Cα-CURVES AND THEIR Cα-FRAME IN CONFORMABLE DIFFERENTIAL GEOMETRY

Year 2024, Volume: 7 Issue: 2, 99 - 112, 31.07.2024

Aykut Has Beyhan Yılmaz

https://doi.org/10.33773/jum.1508243

Abstract

The aim of this study is to redesign the space curve and its Frenet framework, which are extremely important in terms of differential geometry, by using conformable derivative arguments. In this context, conformable counterparts of basic geometric concepts such as angle, vector, line, plane and sphere have been obtained. The advantages of the conformable derivative over the classical (Newton) derivative are mentioned. Finally, new concepts produced by conformable derivative are supported with the help of examples and figures.

Keywords

Fractional calculus, conformable derivative, Frenet frame

Supporting Institution

Kahramanmaraş Sütçü İmam Üniversitesi

Project Number

2022/7-14M

References

• L.R. Bishop, There is more than one way to frame a curve, American Mathematical Monthly, Vol.82, No.3, pp.246-251 (1975).
• H.S.A Aziz, M.K. Saad, On special curves according to Darboux frame in the three dimensional Lorentz space, computers, Materials and Continua, Vol.54, No.3, pp.229-249 (2012).
• S.Senyurt, D-Smarandache curves according to the Sabban frame of the spherical indicatrix curve, Turk. J. Math. Comput. Sci., Vol.9, pp.39-49 (2018).
• R.L. Magin, Fractional calculus in bioengineering, Crit Rev Biomed Eng., Vol.32, No.1, pp.1- 104 (2004).
• V. V. Uchaikin, Fractional derivatives for physicists and engineers, Springer Berlin, Heidelberg, (2013).
• W. Chen, H. Sun, X. Li, Fractional derivative modeling in mechanics and engineering, Springer, Singapore, (2022).
• A. Akgül, S.H.A. Khoshnawb, Application of fractional derivative on non-linear biochemical reaction models, International Journal of Intelligent Networks, Vol.1, pp.52-58 (2020).
• I. Podlubny, Fractional differential equations, Academic Pres, New York, (1999).
• K.B Oldham, J. Spanier, The fractional calculus, Academic Pres, New York, (1974).
• K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, (1993).
• M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, Vol.1, No.2, pp.161-198 (1971).
• R. Khalil, M. Horani, A. Yousef, M. Sababheh, A new deffnition of fractional derivative, Journal of Computational and Applied Mathematics, Vol.264, pp.65-70 (2014).
• U.N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2, (2014).
• J.V.C. Sousa, J., E.C.de Oliveira, E., Mittag-Leffer functions and the truncated V-fractional derivative, Mediterr. J. Math., Vol.14, No.6, pp.244 (2017).
• J.V.C. Sousa, E.C. de Oliveira, On the local M-derivative, Progr. Fract. Differ. Appl., Vol.4, No.4, pp.479-492 (2018).
• T. Yajima, K. Yamasaki, Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional ows, J. Phys. A: Math. Theor., Vol.45, (2012).
• T. Yajima, S. Oiwa, K. Yamasaki, Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas, Fractional Calculus and Applied Analysis, Vol.21, No.6, pp.1493-1505 (2018).
• K.A Lazopoulos, A.K. Lazopoulos, Fractional differential geometry of curves and surfaces. Progress in Fractional Differentiation and Applications, Vol.2, No.3, pp.169-186 (2016).
• M. E. Aydin, Effect of local fractional derivatives on Riemann curvature tensor, Examples and Counterexamples, Vol.5 (2024), https://doi.org/10.1016/j.exco.2023.100134.
• Has A., Yılmaz B., Baleanu D., On the Geometric and Physical Properties of Conformable Derivative, Math. Sci. Appl. E-Notes., Vol.12, No.2, pp.60-70 (2024). doi:10.36753/mathenot.1384280
• U. Gozutok, H.A. Coban, Y. Sagiroglu, Frenet frame with respect to conformable derivative, Filomat, Vol.33, No.6, pp.1541-1550 (2019).
• M.E. Aydin, A. Mihai, A. Yokus, Applications of fractional calculus in equiafine geometry: plane curves with fractional order, Math Meth Appl Sci., Vol.44, No.17, pp.13659-13669 (2021).
• A. Has, B. Yılmaz, Special fractional curve pairs with fractional calculus, International Electronic Journal of Geometry, Vol.15, No.1, pp.132-144 (2022).
• A. Has, B. Yılmaz, A. Akkurt, H. Yildirim, Conformable special curves in Euclidean 3-Space, Filomat, Vol.36, No.14, pp.4687-4698 (2022).
• A. Has, B. Yılmaz, Effect of fractional analysis on magnetic curves, Revista Mexicana de Fisica, Vol.68, No.4, pp.1-15 (2022).
• B. Yılmaz, A. Has, Obtaining fractional electromagnetic curves in optical fiber using fractional alternative moving frame, Optik - International Journal for Light and Electron Optics, Vol.260, No.8 (2022).
• B. Yılmaz, A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus, Optik - International Journal for Light and Electron Optics, Vol.247, No.30 (2021).
• H. Durmaz, Z. Özdemir, Şekerci Y., Fractional approach to evolution of the magnetic field lines near the magnetic null points, Physica Scripta, Vol.99, No.2 (2024). https://doi.org/10.1088/1402-4896/ad1c7e.
• M. Tasdemir, E. O. Canfes, B. Uzun, On Caputo fractional Bertrand curves in E3 and E13, Filomat, Vol.38, No.5, pp.1681-1702 (2024).
• M. Ögrenmiş, Fractional Curvatures of Equiafine Curves in Three-Dimensional Afine Space, Journal of New Theory, Vol.46, pp.11-22 (2024).
• D.J. Struik, Lectures on classical differential geometry, Dover Publications, New York, (1988).
• N.Y. Gozutok, U. Gozutok, Multivariable conformable fractional calculus, Filomat, Vol.32, No.2, pp.45-53 (2018).