Uncertain fractional differential equations on a time scale under Granular differentiability concept

Abstract

In this work, the Caputo q-fractional initial value problem of order $$\alpha \in (0,1)$$ in uncertain environment is investigated. The uncertainties are considered as possibility sets using fuzzy sets. The concepts of horizontal membership function and granular difference are used to give a new definition for fuzzy fractional integral and derivative, called granular Riemann–Liouville q-fractional integral and granular Caputo q-fractional derivative. The existence and uniqueness of solution for the Caputo q-fractional initial value problem under granular differentiability concept is established. Finally, some illustrative examples are also presented.