3 research outputs found
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.Comment: This is a preprint of a paper whose final and definite form is with
'Nonlinear Analysis: Hybrid Systems', ISSN: 1751-570X, available at
[http://www.journals.elsevier.com/nonlinear-analysis-hybrid-systems].
Submitted 21/May/2018; Revised 07/Oct/2018; Accepted for publication
01-Dec-201