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

References

1. D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics 7 (9) (2019) 830 10 pages.
2. 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.
3. 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.
4. 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.
5. U. Gözütok, H. A. Çoban, Y. Sağıroğlu, Frenet frame with respect to conformable derivative, Filomat 33 (6) (2019) 1541-1550.
6. A. Has, B. Yılmaz, Special fractional curve pairs with fractional calculus, International Electronic Journal of Geometry 15 (1) (2022) 132-144.
7. K. Lazopoulos, A. K. Lazopoulos, Fractional differential geometry of curves and surfaces, Progress in Fractional Differentiation and Applications 2 (3) (2016) 169-186.
8. V. E. Tarasov, On chain rule for fractional derivatives, Communications in Nonlinear Science and Numerical Simulation 30 (1) (2016) 1-4.

9. 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.
10. 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.
11. M. A. Dokuyucu, E. Çelik, H. Bulut, H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus 133 (3) (2018) 92.
12. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, Chicago, 2006.
13. T. Yajima, H. Nagahama, Differential geometry of viscoelastic models with fractional-order derivatives, Journal of Physics A: Mathematical and Theoretical 43 (38) (2010) 385207 9 pages.
14. D. Baleanu, S. S. Sajjadi, A. Jajarmi, Ö. Defterli, On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: A new fractional analysis and control, Advances in Difference Equations 2021 (2021) 234 17 pages.
15. T. Yajima, H. Nagahama, Geometric structures of fractional dynamical systems in non-Riemannian space: Applications to mechanical and electromechanical systems, Annalen der Physik 530 (5) (2018) 1700391 9 pages.
16. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering Academic Press, New York, 1999.
17. T. Abdeljawad, On conformable fractional calculus JournalofComputational and Applied Mathematics 279 (2015) 57-66.
18. J. Vanterler da C. Sousa, E. Capelas de Oliveira, Anew truncated m-fractional derivative type unifying some fractional derivative types with classical properties, International Journal of Analysis and Applications 16 (1) (2018) 83-96.
19. J. Vanterler da C. Sousa, E. Capelas de Oliveira, Mittag–Leffler functions and the truncated $\nu$-fractional derivative, Mediterranean Journal of Mathematics 14 (6) (2017) 244 24 pages.
20. A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, New York, 2006.
21. J. N. Clelland, From Frenet to Cartan: The method of moving frames, American Mathematical Society, Providence, 2017.
22. D. Davis, Generic affine differential geometry of curves in $\mathbb{R}^{n}$, Proceedings of the Royal Society of Edinburgh Section A 136 (6) (2006) 1195-1205.
23. H. W. Guggenheimer, Differential geometry, McGraw-Hill, New York, 1963.
24. M. E. Aydın, S. Kaya, Fractional equiaffine curvatures of curves in 3-dimensional affine space, International Journal of Maps in Mathematics 6 (1) (2023) 67-82.