Abstract
The fractional derivative holds historical dependence or non-locality and it becomes a powerful tool in many real-world applications. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. This paper defines the tempered fractional derivative on an isolated time scale and suggests a new method based on the time scale theory for numerical discretization. Some useful properties like composition law and equivalent fractional sum equations are derived for theoretical analysis. Finally, numerical formulas of fractional discrete systems are provided. As a special case for the step size [Formula: see text], a fractional logistic map with two-parameter memory effects is reported.
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