Frenet frame with respect to conformable derivative

Abstract

Conformable fractional derivative is introduced by the authors Khalil at al in 2014. In this study, we investigate the frenet frame with respect to conformable fractional derivative. Curvature and torsion of a conformable curve are defined and the geometric interpretation of these two functions is studied. Also, fundamental theorem of curves is expressed for the conformable curves and an example of the curve corresponding to a fractional differential equation is given.

Special Fractional Curve Pairs with Fractional Calculus

“…Let x be a conformable curve. If D α (x)(t) = 0 for all t ∈ (0, ∞), x is called a conformable regular curve, [7].…”

“…Because in this two fractional derivative, the derivative of the constant is zero. For this reason, Caputo fractional derivative and conformable fractional derivative are used while investigating the effect of fractional derivative on differential geometry [4,13,20,21,2,3,7].…”

Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

“…12,13 In this paper, we contribute to the applications of fractional calculus from a differential geometric viewpoint. These applications have received much attention in the last decade (see 5,7,[14][15][16][17][18][19][20][21] ). More explicitly, we use some techniques from fractional calculus in order to initiate the study of equiaffine differential geometry of plane curves.…”

“…Based on the simplification (), Yajima et al 21 established the notions of curvature and Frenet–Serret formulas for a given Euclidean plane curve in with fractional order and Aydin et al 14 extended these to . Besides the cited references, the investigations of curves using conformable 17 and Leibniz's L‐fractional 18 derivatives appear in the literature. But all of these study focus on the curves in the Euclidean setting, and as far as the authors know, the present approach is a first attempt in this sense.…”

Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

“…us, researchers have paid more attention to conformable derivative and other related local derivatives in modeling scientific phenomena. While there are some recent studies concerning the mathematical analysis of conformable calculus such as the multivariable conformable calculus [15] that was introduced in 2018, the behavior of conformable derivatives of functions in arbitrary Banach spaces [24] that was investigated in 2021, the differential geometry of curves [25] that was investigated in 2019 in the senses of conformable derivatives and integrals, and the behavioral framework for the conformable linear differential systems' stability [26] that was carefully studied in 2020 to utilize the importance of CoV in modeling scenarios of control theory and power electronics, our results in this work provide a comprehensive investigation of α-derivative of a function of SVs and all related properties, the CoV of CR for functions of SVs, and the CoV of IF m involving many numerical examples to validate our obtained results. According to the best of our knowledge, our original investigation in this article provides an essential mathematical analysis tool for researchers working on modeling phenomena in physics and engineering in the sense of conformable calculus because all theorems and properties in this work will be needed in such modeling scenarios.…”