Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative

Abstract: We investigate a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, we reduce the problem to a Volterra type integral equation. This reduction enables the explicit construction of the corresponding Green's function. Based on the obtained Green's function, we derive a closed-form integral representation of the solution and prove its existence and uniqueness. The results extend classical Green-function techniques to a wider class of fractional operators and provide analytical tools for further study of boundary and inverse problems associated with Prabhakar-type fractional differential equations.