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JOURNALS // Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory // Archive

Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 2022 Volume 211, Pages 14–28 (Mi into1022)

On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation

B. Kh. Turmetova, B. J. Kadirkulovb

a Kh. Yasavi International Kazakh-Turkish University
b Tashkent State Institute of Oriental Studies

Abstract: 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.

UDC: 517.956

MSC: 34K37,35A09, 35J25

DOI: 10.36535/0233-6723-2022-211-14-28



© Steklov Math. Inst. of RAS, 2025