Geodesic
Initial value problems for equations with a composition of fractional derivatives
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 269-277.

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We study the unique solvability of initial problems for linear equations in Banach spaces with a composition of two fractional derivatives and with a bounded operator on the right side. It is shown that the compositions of fractional derivatives of Riemann — Liouville and (or) Gerasimov — Caputo are derivatives of Dzhrbashyan — Nersesyan. With the help of the previously obtained general results on the initial problem for a linear equation with the Dzhrbashyan — Nersesyan fractional derivative, statements are formulated about the existence and uniqueness of a solution for initial problems to the equations under study with a composition of two fractional derivatives. The solutions are presented using the Mittag-Leffler functions. The general results are demonstrated by the example of an initial boundary value problem for an equation with polynomials with respect to the Laplace operator.
Keywords: Riemann — Liouville fractional derivative, Gerasimov — Caputo fractional derivative, Dzhrbashyan — Nersesyan fractional derivative, initial value problem, Mittag-Leffler function, initial boundary value problem. 
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A. R. Volkova; E. M. Izhberdeeva; V. E. Fedorov. Initial value problems for equations with a composition of fractional derivatives. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 269-277. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a0/

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