Geodesic
On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 13-23.

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

In this paper, we study a problem with initial functions and boundary conditions for partial differential equations of fractional order with Laplace operators. The boundary conditions of the problem are nonlocal, and the solution is supposed to belong to one of Sobolev classes. The solution of the initial boundary value problem is constructed as the sum of a series of multidimensional spectral problem's eigenfunctions. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system is complete and forms a Riesz basis in the subspaces of Sobolev spaces. Basing on the completeness of the eigenfunctions' system, the uniqueness theorem for the solution of the problem is proved. The existence of a regular solution of the initial boundary value problem is proved in Sobolev subspaces.
Keywords: partial differential equation with delayed argument, fractional time derivative, initial boundary value problem, spectral method, eigenvalues, eigenfunctions, completeness, Riess basis, uniqueness, series, nonlocal boundary conditions, Sobolev class, fractional derivative, mixed problem.
Mots-clés : existence 
@article{SVMO_2020_22_1_a0,
     author = {M. M. Babayev},
     title = {On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and {Laplace} operators with nonlocal boundary conditions in {Sobolev} classes},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {13--23},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a0/}
}
TY  - JOUR
AU  - M. M. Babayev
TI  - On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2020
SP  - 13
EP  - 23
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a0/
LA  - ru
ID  - SVMO_2020_22_1_a0
ER  - 
%0 Journal Article
%A M. M. Babayev
%T On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2020
%P 13-23
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a0/
%G ru
%F SVMO_2020_22_1_a0
M. M. Babayev. On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 13-23. http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a0/

[1] A. N. Tikhonov, A. A. Samarskiy, Equations of Mathematical Physics, 3th ed., Nauka Publ., Moscow, 1977, 736 pp. (In Russ.) | MR

[2] L. Raleigh, Sound Theory, v. 1, 1955, 504 pp.

[3] S.P. Tymoshenko, D.H. Yang, U. Weaver, Fluctuations in engineering, Mashinostroenie Publ., Moscow, 1985, 472 pp. (In Russ.)

[4] L. E. Elsholz, S. B. Norkin, “Introduction to the theory of differential equations with a deviating argument”, Nauka Publ., M., 1971, 296 pp. (In Russ.) | MR

[5] M. A. Nimark, Linear differential operators, Nauka, Moscow, 1969, 528 pp. (In Russ.)

[6] A. V. Pskhu, Equations in private derivatives fractional order, Nauka, Moscow, 2005, 200 pp. (In Russ.)

[7] Igor Podlubny, “Fractional Differential Equations”, Mathematics in science and engineering, 198 (1999), 340 (In Eng.) | MR | Zbl

[8] Sh. G. Kasimov, Sh. K. Ataev, “On solvability of the mixed problem for a partial equation of a fractional order with Laplace operators and nonlocal boundary conditions in the Sobolev classes”, Uzbek mathematical journal, 1 (2018), 73–89 (In Eng.) | DOI | MR