Solution of the boundary problem for the generalized Laplace equation with a fractional derivative

In this paper, we study the Dirichlet boundary value problem in the upper halfplane for a second-order partial differential equation containing a composition of Riemann-Liouville fractional differentiation operators with respect to one of two independent variables. The equation under consideration, for an integer value of the order of fractional differentiation, passes into the Laplace equation in two independent variables. An explicit representation of the solution of the problem under study (in terms of a function of the Mittag-Leffler type) is obtained by the method of the integral Fourier transform. Asymptotic estimates for a particular solution and its derivatives are found. Theorems on the existence and uniqueness of a regular solution are proved. The existence of a solution is proved in the class of continuous functions with weight in a closed half-plane. The uniqueness of the solution is proved in the class of continuously differentiable functions with respect to the spatial variable and having a corresponding continuous fractional derivative with weight with respect to the time variable in a closed half-plane.