Local non-integer order dynamic problems on time scales revisited

Abstract

This paper deals with a non-integer order calculus on time scales in a local sense. Taking advantage of a new field structure over real and complex numbers, the researchers have normalized the proposed derivative. Non-integer order mean value theorem on time scales is stated. In addition to a hypothesis on the existence of a H\( \ddot{o} \)lder continuous function h, a given time scale \( \mathbb {T} \) is explored. This consideration provided an extension to a recently proposed local fractional differentiation theory and it paved the ground to a local fractional integration on time scales. Also, making use of the presented non-integer order calculus, a class of dynamic initial value problems of fractional orders on an arbitrary time scale is studied. Eventually, an approximate solution to the potential-free Schrdinger equation of time-fractional order is obtained and illustrated.