Geodesic
Generalized polynomial method for solving a Cauchy-type problem for one fractional differential equation
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 2, Tome 176 (2020), pp. 80-90.

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

In this paper, we examine a Cauchy-type problem for one ordinary differential equation with Riemann–Liouville fractional derivatives. For this problem, based on the Lebesgue space of functions summable with an arbitrarily fixed degree, we propose a family of pairs of spaces of elements and right-hand sides that yield its well-posed statement. In these pairs of spaces, we develop a generalized polynomial projection method for solving the problem considered and justify it from the functional-theoretic point of view. Based on the general results obtained, we prove the convergence of the “polynomial” Galerkin method, the collocation method, and the subdomain method for solving the corresponding Cauchy-type problem.
Mots-clés : Lebesgue space, convergence.
Keywords: differential equation, fractional derivative, Cauchy-type problem, well-posed problem, projection method 
@article{INTO_2020_176_a7,
     author = {Yu. R. Agachev and A. V. Guskova},
     title = {Generalized polynomial method for solving a {Cauchy-type} problem for one fractional differential equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {80--90},
     publisher = {mathdoc},
     volume = {176},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_176_a7/}
}
TY  - JOUR
AU  - Yu. R. Agachev
AU  - A. V. Guskova
TI  - Generalized polynomial method for solving a Cauchy-type problem for one fractional differential equation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 80
EP  - 90
VL  - 176
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_176_a7/
LA  - ru
ID  - INTO_2020_176_a7
ER  - 
%0 Journal Article
%A Yu. R. Agachev
%A A. V. Guskova
%T Generalized polynomial method for solving a Cauchy-type problem for one fractional differential equation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 80-90
%V 176
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_176_a7/
%G ru
%F INTO_2020_176_a7
Yu. R. Agachev; A. V. Guskova. Generalized polynomial method for solving a Cauchy-type problem for one fractional differential equation. 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 2, Tome 176 (2020), pp. 80-90. http://geodesic.mathdoc.fr/item/INTO_2020_176_a7/

[1] Agachev Yu. R., Lipachev E. K., Pryamye polinomialnye i splainovye metody resheniya integralnykh uravnenii vtorogo roda, Kazan, 2017

[2] Butkovskii A. G, Postnov S. S., Postnova E. A., “Drobnoe integro-differentsialnoe ischislenie i ego prilozheniya v teorii upravleniya”, Avtomat. telemekh., 2013, no. 4, 3–42 | Zbl

[3] Gabdulkhaev B. G., Optimalnye approksimatsii reshenii lineinykh zadach, Izd-vo Kazan. un-ta, Kazan, 1980 | MR

[4] Gabdulkhaev B. G., “Nekotorye voprosy teorii priblizhennykh metodov”, Izv. vuzov. Mat., 1968, no. 2

[5] Daugavet I. K., Vvedenie v teoriyu priblizheniya funktsii, Izd-vo LGU, L., 1977

[6] Lyusternik L. A, Sobolev V. I., Elementy funktsionalnogo analiza, Nauka, M., 1965 | MR

[7] Natanson I. P., Konstruktivnaya teoriya funktsii, Gostekhizdat, M.-L., 1949 | MR

[8] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003

[9] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987

[10] Sege G., Ortogonalnye mnogochleny, GIFML, M., 1962

[11] Uchaikin V. V., Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008

[12] Pollard H., “The mean convergence of orthogonal series”, Duke Math. J., 16:1 (1949) | DOI | MR | Zbl