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EN
In the paper the linear time-invariant fractional-order models of the separated wheel closed-loop electrical drive of the autonomous platform are considered. As a reference model one considers the classical model described by the second-order linear difference equation. Two discrete-time fractional-order models are considered: non-commensurate and commensurate. According to the sum of the squared error criterion (SSE) one compares two-parameter integer-order model with the four-parameter non-commensurate and three-parameter commensurate fractional-order ones. Three mathematical models are built and simulated. The computer simulation results are compared with measured velocity of the real autonomous platform separate wheel closed-loop electrical drive.
EN
This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ) β (k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ) . This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ) β (k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.
3
Content available remote Meromorphic solutions of linear difference equations with polynomial coefficients
EN
We study the growth of the transcendental meromorphic solution f(z) of the linear difference equation: [formula] where q(z), p0(z), . . ., pn(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≠ 0, and obtain some necessary conditions guaranteeing that the order of ƒ(z) satisfies σ(ƒ) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of ƒ(z) with two Borel exceptional values when two of p0(z), . . ., pn(z) have the maximal degrees.
4
Content available remote Global behavior of higher order a rational difference equation
EN
The main objective of this paper is to study the global asymptotic stability and the periodic character of the rational difference equation …[wzór], n – 0,1,…, where the parameters a, β, ϒ, p, q are nonnegative real numbers and initial conditions are nonnegative real numbers l, r, k are nonnegative integers, such that l < k and r < k. Also, we give some numerical simulations to the equation to illustrate our results.
5
Content available remote On a system of rational difference equation
EN
In this paper, we study local asymptotic stability, global character and periodic nature of solutions of the system of rational difference equations given by (…) , where the parameters (…) , and with initial conditions (…). Some numerical examples are given to illustrate our results.
6
EN
Our aim in this paper is to obtain sufficient conditions for the oscillation of every solution of first order difference equations ...[wzór] where pi, qi ∈ R+ and ki, li ∈ N for i = 1, 2,..., m.
EN
The paper focuses on a linear diffrence equation depending on parameters. The equation is related to Good win’s theory of extrapolative expectations. The stability region of the equation is investigated. Conditions for asymptotic stability are formulated and presented as an optimisation problem, which is further analysed. Despite employing state-of-the-art solvers, numerical results have turned out to be too ambiguous to provide the basis for definite conclusions about the investigated stability region.
EN
In this paper we explore the linear difference equations with fractional orders, which are functions of time. A description of closed-loop dynamical systems described by such equations is proposed. In a numerical example a simple control strategy based on time-varying fractional orders is presented.
EN
In this paper the problem of the influence of fixed point computation on numerical solutions of linear differential equations of fractional order is considered. It is a practically important problem, because of potential possibilities of using dynamical systems of fractional order in the tasks of control and filtering. Discussion includes numerical method is based on the Grünwald-Letnikov fractional derivative and how the application of fixed-point architecture influences its operation. Conclusions are illustrated with results of floating-point arithmetic with double precision and fixed point arithmetic with dif- ferent word lengths.
10
Content available Approximation of Fractional Diffusion-Wave Equation
EN
In this paper we consider the solution of the fractional differential equations. In particular, we consider the numerical solution of the fractional one dimensional diffusion-wave equation. Some improvements of computational algorithms are suggested. The considerations have been illustrated by examples.
EN
In this paper we consider two ordinary fractional differential equations with composition of the left and the right Caputo derivatives. Analytical solution of this type of equations is known for particular cases, having a complex form, and therefore is difficult in practical calculations. Here, we present two numerical schemes being dependent on a fractional order of equation. The results of numerical calculations are compared with analytical solutions and then we illustrate convergence of our schemes. Finally, we show an application of the considered equation.
PL
W pracy rozważa się zagadnienie początkowe dla liniowego równania różnicowego o stałych współczynnikach określonego w przestrzeni ciągów dwustronnych C (Z) . Problem ten przedstawiono w ujęciu nieklasycznego rachunku operatorów Bittnera. Wykorzystując model nabla tego rachunku z pochodną rozumianą jako różnica wsteczna, rozpatrywane zagadnienie rozwiązano w tzw. przestrzeni wyników. Wyniki powstają przez dzielenie elementów przestrzeni C (Z) przez injekcyjne endomorfizmy tej przestrzeni. Przedstawione rozważania dają początek nabla-rachunkowi, który może być konkurencyjny w stosunku do rachunku opartego na dwustronnym przekształceniu.
EN
The paper analyses the initial value problem for a linear difference equation with constant coefficients, defined in the space of two-sided sequences. The above problem has been C (Z) presented using the non-classical Bittner operational calculus approach. Using the -model of that calculus with its derivative understood as a backward difference, the issue in question has been solved in a so-called results' space. The results are obtained by dividing the elements of the C (Z) space by the injective endomorphisms of that space. The described analysis gives rise to a nabla-calculus that can be considered competitive to the calculus based on the bilateral -transform.
EN
An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.
14
Content available remote On the solutions of the difference equation xn+1 = max ...[wzór]
EN
We study the solutions of the following difference equation ...[wzór] where initial conditions x-1and x0 are nonzero real numbers. In most of the cases we determine the solutions in function of the initial conditions x-1 and x0.
15
Content available remote On the solutions of a rational system of difference equations
EN
In this paper we deal with the solutions of the system of the difference equations xn+1 = ...[wzór], yn+1 = ...[wzór], with a nonzero real numbers initial conditions.
EN
The paper contains connections between oscillation of solutions of iterative functional equations, difference equations and differential equations with advanced or delayed arguments. New oscillatory criteria for these equations are given.
17
Content available remote On a class of rational difference equations
EN
In this paper we study the behavior of the positive solutions of the following nonlinear difference equation ...[wzór], n = 0, 1, 2, ... where the initial values ...[wzór] and k = 0, 1, 2, . . ..
18
Content available remote Further instances of periodicity in May's host parasitoid equation
EN
May’s host parasitoid equation is the difference equation (1) ...[wzór] We show that for each &slpha; there is a number k such that, whenever n > k, equation (1) has a one cycle periodic solution of period n. We also give some results on two cycle periodic solutions.
19
EN
In this paper three-dimensional nonlinear difference system with delays ...[wzór] is investigated. The classification of nonoscillatory solutions of the considered system are presented. Next, the sufficient conditions under which nonoscillatory solution of considered system is bounded or is unbounded are given. Key words: difference equation, nonlinear system, nonoscillatory, bounded, unbounded solution.
EN
In this paper, necessary and sufficient condition are obtained so that every bounded solution of Δ(yn - yn-k) + qnG(yσ(n)) = 0 is oscillatory, under a condition weaker than ...[wzór]
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