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Sibirskii Matematicheskii Zhurnal, 2021, Volume 62, Number 5, Pages 1143–1162
DOI: https://doi.org/10.33048/smzh.2021.62.514
(Mi smj7620)
 

This article is cited in 22 scientific papers (total in 22 papers)

The defect of a Cauchy type problem for linear equations with several Riemann–Liouville derivatives

V. E. Fedorovab, M. M. Turova

a Chelyabinsk State University, Chelyabinsk, Russia
b Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia
References:
Abstract: We consider the existence and uniqueness of solutions to initial value problems for general linear nonhomogeneous equations with several Riemann–Liouville fractional derivatives in Banach spaces. Considering the equation solved for the highest fractional derivative $D^\alpha_t$, we introduce the concept of the defect $m^*$ of a Cauchy type problem which determines the number of the zero initial conditions $D^{\alpha-m+k}_tz(0)=0$, $k=0,1,\dots,m^*-1$, necessary for the existence of the finite limits $D^{\alpha-m+k}_tz (t)$ as $t\to0+$ for all $k=0,1,\dots,m-1$. We show that the defect $m^*$ is uniquely determined by the set of orders of the Riemann–Liouville fractional derivatives in the equation. Also we prove the unique solvability of the incomplete Cauchy problem $D^{\alpha-m+k}_tz (0)=z_k$, $k=m^*,m^*+1,\dots, m-1$, for the equation with bounded operator coefficients solved for the highest Riemann–Liouville derivative. The obtained result allowed us to investigate initial problems for a linear nonhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that the operator at the second highest order derivative is $0$-bounded with respect to this operator, while the cases are distinguished that the fractional part of the order of the second derivative coincides or does not coincide with the fractional part of the order of the highest derivative. The results for equations in Banach spaces are used for the study of initial boundary value problems for a class of equations with several Riemann–Liouville time derivatives and polynomials in a selfadjoint elliptic differential operator of spatial variables.
Keywords: fractional order differential equation, Riemann–Liouville fractional derivative, degenerate evolution equation, Cauchy type problem, initial-boundary value problem.
Funding agency Grant number
Russian Foundation for Basic Research 21-51-54003
Ministry of Science and Higher Education of the Russian Federation 075-02-2021-1383
The authors were supported by the Russian Foundation for Basic Research (Grant 21–51–54003) and the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement 075–02–2021–1383).
Received: 31.05.2021
Revised: 31.05.2021
Accepted: 11.06.2021
English version:
Siberian Mathematical Journal, 2021, Volume 62, Issue 5, Pages 925–942
DOI: https://doi.org/10.1134/S0037446621050141
Bibliographic databases:
Document Type: Article
UDC: 517.9
Language: Russian
Citation: V. E. Fedorov, M. M. Turov, “The defect of a Cauchy type problem for linear equations with several Riemann–Liouville derivatives”, Sibirsk. Mat. Zh., 62:5 (2021), 1143–1162; Siberian Math. J., 62:5 (2021), 925–942
Citation in format AMSBIB
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\by V.~E.~Fedorov, M.~M.~Turov
\paper The defect of a~Cauchy type problem for linear~equations with several Riemann--Liouville derivatives
\jour Sibirsk. Mat. Zh.
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\vol 62
\issue 5
\pages 1143--1162
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\crossref{https://doi.org/10.33048/smzh.2021.62.514}
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\transl
\jour Siberian Math. J.
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\issue 5
\pages 925--942
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  • This publication is cited in the following 22 articles:
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