Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

Abstract

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.