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1

Anisotropic p-Laplace Equations on long cylindrical domain

Jana Purbita

Opuscula Mathematica

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2024

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Vol. 44, no. 2

249--265

EN

The main aim of this article is to study the Poisson type problem for anisotropic p-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo p-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension.

2

Asymptotic analysis of the steady advection-diffusion problem in axial domains

Morales Fernando A.

Opuscula Mathematica

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2023

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Vol. 43, no. 2

199--220

EN

We present the asymptotic analysis of the steady advection-diffusion equation in a thin tube. The problem is modeled in a mixed-type variational formulation, in order to separate the phenomenon in the axial direction and a transverse one. Such formulation makes visible the natural separation of scales within the problem and permits a successful asymptotic analysis, delivering a limiting form, free from the initial geometric singularity and suitable for approximating the original one. Furthermore, it is shown that the limiting problem can be simplified to a significantly simpler structure.

3

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.

4

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.

5

Heat flow through a thin cooled pipe filled with a micropolar fluid

Beneš M., Pažanin I., Suárez-Grau F. J.

Journal of Theoretical and Applied Mechanics

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2015

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Vol. 53 nr 3

569--579

EN

In this paper, a non-isothermal flow of a micropolar fluid in a thin pipe with circular cross- -section is considered. The fluid in the pipe is cooled by the exterior medium and the heat exchange on the lateral part of the boundary is described by Newton’s cooling condition. Assuming that the hydrodynamic part of the system is provided, we seek for the micropolar effects on the heat flow using the standard perturbation technique. Different asymptotic models are deduced depending on the magnitude of the Reynolds number with respect to the pipe thickness. The critical case is identified and the explicit approximation for the fluid temperature is built improving the known result for the classical Newtonian flow as well. The obtained results are illustrated by some numerical simulations.

6

Modeling of Stiff Interfaces : from Statics to Dynamics

Dumont S., Lebon F., Rizzoni R.

Machine Dynamics Research

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2013

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Vol. 37, No. 1

37-50

EN

In this paper, some results on the asymptotic behavior of stiff thin interfaces in elasto-statics are recalled. A specific study of stiff interfaces in elastodynamics is presented and a numerical procedure is given.

7

The asymptotic analysis of the income in closed HM-structure with priority messages

Matalytski M., Kiturko O.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 2

51--59

EN

In the article the asymptotic analysis of closed exponential queueing HM-structure with priority messages is carried out with a large total number of messages, depending on time. The number of service lines in systems, the intensity of service messages in them, the probabilities of message transitions between systems also depends on time. It is proved that the density of the income distribution in the network systems in asymptotic satisfies differential equations in partial derivatives. This provided the inhomogeneous differential equation for the expected incomes system structure. An example of transport logistics shows how to solve such equations.

8

Asymptotic analysis of the closed queueing structure with time-dependent service parameters and single-type messages

Matalytski M., Rusilko T., Pankov A.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 2

73--80

EN

A closed queueing structure is considered in the paper; the number of single-type messages is not constant and depends on time. The route of messages is given by an arbitrary stochastic matrix of transition probabilities. An asymptotic analysis of this structure in case of large number of service requests is conducted. The service parameters of each queueing system of this structure, as well as the probability of messages transition between systems, depend on time. A system of ordinary differential equations to calculate the average relative number of messages in each queueing system, depending on the time, was obtained.

9

Shape and topology optimization of distributed parameter systems

Sokołowski J., Żochowski A.

Control and Cybernetics

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2013

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Vol. 42, no. 1

217--226

EN

The energy functional for an elliptic boundary value problem in two spatial dimensions is considered. The variations of shape functional resulting from the small shape-topological domain perturbations with the holes and inclusions in elastic body are determined. The exact representation of solutions to the boundary value problem is exploited for the purposes of asymptotic analysis. To this end the perturbed solutions of the boundary value problem are Expressem as the minimizers of perturbed energy functionals. The proposed method of asymptotic analysis results in the double asymptotic expansions, with respect to the size of a hole and to the contrast parameter of an inclusion with respect to the matrix, of solutions to the boundary value problems as well as of the associated energy functional. The shape sensitivity analysis of the energy functional with respekt of the boundary variations of an inclusion is performed. The further asymptotic analysis allows for the limit passage with the size of inclusion to zero. In this way the topological derivative of the energy functional is obtained. The proposed analysis can be used in the shape and topology optimum design for elastic bodies governed by the stationary as well as by the time dependent elasticity boundary value problems in the framework of selfadjoint extensions of elliptic operators.

10

Asymptotic guarantee of success for multi-agent memetic systems

Byrski A., Schaefer R., Smołka M., Cotta C.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 1

257-278

EN

The paper introduces a stochastic model for a class of population-based global optimization meta-heuristics, that generalizes existing models in the following ways. First of all, an individual becomes an active software agent characterized by the constant genotype and the meme that may change during the optimization process. Second, the model embraces the asynchronous processing of agent’s actions. Third, we consider a vast variety of possible actions that include the conventional mixing operations (e.g. mutation, cloning, crossover) as well as migrations among demes and local optimization methods. Despite the fact that the model fits many popular algorithms and strategies (e.g. genetic algorithms with tournament selection) it is mainly devoted to study memetic algorithms. The model is composed of two parts: EMAS architecture (data structures and management strategies) allowing to define the space of states and the framework for stochastic agent actions and the stationary Markov chain described in terms of this architecture. The probability transition function has been obtained and the Markov kernels for sample actions have been computed. The obtained theoretical results are helpful for studying metaheuristics conforming to the EMAS architecture. The designed synchronization allows the safe, coarse-grained parallel implementation and its effective, sub-optimal scheduling in a distributed computer environment. The proved strong ergodicity of the finite state Markov chain results in the asymptotic stochastic guarantee of success, which in turn imposes the liveness of a studied metaheuristic. The Markov chain delivers the sampling measure at an arbitrary step of computations, which allows further asymptotic studies, e.g. on various kinds of the stochastic convergence.

11

Asymptotic analysis of the closed network with time-dependent service parameters and single-type messages

Matalytski M., Rusilko T., Pankov A.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 4

105-112

EN

A queueing network of any structure with single-type messages is considered in the paper. An asymptotic analysis of the network in case of large number of service requests conducted. It is suggested that the service parameters of each queueing system of the network, as well as the probability of messages transition between systems, depend on time. A system of ordinary differential equations to calculate the average relative number of messages in each queueing system, depending on the time, was obtained. There is one calculated example in the article.

12

Isospectral integrability analysis of dynamical systems on discrete manifolds

Blackmore D., Prykarpatsky A. K., Prykarpatsky Y. A.

Opuscula Mathematica

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2012

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Vol. 32, no. 1

41-66

EN

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems.

13

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.

14

Asymptotic Analysis of an Adhesive Joint : A Focus on Cylindrical Coordinates

Lebon F., Rizzoni R.

Machine Dynamics Research

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2011

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Vol. 35, No. 1

97-107

EN

In this paper, some results on the asymptotic behavior of hard and soft thin interfaces are recalled. A specific study of soft interfaces in cylindrical coordinates is presented and an analytical example is studied.

15

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.

16

Topological sensitivity analysis for elliptic problems on graphs

Leugering G., Sokolowski J.

Control and Cybernetics

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2008

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Vol. 37, no 4

971-997

EN

We consider elliptic problems on graphs under given loads and bilateral contact conditions. We ask the question: which graph is best suited to sustain the loads and the constraints. More precisely, given a cost function we may look at a multiple node of the graph with edge degree q and ask as to whether that node should be resolved into a number of nodes of edge degree less than q, in order to decrease the cost. With this question in mind, we are looking into the sensitivity analysis of a graph carrying a second order elliptic equation with respect to changing its topology by releasing nodes with high edge degree or including an edge. With the machinery at hand developed here, we are in the position to define the topological gradient of an elliptic problem on a graph.

17

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.

18

The choice of the forms of Lyapunov functions for a positive 2D Roesser model

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 4

471-475

EN

The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix ATPA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.

19

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.

20

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.