Geodesic
Discrete approximation of solutions of the Cauchy problem for a linear homogeneous differential-operator equation with a fractional Caputo derivative in a Banach space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the XVII All-Russian Youth School-Conference «Lobachevsky Readings-2018», November 23-28, 2018, Kazan. Part 1, Tome 175 (2020), pp. 79-104

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In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order $\alpha \in (0,1)$ in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented.
Keywords: Cauchy problem, Caputo derivative, Banach space, finite-difference scheme, error estimate, Mittag-Leffler function, hypergeometric function, sectorial operator. 
@article{INTO_2020_175_a7,
     author = {M. M. Kokurin},
     title = {Discrete approximation of solutions of the {Cauchy} problem for a linear homogeneous differential-operator equation with a fractional {Caputo} derivative in a {Banach} space},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {79--104},
     publisher = {mathdoc},
     volume = {175},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_175_a7/}
}
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M. M. Kokurin. Discrete approximation of solutions of the Cauchy problem for a linear homogeneous differential-operator equation with a fractional Caputo derivative in a Banach space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the XVII All-Russian Youth School-Conference «Lobachevsky Readings-2018», November 23-28, 2018, Kazan. Part 1, Tome 175 (2020), pp. 79-104. http://geodesic.mathdoc.fr/item/INTO_2020_175_a7/