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Fractional-order operator, is defined

(2.1)

where t₁ and t₂ are the upper and lower time limits for the operator.

The term λe is the fractional order. It is an arbitrary complex number. Real(λe) is the real part of λe.

The Grnwald-Letniknov (GL) fractional-order derivative of the function f(t) is defined

(2.2)

where −1 is the rounding operation, c is the calculation step, and is the binomial coefficients defined as ϵ⁰.

Integration and differential denoted by a uniform expression.

(2.3)

The fractional-order operator can be done by using the following equation [8]:

(2.4)

where

(2.5)

(2.6)

By ignoring the very old data, an approximate fractional-order approximation is obtained by

(2.7)

where and L is the memory length.