Fractional Curvatures of Equiaffine Curves in Three-Dimensional Affine Space
Abstract
This paper presents a method for computing the curvatures of equiaffine curves in three-dimensional affine space by utilizing local fractional derivatives. First, the concepts of $\alpha$-equiaffine arc length and $\alpha$-equiaffine curvatures are introduced by considering a general local involving conformable derivative, V-derivative, etc. In fractional calculus, equiaffine Frenet formulas and curvatures are reestablished. Then, it presents the relationships between the equiaffine curvatures and $\alpha$-equiaffine curvatures. Furthermore, graphical representations of equiaffine and $\alpha$-equiaffine curvatures illustrate their behavior under various conditions.
Keywords
Fractional derivative, equiaffine curvatures, affine space
Supporting Institution
Çalışma hazırlanırken herhangi bir kurum tarafından maddi destek sağlanmamıştır.
Ethical Statement
Çalışmada etik beyana gerek duyulacak bir veri kullanılmamıştır.
References
- D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics 7 (9) (2019) 830 10 pages.
- M. E. Aydin, A. Mihai, A. Yokus, Applications of fractional calculus in equiaffine geometry: Plane curves with fractional order, Mathematical Methods in the Applied Sciences 44 (17) (2020) 13659-13669.
- T. Yajima, S. Oiwa, K. Yamasaki, Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas, Fractional Calculus and Applied Analysis 21 (6) (2018) 1493-1505.
- M. E. Aydın, M. Bektaş, A. O. Öğrenmis, A. Yokuş, Differential geometry of curves in Euclidean 3-space with fractional order, International Electronic Journal of Geometry 14 (1) (2021) 132-144.
- U. Gözütok, H. A. Çoban, Y. Sağıroğlu, Frenet frame with respect to conformable derivative, Filomat 33 (6) (2019) 1541-1550.
- A. Has, B. Yılmaz, Special fractional curve pairs with fractional calculus, International Electronic Journal of Geometry 15 (1) (2022) 132-144.
- K. Lazopoulos, A. K. Lazopoulos, Fractional differential geometry of curves and surfaces, Progress in Fractional Differentiation and Applications 2 (3) (2016) 169-186.
- V. E. Tarasov, On chain rule for fractional derivatives, Communications in Nonlinear Science and Numerical Simulation 30 (1) (2016) 1-4.
- H. Bulut, H. M. Baskonus, Y. Pandir, The modified trial equation method for fractional wave equation and time fractional generalized burgers equation, Abstract and Applied Analysis 2013 (2013) 636802 8 pages.
- H. F. Ismael, H. M. Baskonus, H. Bulut, W. Gao, Instability modulation and novel optical soliton solutions to the Gerdjikov–Ivanov equation with m-fractional, Optical and Quantum Electronics 55 (4) (2023) 303.