Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

TL;DR

This paper develops a Green's function framework for boundary value problems involving the regularized Prabhakar fractional derivative, providing explicit solutions and representations in terms of Mittag-Leffler functions.

Contribution

It introduces a novel approach to solve sub-diffusion equations with the regularized Prabhakar derivative using Green's functions and superposition.

Findings

Explicit Green's function expressed via bivariate Mittag-Leffler functions

Solution representation verified as valid for the boundary value problem

Reduction of the problem to two initial-boundary value problems

Abstract

In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the superposition method. An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. Then, it is proved that the obtained solution indeed constitutes the solution of the considered problem.