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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - M. M. Kokurin
TI  - Discrete approximation of solutions of the Cauchy problem for a linear homogeneous differential-operator equation with a fractional Caputo derivative in a Banach space
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 79
EP  - 104
VL  - 175
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_175_a7/
LA  - ru
ID  - INTO_2020_175_a7
ER  - 
%0 Journal Article
%A M. M. Kokurin
%T Discrete approximation of solutions of the Cauchy problem for a linear homogeneous differential-operator equation with a fractional Caputo derivative in a Banach space
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 79-104
%V 175
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_175_a7/
%G ru
%F INTO_2020_175_a7
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/

[13] Abrashina-Zhadaeva N. G., Timoschenko I. A., “Konechno-raznostnye metody dlya uravneniya diffuzii s proizvodnymi drobnykh poryadkov v mnogomernoi oblasti”, Differ. uravn., 49:7 (2013), 819–825 | MR | Zbl

[14] Alikhanov A. A., “Ustoichivost i skhodimost raznostnykh skhem dlya kraevykh zadach uravneniya diffuzii drobnogo poryadka”, Zh. vychisl. mat. mat. fiz., 56:4 (2016), 572–586 | DOI | MR | Zbl

[15] Bakushinskii A. B., Kokurin M. M., Kokurin M. Yu., “O skheme polnoi diskretizatsii nekorrektnoi zadachi Koshi v banakhovom prostranstve”, Tr. in-ta mat. mekh. UrO RAN., 18:1 (2012), 96–-108

[16] Bakushinskii A. B., Kokurin M. M., Kokurin M. Yu., “Ob odnom klasse raznostnykh skhem resheniya nekorrektnoi zadachi Koshi v banakhovom prostranstve”, Zh. vychisl. mat. mat. fiz., 52:3 (2012), 483–498 | MR | Zbl

[17] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1965

[18] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972

[19] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966

[20] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[21] Kokurin M. M., “Ob optimizatsii otsenok skorosti skhodimosti nekotorykh klassov raznostnykh skhem resheniya nekorrektnoi zadachi Koshi”, Vychisl. met. program., 14 (2013), 58–76

[22] Kokurin M. M., “O edinstvennosti resheniya obratnoi zadachi Koshi dlya differentsialnogo uravneniya s drobnoi proizvodnoi v banakhovom prostranstve”, Izv. vuzov. Mat., 2013, no. 12, 19–35 | Zbl

[23] Kokurin M. M., “Raznostnye skhemy resheniya zadachi Koshi dlya lineinogo differentsialno-operatornogo uravneniya vtorogo poryadka”, Zh. vychisl. mat. mat. fiz., 54:4 (2014), 569–584 | DOI | Zbl

[24] Kokurin M. M., “Neobkhodimye i dostatochnye usloviya stepennoi skhodimosti metoda kvaziobrascheniya i raznostnykh metodov resheniya nekorrektnoi zadachi Koshi v usloviyakh tochnykh dannykh”, Zh. vychisl. mat. mat. fiz., 55:12 (2015), 2027–2041 | DOI | Zbl

[25] Kokurin M. M., “Otsenki skorosti skhodimosti i pogreshnosti raznostnykh skhem resheniya lineinoi nekorrektnoi zadachi Koshi vtorogo poryadka”, Vychisl. met. program., 18 (2017), 322–347

[26] Kochubei A. N., “Zadacha Koshi dlya evolyutsionnykh uravnenii drobnogo poryadka”, Differ. uravn., 25:8 (1989), 1359–1368 | MR

[27] Lafisheva M. M., Shkhanukov-Lafishev M. Kh., “Lokalno-odnomernaya raznostnaya skhema dlya uravneniya diffuzii drobnogo poryadka”, Zh. vychisl. mat. mat. fiz., 48:10 (2008), 1878–1887 | MR | Zbl

[28] Lyu Zh., Li M., Pastor Kh., Piskarev S. I., “Ob approksimatsii drobnykh razreshayuschikh semeistv”, Differ. uravn., 50:7 (2014), 937–946 | DOI

[29] Piskarev S. I., “Otsenki skorosti skhodimosti pri reshenii nekorrektnykh zadach dlya evolyutsionnykh uravnenii”, Izv. AN SSSR. Ser. mat., 51:3 (1987), 676-–687 | Zbl

[30] Popov A. Yu., Sedletskii A. M., “Raspredelenie kornei funktsii Mittag-Lefflera”, Sovr. mat. Fundam. napravl., 40 (2011), 3–171

[31] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Spetsialnye funktsii. Dopolnitelnye glavy, Fizmatlit, M., 2003 | MR

[32] Radzievskaya E. I., Radzievskii G. V., “Dlya golomorfnoi v oblasti funktsii ostatochnyi chlen v formule Teilora dopuskaet zapis v forme Lagranzha”, Sib. mat. zh., 44:2 (2003), 402–414 | MR | Zbl

[33] Taukenova F. I., Shkhanukov-Lafishev M. Kh., “Raznostnye metody resheniya kraevykh zadach dlya differentsialnykh uravnenii drobnogo poryadka”, Zh. vychisl. mat. mat. fiz., 46:10 (2006), 1871–1881 | MR

[34] Trenogin V. A., Funktsionalnyi analiz, Fizmatlit, M., 2007

[35] Bajlekova E. G., Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Pleven, 2001 | MR | Zbl

[36] Haase M., The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006 | MR | Zbl

[37] Jin B., Lazarov R., Zhou Z., “Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data”, SIAM J. Sci. Comput., 38:1 (2016), A146–A170 | DOI | MR | Zbl

[38] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006 | MR | Zbl

[39] Li C., Zeng F., “The finite difference methods for fractional ordinary differential equations”, Numer. Funct. Anal. Optim., 34:2 (2013), 149–179 | DOI | MR | Zbl

[40] Lubich C., “Discretized fractional calculus”, SIAM J. Math. Anal., 17:3 (1986), 704–719 | DOI | MR | Zbl

[41] Sakamoto K., Yamamoto M., “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems”, J. Math. Anal. Appl., 382 (2011), 426-–447 | DOI | MR | Zbl