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.
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Neamaty, A., Nategh, M. & Agheli, B. Local non-integer order dynamic problems on time scales revisited. Int. J. Dynam. Control 6, 486–498 (2018). https://doi.org/10.1007/s40435-017-0322-x
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DOI: https://doi.org/10.1007/s40435-017-0322-x
Keywords
- Time Scales
- Fractional derivative
- Pre-differentiability
- Banach Valued Function
- Mean value theorem
- H\( \ddot{o} \)lder continuous
- Fractional integration on time scales