild assumption the resulting Markov chain is ergodic and the sequence of the related sampling measures converges to some invariant measure. The asymptotic guarantee of success is also obtained as a simple issue of ergodicity. Moreover, if the cardinality of each island population grows to infinity, then the sequence of the limit invariant measures contains a weakly convergent subsequence. The formal description of the island model obtained for the case of solving a single-objective problem can also be extended to the multi-objective case.
4
Content available remote Topological derivatives for semilinear elliptic equations
Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
5
Content available remote Vibrations of strongly degenerated 3D-1D multi-structures: Asymptotic analysis
Aslanyan A., Dronka J.
Journal of Mathematics and Applications
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2007
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Vol. 29
53-73
EN
We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with some thin elastic cyliriders. An asyrnptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae for various types of degeneracy are given.
6
Content available remote A level set method in shape and topology optimization for variational inequalities
Fulmański P., Laurain A., Scheid J. F., Sokołowski J.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 3
413-430
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
7
Content available remote An asymptotic analysis of convection in boundary layer flows in the presence of a chemical reaction
Shateyi S., Sibanda P., Motsa S. S.
Archives of Mechanics
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2005
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Vol. 57, nr 1
25--41
EN
In this study we investigate the stability of two-dimensional disturbances imposed on a boundary layer flow over a semi-infinite flat plate in the presence of a reacting chemical species. Species concentration levels are assumed to be small, what is typical for many processes in water and in atmospheric air. We exploit the multi-deck structure of the flow in the limit of large Reynolds numbers to analyze asymptotically the perturbed flow. The neutral eigenrelations are obtained implicitly and limiting cases for large buoyancy and reaction kinematics are investigated. The results show some interesting effects of the Damkohler number on the wave number and wave speed of the disturbed flow.
8
Content available remote Linear stability of two-dimensional flow to three-dimensional perturbations in a channel with a flexible wall
Sibanda P., Motsa S. S., Shateyi S.
Archives of Mechanics
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2004
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Vol. 56, nr 4
293-311
EN
In this paper we investigate the effects of three-dimensional disturbance waves on the stability of a two-dimensional channel flow with one compliant surface. The study exploits the multideck structure of the flow in the limit of large Reynolds numbers to make an asymptotic analysis of the flow and to derive linear neutral stability results. The study shows that for a flow over flexible surfaces, three-dimensional disturbances may be more unstable than two-dimensional modes for a given set of wall properties.
9
Content available Asymptotic analysis of three random branching walk models arising in molecular biology
Świerniak A., Polański A., Śmieja J., Kimmel M., Rzeszowska-Wolny J.
Control and Cybernetics
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2003
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Vol. 32, no 1
147-162
EN
Using asymptotic techniques based on Laplace transforms, spectral analysis and theory of feedback systems, we characterise the asymptotic behaviour of the repeat loci in microsatellite DNA and cancer cells with increasing number of copies of genes responsible for coding proteins causing drug removal or metabolisation as well as telomeres shortening, which is supposed to be the mechanism of ageing and death. These three problems are described by models in the form of infinitely many differential linear or bilinear first order equations, resulting from branching random walk processes used to represent the evolution of particles in these problems.
PL
Wykorzystując techniki asymptotyczne oparte na transformatach Laplace'a, analizę spektralną oraz teorię układów ze sprzężeniem zwrotnym w artykule scharakteryzowano zachowanie asymptotyczne powtórek w DNA mikrosatelitarnym oraz w komórkach rakowych z rosnącą liczbą kopii genów odpowiedzialnych za kodowanie białek powodujących usuwanie lub przemianę metaboliczną leków, a także skracanie telomerów, o którym się sądzi, że jest mechanizmem starzenia się i śmierci. Te trzy zagadnienia są opisywane przy pomocy modeli w postaci nieskończonej liczby liniowych lub biliniowych równań pierwszego rzędu, wynikających z procesów błądzenia, stosowanych do opisu ewolucji cząstek w tych zagadnieniach.
10
Content available remote A nonlinear model of a turbine blade by asymptotic analysis
Rodriguez J. M.
International Journal of Applied Mathematics and Computer Science
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2002
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Vol. 12, no 1
101-113
EN
In this paper we obtain a limit model for a turbine blade fixed to a 3D solid. This model is a three-dimensional linear elasticity problem in the 3D part of the piece (the rotor) and a two-dimensional problem (the nonlinear shallow shell equations) in the 2D part (the turbine blade), with junction conditions in the part of the turbine blade fixed to the rotor. To obtain this model, we perform an asymptotic analysis, starting with the nonlinear three-dimensional elasticity equations on all the pieces and taking as a small parameter the thickness of the blade.
11
Analiza efektywności wybranych drgań mechanicznych
Wernerowski K.
Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
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2000
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z. 14
153-161
PL
Znaczenie analizy techniczno-ekonomicznej drgań mechanicznych w gospodarce jest duże. Analizowano wibracje o dużej energii i charakterze nieliniowym. Przedstawiono metodykę obliczeń asymptotycznych i komputerową programem Mathcad. Opracowano równanie stosowania optymalnego uwzględniające realny efekt działania technicznego i wysiłek ekonomiczny. Na podstawie pochodnych uzyskano maksimum efektu technicznego i minimum kosztów.
EN
Significance of techno-economic vibrations analysis in the management is great. Oscillations with great energy and nonlinear action were analyzed. Method of asymptotic calculation and computation with the Mathcad program are presented. Optimum control equation taking into account the real effect of technical action and the economic functioning was elaborated. Differentiating the effectiveness the technical maximum and costs minimum was obtained.
12
Content available remote Recent trends in asymptotical techniques with applications to mechanics
Awrejcewicz J.
Vibrations in Physical Systems
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2000
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Vol. 19
34--46
EN
Three important problems of mechanics are addressed. The first corresponds to analytical prediction of a suitable choice of a delay loop parameters in order to control a transitional (perturbed) behaviour of a periodic vibro-impact process modelled by one degree-of-freedom system. The second presents an application of the Melnikov's method to stick-slip chaos prediction in periodically driven self-excited oscillator. The third one illustrates an application of a so called „small 5 method” to analysis of strong non-linear oscillations using example of a mathematical pendulum.
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