Jan Čermák

ON (q, h)-ANALOGUE OF FRACTIONAL CALCULUS

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.

Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and 𝑞-difference calculus. Some of our results are new also in these particular discrete settings.

On explicit stability conditions for a linear fractional difference system

Abstract The paper describes the stability area for the difference system (Δαy)(n + 1 − α) = Ay(n), n= 0, 1, . . . , with the Caputo forward difference operator Δα of a real order α ∈ (0, 1) and a real constant matrix A. Contrary to the existing result on this topic, our stability conditions are fully explicit and involve the decay rate of the solutions. Some comparisons with a difference system of the Riemann- Liouville type are discussed as well, including related consequences and illustrating examples.

On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations

This paper discusses two explicit forms of necessary and sufficient conditions for the asymptotic stability of the autonomous linear difference equation where are real numbers and is an integer. These conditions are derived by use of the Schur–Cohn criterion converted into a more applicable form. We compare our stability conditions with related results obtained by other authors and apply them to a problem of numerical discretization of delay differential equations.

Stability and asymptotic properties of a linear fractional difference equation

AbstractThis paper discusses qualitative properties of the two-term linear fractional difference equation ∇α0y(n)=λy(n), where α,λ∈R, 0<α<1, λ≠1 and ∇α0 is the α th order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions.MSC:39A30, 39A12, 26A33.

A change of variables in the asymptotic theory of differential equations with unbounded delays

In this paper, the author investigates the asymptotic properties of solutions of the nonhomogeneous linear differential equation X(t) = ax(τ(t)) + bx(t) + f(t) with nonzero real scalars a,b and the unbounded lag. Using the change of the independent and dependent variable he relates the asymptotic behaviour of solutions of this equation to the asymptotic behaviour of solutions of auxiliary functional (nondifferential) equations.

Asymptotic Bounds for Linear Difference Systems

This paper describes asymptotic properties of solutions of some linear difference systems. First we consider system of a general form and estimate its solutions by use of a solution of an auxiliary scalar difference inequality assuming that this solution admits certain properties. Then applying this result to linear difference systems of a variable order with constant (or bounded) coefficients we derive effective asymptotic criteria for such systems. Beside it, we give applications of these results to numerical analysis of vector differential equations with infinite lags.

Delay equations on time scales: Essentials and asymptotics of the solutions

The paper discusses the notion of a delay dynamic equation on time scales and describes some asymptotic properties of its solutions. The application of the derived results to continuous and discrete time scales presents new qualitative results for delay differential and difference equations. In particular, our approach faciliates the joint investigation of stability properties of the exact equations and their numerical discretizations.

Blind Source Separation Based on a Beamformer Array and Time Frequency Binary Masking

This paper deals with a new technique for blind source separation (BSS) from convolutive mixtures. We present a three-stage separation system employing time-frequency binary masking, beamforming and a non-linear post processing technique. The experiments show that this system outperforms conventional time-frequency binary masking (TFBM) in both (over-)determined and underdetermined cases. Moreover it removes the musical noise and reduces interference in time-frequency slots extracted by TFBM.

Exact and discretized stability of the Bagley–Torvik equation

Abstract The paper discusses stability and asymptotic properties of a two-term linear fractional differential equation involving the Bagley–Torvik equation as the particular case. These properties are analysed for the exact as well as numerical solutions obtained from the Grunwald–Letnikov discretization of the studied differential equation. As the main results, precise descriptions of the exact and discretized stability regions are presented, including the decay rate of the solutions. These results enable us, among others, to observe similarities and distinctions between the asymptotic behaviour of the classical and fractional damping models.