Analytical and numerical representations for discrete Grünwald–Letnikov fractional calculus

摘要

This paper focuses on the new representation of discrete Grünwald-Letnikov fractional calculus. By resorting the classical nabla Taylor formula and nabla Taylor series, the representations of Grünwald-Letnikov difference/sum are established. Changing the expanded point from the initial instant to the current time, another series like representation is developed. To improve the practicability, the results are extended to the variable order case and the fixed memory step case. With the developed representations, the corresponding Leibniz rules are built subsequently.