“Conformable fractional” derivatives and integrals are integer-order operators: Physical and geometrical interpretations, applications to fractal physics

Vasily E. Tarasov

Chaos, Solitons & Fractals, 2025, vol. 192, issue C

Abstract: Operators, which Khalil, Al Horani, Yousef, Sababhe called “conformable fractional” derivatives and integrals in 2014, were proposed in 2005 as differential operators of the integer order. Since 2005, these operators have been used in various articles to describe fractal distributions of matter and fractal media in the framework of continuum models with fractal density of states. In this paper, we proved that the “conformable fractional” derivatives and integrals cannot be considered as derivatives and integrals of non-integer orders, nor as fractional derivatives or integrals. The “conformable fractional” derivatives and integrals are operators of the integer orders. We demonstrate that “conformable fractional” operators suggested in 2014, have been proposed and applied since at least in 2005–2014 to describe fractal media. We prove that these integer-order operators can be interpreted and applied as operators in non-integer dimensional space (NIDS), and therefore it can be called NIDS operators. It can be used to describe fractal media and fractal distributions of matter in the framework of continuum models with fractal density of states. The proposed interpretation allows us to avoid errors in constructing mathematical models and interpreting the results obtained when using these operators. To do this, it is necessary to remove the incorrect term “conformable fractional” and use in these operators numerical multipliers with parameters describing the dimensions (fractality) along the axes.

Keywords: Fractional calculus; Fractal media; Non-integer dimensional space; Fractal density of states; Conformable fractional derivative (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000797

DOI: 10.1016/j.chaos.2025.116066

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