Hilfer Fractional Quantum Derivative and Boundary Value Problems

Phollakrit Wongsantisuk,Jessada Tariboon,Sotiris K Ntouyas,Donny Passary

doi:10.3390/math10060878

Abstract

In this paper, we introduce an extension of the Hilfer fractional derivative, the “Hilfer fractional quantum derivative”, and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach’s contraction mapping principle. Examples illustrating the obtained results are also presented.

Highlights

Fractional Quantum Derivative and Quantum calculus, known as q-calculus, is a branch of mathematics that study calculus without the notion of limits
We introduce the Hilfer fractional quantum derivative, which generalizes the Hilfer fractional derivative created by R
Hilfer created a generalization that combined both aspects of Riemann–Liouville and Caputo fractional derivatives, known as the Hilfer fractional derivative of order α ∈ (0, 1) and a type β ∈ [0, 1], which can be reduced to the Riemann–Liouville derivative when β = 0, and to the Caputo fractional derivative when β = 1

Summary

Introduction

Fractional Quantum Derivative and Quantum calculus, known as q-calculus, is a branch of mathematics that study calculus without the notion of limits. The fractional q-difference calculus, which further generalizes the idea of q-derivatives and q-integrals with non-integer orders, was developed in the works by AlSalam [5] and Agarwal [6]. Recent treatment on such material can be found in [7]. Tariboon and Ntouyas [24] introduced the idea of quantum calculus on finite intervals, in which they obtained q-analogues of several well-known mathematical objects and opened a new avenue of research.

Preliminaries

The Hilfer Fractional Quantum Derivative

Applications

Boundary Value Problems of Order 0 < α < 1

Boundary Value Problems of Order 1 < α < 2

Conclusions