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1

Difference scheme for differential-difference problems with small shifts arising in computational model of neuronal variability

Kodipaka Mamatha, Emineni Siva Prasad, Kolloju Phaneendra

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 1

91--106

EN

The solution of differential-difference equations with small shifts having layer behaviour is the subject of this study. A difference scheme is proposed to solve this equation using a non-uniform grid. With the non-uniform grid, finite - difference estimates are derived for the first and second-order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the tridiagonal system algorithm. Convergence of the scheme is examined. Various numerical simulations are presented to demonstrate the validity of the scheme. In contrast to other techniques, maximum errors in the solution are organized to support the method. The layer behaviour in the solutions of the examples is depicted in graphs.

2

Entire and meromorphic solutions for systems of the differential difference equations

Xu Hong Yan, Li Hong, Ding Xin

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

676--694

EN

With the help of the Nevanlinna theory of meromorphic functions, the purpose of this article is to describe the existence and the forms of transcendental entire and meromorphic solutions for several systems of the quadratic trinomial functional equations: {f(z)2+2αf(z)g(z+c)+g(z+c)2=1,g(z)2+2αg(z)f(z+c)+f(z+c)2=1, {f(z+c)2+2αf(z+c)g'(z)+g'(z)2=1,g(z+c)2+2αg(z+c)f'(z)+f'(z)2=1, and {f(z+c)2+2αf(z+c)g′′(z)+g′′(z)2=1,g(z+c)2+2αg(z+c)f′′(z)+f′′(z)2=1. We obtain a series of results on the forms of the entire solutions with finite order for such systems, which are some improvements and generalizations of the previous theorems given by Gao et al. Moreover, we provide some examples to explain the existence and forms of solutions for such systems in each case.

3

Polynomial quasisolutions of linear differential-difference equations

Cherepennikov V. B., Ermolaeva P. G.

Opuscula Mathematica

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2006

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Vol. 26, no. 3

431-443

EN

The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment t = 0 the value of the desired function x(t) is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree N is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution) is understood in the sense of appearance of the residual Δ(t) = O(tN), when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.

4

An M/G/1 queue with second optional service and Bernoulli schedule server vacations

Madan K.C., Abu-Dayyeh W., Saleh M.F.

Systems Science

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2002

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Vol. 28, nr 3

51-62

EN

We study an M/G/1 queue with second optional service and Bernoulli schedule server vacations. Poisson arrivals with mean arrival rate lambda ([right angle bracket] 0), all demand the first 'essential' service, whereas only some of them demand the second 'optional' service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B( nu ) and that of the second optional service are exponential with mean service time 1/ mu /sub 2/ ( mu /sub 2/ [right angle bracket] 0). We have assumed that after completion of a service, the server takes Bernoulli schedule server vacations. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly in closed form. Some known results have been derived as particular cases.

5

Metoda analizy dynamiki układów napędowych z parametrami rozłożonymi

Sikora-Iliew R., Potrawka S.

Elektrotechnika i Elektronika

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2002

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T. 21, z. 1

82--89

PL

W artykule przedstawiono metodę analizy dynamiki układów elektromechanicznych z parametrami rozłożonymi. Dynamikę układu napędowego z parametrami rozłożonymi zapisano w oparciu o równanie różniczkowe cząstkowe-równanie falowe. Model matematyczny opisujący w sposób ogólny dynamikę napędu stworzono w oparciu o energetyczna funkcję stanu Lagrange'a przy uwzględnieniu energii potencjalnych i koenergii kinetycznych układu elektromechanicznego. Równania opisujące przemianę elektromechaniczną zapisano w oparciu o zasadę najmniejszego działania Hamiltona i równania Eulera—Lagrange'a. Analiza układu poparta została badaniami symulacyjnymi wybranych systemów elektromechanicznych. Przypadek systemu elektromechanicznego powiązanego materiałowo rozważono dla układu napędowego zwijarki. Uzyskane wyniki i wnioski końcowe mogą być przydatne również dla dydaktyki.

EN

The dynamics analysis for distributed-parameters and concentrated- parameters system drive's are presented in this paper. The distributed-parameters system drive's dynamics has described by use partial differential equation - wave equation. This equation for system drive's has solved by change to differential-difference equation. A mathematical model describing drive's system dynamics universally has given in this paper. This model has created by use Lagrange function of state. Equations describing electromechanical conversion have derived by use Hamilton' s principle of least action and Euler - Lagrange equations. The analysis has been backed up by simulation of selected electromechanical systems. The case of material-connected electromechanical system have been also considered by simulation of a coiling machine drive. Obtained results allowed for final conclusions, which can be useful for educational purposes too.