Backward Continuation of the Solutions of the Cauchy Problem for Linear Fractional System with Deviating Argument

Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev ()
Additional contact information
Hristo Kiskinov: Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Mariyan Milev: Faculty of Economics and Business Administration, Sofia University, 1504 Sofia, Bulgaria
Milena Petkova: Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Andrey Zahariev: Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria

Mathematics, 2024, vol. 13, issue 1, 1-16

Abstract: Fractional calculus provides tools to model systems with memory effects; when coupled with delays, they model process histories inspired by two independent sources—the memory of the fractional derivative and the impact conditioned by the delays. This work considers a Cauchy (initial) problem for a linear delayed system with derivatives in Caputo’s sense of incommensurate order, distributed delays, and piecewise initial functions. For this initial problem, we study the important problem of the backward continuation of its solutions. We consider the backward continuation of the solutions as a problem of the renewal of a process with aftereffect under given final observation. Sufficient conditions for backward continuation of the solutions of these systems have been obtained. As application, a formal (Lagrange) adjoint system for the studied homogeneous system is introduced, and using the backward continuation, it is proved that for this system there exists a unique matrix solution called by us as the formal adjoint fundamental matrix, which can play the same role as the fundamental matrix in the forward case.

Keywords: distributed delay; linear fractional system; backward continuation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/1/76/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/1/76/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2024:i:1:p:76-:d:1555391

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().