Caputo q-Fractional Initial Value Problems and a q-Analogue Mittag-Leffler Function

TL;DR

This paper introduces Caputo q-fractional derivatives, solves initial value problems using a new q-Mittag-Leffler function, and discusses open problems in q-fractional integrals.

Contribution

It presents the first study of Caputo q-fractional derivatives and develops a novel q-Mittag-Leffler function for solving related initial value problems.

Findings

Solution expressed via new q-Mittag-Leffler function

Introduction of Caputo q-fractional derivatives

Discussion of open problems in q-fractional integrals

Abstract

Caputo q-fractional derivatives are introduced and studied. A Caputo -type q-fractional initial value problem is solved and its solution is expressed by means of a new introduced q-Mittag-Leffler function. Some open problems about q-fractional integrals are proposed as well.