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2.2 ELECTRICAL, ELECTRONIC, INFORMATION ENGINEERING, Electrical and electronic engineering
A brief review of fractional differentiation, frac-tional integration and differo-integral operators is given basedon the properties of the Fourier transform. This is undertakento provide the reader with a quick-guide and a short back-ground to the fractional calculus and includes a brief discussionon some of the principal characteristics of fractional differo-integral operators. The paper then presents a new definitionfor a fractional differo-integral based on the properties ofthe sign function and explores some related results. Usingthe properties of the Dirac delta function, a generalisation isdeveloped in order to quantify the issue as to whether there isan upper bound to the number of definitions for a fractionaldiffero-integral operator that can be developed. Finally, anapproximation of a fractional differential is considered andused in the construction of a self-affine stochastic time seriesmodel based on the Kolmogorov-Feller equation for the memoryfunctiont−α/Γ(1−α), α∈(0,1).
Blackledge, J. (2021) "A New Definition, a Generalisation and an Approximation for a Fractional Derivative with Applications to Stochastic Time Series Modeling, Engineering Letters, vol. 29, no. 1, pp138-150, 2021.