Special Fractional Curve Pairs with Fractional Calculus

Yıl 2022, Cilt: 15 Sayı: 1, 132 - 144, 30.04.2022

Aykut Has Beyhan Yılmaz

https://doi.org/10.36890/iejg.1010311

Cited By: 6

Öz

In this study, the effect of fractional derivatives, whose application area is increasing day by day, on curve pairs is investigated. As it is known, there are not many studies on a geometric interpretation of fractional calculus. When examining the effect of fractional analysis on a curve, the Conformable fractional derivative that fits the algebraic structure of differential geometry derivative is used. This effect is examined with the help of examples consistent with the theory and visualized for different values of the Conformable fractional derivative. The difference of this study from others is the use of Conformable fractional derivatives and integrals in calculations. Fractional calculus has applications in many fields such as physics, engineering, mathematical biology, fluid mechanics,signal processing, etc. Fractional derivatives and integrals have become an extremely important and new mathematical method in solving various problems in many sciences.

Anahtar Kelimeler

Fractional calculus, conformable fractional derivatives, special curve pairs, curvatures

Kaynakça

• [1] Abdeljawad, T.: On conformable Fractional Calculus. Journal of Computational and Applied Mathematics. 27(9), 57-66 ( 2015).
• [2] Aydın M.E., Bektaş M., Öğrenmiş A.O., YokuşA.:Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. International Electronic Journal of Geometry. 14(1), 132-144 (2021).
• [3] Aydin ME, Mihai A, Yokus A.: Applications of fractional calculus in equiaffinegeometry: Plane curves with fractional order. Math Meth Appl Sci. 1-11 (2021). https://doi.org/10.1002/mma.7649
• [4] Baleanu, D., Vacaru, S.I.: Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics. Cent. Eur. J. Phys. 9 (5), 1267-1279 (2011).
• [5] Camci, Ç., Uçum, A., İlarslan, K.: A new approach to Bertrand curves in Euclidean 3-space. J. Geom. 111(49) (2020). https://doi.org/10.1007/s00022-020-00560-5
• [6] Dalir, M., Bashour, M.: Applications of Fractional Calculus. Applied Mathematical Sciences. 21, 1021-1032 (2010).
• [7] Gozutok, U., Coban, H.A., Sagiroglu, Y.: Frenet frame with respect to conformable derivative. Filomat. 33 (6), 1541-1550 (2019).
• [8] Hacısalihoglu H. H.: Differential Geometri. Inonu University Publications. Malatya (1983).
• [9] Has, A, Yılmaz, B.: Various Characterizations for Quaternionic Mannheim Curves in Three-Dimensional Euclidean Space. Journal of Universal Mathematics. 4 (1), 62-72 (2021). doi:10.33773/jum.856869
• [10] Hilfer, R.: Applications of fractional calculus in physics. World Scientific. Singapore (2000).
• [11] Kilbas, A., Srivastava H., Trujillo J.: Theory and Applications of Fractional Differential Equations. in:Math. Studies. North-Holland. New York (2006).
• [12] Khalil, R., Horani, M. Al., Yousef, A., Sababheh, M.: A New Definition of Fractional Derivative. Journal of Computational and Applied Mathematics. 264, 65-70 (2014).
• [13] Lazopoulos, K.A., Lazopoulos, A.K.: Fractional differential geometry of curves & surfaces. Progr. Fract. Differ. Appl. 2 (3), 169-186 (2016).
• [14] Liu, H., Wang, H. F.: Mannheim partner curves in 3-space. Journal of Geometry. 88, 120-126 (2008).
• [15] Loverro, A.: Fractional Calculus. History, Defination and Applications for the Engineer. USA (2004).
• [16] Magin, R. L.: Fractional Calculus in Bioengineering. Critical Reviews in Biomedical Engineering. 32(1), 1-104 (2004).
• [17] Oldham, K. B., Spanier, J.: The fractional calculus. Academic Pres. New York (1974).
• [18] Sabuncuoglu A.: Diferansiyel Geometri. Nobel Academic Publishing. Ankara (2014).
• [19] Struik, D.J.: Lectures on classical diferential geometry.Addison Wesley. Boston (1988).
• [20] Yajima, T., Yamasaki, K.: Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows. J. Phys. A: Math. Theor. 45(6), 065201 (2012). doi:10.1088/17518113/45/6/065201.
• [21] Yajima, T., Oiwa, S., Yamasaki, K.: Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas. Fract. Calc. Appl. Anal. 21 (6), 1493-1505 (2018).
• [22] Yıldırım A., Kaya F.: Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3􀀀space E3. Fundamentals of Contemporary Mathematical Sciences. 1(2), 63-70 (2020).
• [23] Yılmaz, B.: 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. 247(30), 168026 (2021). 143