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

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.