arXiv subjects

Dorota Mozyrska

Publications and source records attributed to Dorota Mozyrska.

11 recordsLinked to original sources

arXiv2018

Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales

Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.

math.CA↗

arXiv2013

Fractional Discrete Systems with Sequential h-differences

In the paper we study the subject of positivity of systems with sequential fractional difference. We give formulas for the unique solutions of systems in linear and semi-linear cases. The positivity of systems is considered.

math.DS↗

arXiv2010

Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the inverse Laplace transform on time scales. Useful properties of the new fractional operators are proved.

math.CA↗

arXiv2010

Modified Optimal Energy and Initial Memory of Fractional Continuous-Time Linear Systems

Fractional systems with Riemann-Liouville derivatives are considered. The initial memory value problem is posed and studied. We obtain explicit steering laws with respect to the values of the fractional integrals of the state variables. The Gramian is generalized and steering functions between memory values are characterized.

math.OC↗

arXiv2010

Inequalities and majorisations for the Riemann-Stieltjes integral on time scales

We prove dynamic inequalities of majorisation type for functions on time scales. The results are obtained using the notion of Riemann-Stieltjes delta integral and give a generalization of [App. Math. Let. 22 (2009), no. 3, 416--421] to time scales.

math.CA↗

arXiv2010

Minimal modified energy control for fractional linear control systems with the Caputo derivative

Fractional control systems with the Caputo derivative are considered. The modified controllability Gramian and the minimum energy optimal control problem are investigated. Construction of minimizing steering controls for the modified energy functional are proposed.

math.OC↗

arXiv2009

The Riemann-Stieltjes integral on time scales

We study the process of integration on time scales in the sense of Riemann-Stieltjes. Analogues of the classical properties are proved for a generic time scale, and examples are given.

math.CA↗

arXiv2009

A Study of Diamond-alpha dynamic equations on regular time scales

We introduce the diamond-alpha exponential function on time scales. As particular cases, one gets both delta and nabla exponential functions. A method of solution of a homogenous linear dynamic diamond-alpha equation on a regular time scale is investigated, and examples of diamond-alpha exponential functions are presented.

math.CA↗

arXiv2008

The Natural Logarithm on Time Scales

We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. {\bf 11} (2005), no. 15, 1305--1306].

math.CA↗

arXiv2008

Diamond-alpha Polynomial Series on Time Scales

The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combination of delta and nabla generalized series. Main results are formulated for homogenous time scales. As an example, we compute diamond-alpha derivatives on time scales for delta and nabla exponential functions.

math.CA↗

arXiv2006

Systems of germs and theorems of zeros in infinite-dimensional spaces

Systems of germs of sets in infinite-dimensional spaces are introduced and studied. Such a system corresponds to a local zero-set of an ideal of the ring of analytic functions of infinite number of variables. Conversely, this system of germs defines the ideal of germs of analytic functions vanishing on it. A theorem of zeros is proved, stating that this ideal is the radical (in the complex case) or real radical (in the real case) of the initial ideal.

math.CV↗