Effect of fractional analysis on magnetic curves

Authors

• Aykut Has Kahramanmaraş Sütçü İmam University
• Beyhan Yılmaz Kahramanmaraş Sütçü İmam University

Keywords:

Magnetic curve, vector fields, fractional derivative, conformable fractional derivative

Abstract

In this present paper, the effect of fractional analysis on magnetic curves is researched. A magnetic field is defined by the property that its divergence is zero in a three dimensional Riemannian manifold. We investigate the trajectories of the magnetic fields called as t-magnetic, n-magnetic and b-magnetic curves according to fractional derivative and integral. 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 magnetic curve, the conformable fractional derivative that best 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.

References

M. Barros, A. Romero, Magnetic vortices. EPL, 77 (2007) 1. https://doi.org/10.1209/0295-5075/77/34002.

H. Ceyhan et al., Electromagnetic curves and rotation of the polarization plane through alternative moving frame, Eur. Phys. J. Plus, 135 (2020) 867. https://doi.org/10.1140/epjp/s13360-020-00881-z.

T. Körpinar and R.C. Demirkol, Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D semi-Riemannian manifold, J. Mod. Optik, 66(8) (2019) 857. https://doi.org/10.1080/09500340.2019.1579930.

T. Körpinar and R.C. Demirkol, Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D Riemannian manifold with Bishop equations, J. Mod. Optik, 200 (2020) 163334. https://doi.org/10.1080/09500340.2019.1579930.

T. Körpinar and R.C. Demirkol, Electromagnetic curves of the polarized light wave along the optical fiber in De-Sitter 2-space S12, Indian J. Phys., 95 (2021) 147. https://doi.org/10.1007/s12648-019-01674-6.

T. Körpinar, Geometric magnetic phase for timelike spherical optical ferromagnetic model, Int. J. Geom. Methods Mod. Phys., 18 (2021) 2150099. https://doi.org/10.1142/S0219887821500997.

T. Körpinar, R.C. Demirkol, Z. Körpinar and V. Asil, New magnetic flux flows with Heisenberg ferromagnetic spin of optical quasi velocity magnetic flows with flux density, Rev. Mex. Fis., 67 (2021) 378. https://doi.org/10.31349/RevMexFis.67.378.

T. Körpinar, R.C. Demirkol, Z. Körpinar and V. Asil, Fractional solutions for the inextensible Heisenberg antiferromagnetic flow and solitonic magnetic flux surfaces in the binormal direction, Rev. Mex. Fis., 67 (2021) 452. https://doi.org/10.31349/RevMexFis.67.452.

Z. Ozdemir, I. Gok, Y. Yaylıand F.N. Ekmekci, Notes on magnetic curves in 3D semi-Riemannian manifolds, Turkish Journal of Mathematics, 39(3) (2015) 412, https://doi.org/10.3906/mat-1408-31.

A. Loverro, Fractional Calculus: History, definitions and applications for the engineer, (USA, 2004)

R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol., 27(3) (1983) 201. https://doi.org/10.1122/1.549724.

K.B. Oldham and J. Spanier, The fractional calculus, (Academic Pres, New York, 1974)

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc., 13(5) (1967) 529. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.

R. Hilfer, Applications of fractional calculus in physics, (World Scientific, Singapore, 2000), https://doi.org/10.1142/3779.

A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, (North-Holland, New York, 2006).

D. Baleanu and S.I. Vacaru, Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics, Cent. Eur. J. Phys., (5) (2011) 1267. https://doi.org/10.2478/s11534-011-0040-5.

K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, (Wiley, New York, 1993)

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, 247 (2021) 168026.

K.A. Lazopoulos and A.K. Lazopoulos, Fractional differential geometry of curves and surfaces, Progr. Fract. Differ. Appl., 2(3) (2016) 169. https://doi.org/10.18576/pfda/020302.

T. Yajima, S. Oiwa and K. Yamasaki, Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas, Fractional Calculus and Applied Analysis, 21(6) (2018), 1493. https://doi.org/10.1515/fca-2018-0078.

D.R. Anderson and D.J. Ulness, Results for conformable differential equations, Preprint (2016). 22. D.R. Anderson, E. Camrud and D.J. Ulness, On the nature of the conformable derivative and its applications to physics, Journal of Fractional Calculus ans Applications, 10(2) (2019), 92. 23. T. Yajima and K. Yamasaki, Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows, J. Phys. A: Math. Theor., 45(6) (2012) 065201. https://doi.org/10.1515/fca-2018-0078.

M.E. Aydın, M. Bektas¸, A.O. Ogrenmis¸, A. Yokus¸, Differential geometry of curves in Euclidean 3-space with fractional order, International Electronic Journal of Geometry, 14(1) (2021) 132. https://doi.org/10.36890/iejg.751009.

M.E. Aydin, A. Mihai and A. Yokus, Applications of fractional calculus in equiaffine geometry: Plane curves with fractional order, Math Meth Appl Sci., 44 (2021) 13659. https: //doi.org/10.1002/mma.764913669.

D.J. Struik, Lectures on classical diferential geometry, (2nd edn. Addison Wesley, Boston, 1988)

M. Barros, General helices and a theorem of Lancret, Proc, Am. Math. Soc., 125(5) (1997) 1503.

S. Izumiya , N. Takeuchi, New special curves and developable surfaces, Turk J Math, 28 (2004) 153.

R. Khalil, M.A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264 (2014) 65. https://doi.org/10.1016/j.cam.2014.01.002.

T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 27(9) (2015) 57. https://doi.org/10.1016/j.cam.2014.10.016.

M. Barros, J.L. Cabrerizo, M. Fernández and A. Romero, Magnetic vortex filament flows, Journal of Mathematical Physics, 48(8) (2007) 082904. https://doi.org/10.1063/1.2767535.

Z. Bozkurt, I. Gok, Y. Yaylı and F.N. Ekmekc¸i, A new approach for magnetic curves in 3D Riemannian manifolds, Journal of Mathematical Physics, 55 (2014) 053501. https://doi. org/10.3906/mat-1408-31.

U. Gozutok, H.A. Coban and Y. Sagiroglu, Frenet frame with respect to conformable derivative, Filomat, 33(6) (2019) 1541. https://doi.org/10.2298/FIL1906541G

2022-06-07

How to Cite

[1]
A. Has and B. Yılmaz, “Effect of fractional analysis on magnetic curves”, Rev. Mex. Fís., vol. 68, no. 4 Jul-Aug, pp. 041401 1–, Jun. 2022.

Section

14 Other areas in Physics