Geodesic
On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 14-28

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In this paper, we examine methods for solving the Dirichlet boundary-value problem and the periodic boundary-value problem for one class of nonlocal second-order partial differential equations with involutive argument mappings. The concept of a nonlocal analog of the Laplace equation is introduced. A method for constructing eigenfunctions and eigenvalues of the spectral problem based on separation of variables is proposed. The completeness of the system of eigenfunctions is examined. The concept of a fractional analog of the nonlocal Laplace equation is introduced. For this equation, boundary-value problems with the Dirichlet and periodic conditions are considered. The well-posedness of these problems is verified and the existence and uniqueness of the solution of boundary-value problems are proved.
Keywords: Gerasimov–Caputo fractional derivative, nonlocal differential equation, involution, Dirichlet problem, periodic boundary-value problem, eigenfunction, Mittag-Leffler function, Fourier series. 
@article{INTO_2022_211_a1,
     author = {B. Kh. Turmetov and B. J. Kadirkulov},
     title = {On the solvability of some boundary-value problems for the fractional analog of the nonlocal {Laplace} equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {14--28},
     publisher = {mathdoc},
     volume = {211},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/}
}
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B. Kh. Turmetov; B. J. Kadirkulov. On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 14-28. http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/