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

TL;DR

This paper develops Green's function techniques for a boundary value problem involving the Prabhakar fractional derivative, providing explicit solutions and extending classical methods to fractional operators.

Contribution

It introduces a method to explicitly construct Green's functions for Prabhakar fractional PDEs, extending classical techniques to this new class of fractional derivatives.

Findings

Explicit Green's function derived for the problem.

Existence and uniqueness of the solution proved.

Provides analytical tools for further fractional PDE studies.

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.