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Year 2021, Volume: 14 Issue: 1, 132 - 144, 15.04.2021

Abstract

References

  • [1] Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 (3), 201–210 (1983).
  • [2] Baleanu, D., Fernandez, A.: On fractional operators and their classifications. Mathematics 2019, 7, 830; doi:10.3390/math7090830.
  • [3] 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).
  • [4] Baleanu, D., Vacaru, S.I.: Fractional almost Kähler-Lagrange geometry. Nonlinear Dyn. 64 (4), 365–373 (2011).
  • [5] Baleanu, D., Diethelm, K., Scalas, E., Trujillo J.J.: Fractional Calculus: Models and Numerical Methods.World Scientific, New Jersey (2012).
  • [6] Baleanu, D., Trujillo, J.J.: A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. Comm. Nonlin. Sci. Numer. Simul. 15 (5), 1111–1115 (2010).
  • [7] Bas, E., Ozarslan, R.: Real world applications of fractional models by Atangana–Baleanu fractional derivative. Chaos Solit. Fractals 116, 121–125 (2018).
  • [8] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 13 (5), 529-539 (1967).
  • [9] do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976).
  • [10] Dokuyucu, M.A., Celik, E., Bulut, H., Baskonus, H.M.: Cancer treatment model with the Caputo-Fabrizio fractional derivative. Eur. Phys. J. Plus 133 (92) (2018). doi:10.1140/epjp/i2018-11950-y.
  • [11] De Espíndola, J., Bavastri, C., De Oliveira Lopes, E.: Design of optimum systems of viscoelastic vibration absorbers for a given material based on the fractional calculus model. J. Vib. Control 14 (9–10), 1607-1630 (2008).
  • [12] El-Nabulsi, R.A.: Fractional derivatives generalization of Einstein’s field equations. Ind. J. Phys. 87 (2), 195-200 (2013).
  • [13] El-Nabulsi, R.A.: Modifications at large distances from fractional and fractal arguments. Fractals 18 (02), 185-190 (2010).
  • [14] El-Nabulsi, R.A.: Fractional nonlocal Newton’s law of motion and emergence of Bagley-Torvik equation. J. Peridyn Nonlocal Model. 2, 50-58 (2020).
  • [15] El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. Royal Society A 476, 20190729, (2020).
  • [16] Gozutok, U., Coban, H.A., Sagiroglu, Y.: Frenet frame with respect to conformable derivative. Filomat 33 (6), 1541-1550 (2019).
  • [17] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. Second Ed. CRC Press, (1998).
  • [18] Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91 (3) (2003), 034101; doi: 10.1103/PhysRevLett.91.034101.
  • [19] Haba, T.C., Ablart, G., Camps, T., Olivie, F.: Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos, Solit. Fractals 24 (2), 479-490 (2005).
  • [20] Herrmann, R.: Towards a geometric interpretation of generalized fractional integrals-Erdelyi-Kober type integrals on RN as an example. Fract. Calc. Appl. Anal. 17 (2), 361-370 (2014).
  • [21] Heymans, N.: Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state. J. Vib. Control 14 (9–10), 1587–1596 (2008).
  • [22] Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk. J. Math. 28 153-163, (2004).
  • [23] Kilbas, A., Srivastava, H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Math. Studies. North-Holland, New York (2006).
  • [24] Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51 (2), 299–307 (1984).
  • [25] Kula, L., Ekmekci, N., Yayli, Y., Ilarslan, K.: Characterizations of slant helices in Euclidean 3-space. Turk. J. Math. 34, 261–273 (2010).
  • [26] Lancret, M.A.: Mémoire sur les courbes à double courbure. Mémoires présentés à l’Institut 1, 416-454 (1806).
  • [27] Lazopoulos, K.A., Lazopoulos, A.K.: Fractional differential geometry of curves & surfaces. Progr. Fract. Differ. Appl. 2 (3), 169–186 (2016).
  • [28] Lazopoulos, K.A., Lazopoulos, A.K.: Fractional vector calculus and fractional continuum mechanics. Progr. Fract. Differ. Appl. 2 (2), 85–104 (2016).
  • [29] Lazopoulos, K.A., Lazopoulos, A.K.: On the mathematical formulation of fractional derivatives. Progr. Fract. Differ. Appl. 5 (4), 261–267 (2019).
  • [30] Magin, R. L.: Fractional Calculus in Bioengineering. Begell House, Connecticut, UK, (2006).
  • [31] Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Franklin Inst. 335 B(6), 1077-1086 (1998).
  • [32] Ozarslan, R., Ercan, A., Bas, E.: Novel fractional models compatible with real world problems. Fractal Fract. 3 (15), (2019) doi:10.3390/fractalfract3020015.
  • [33] Pressley, A.: Elementary Differential Geometry, Springer Verlag, Berlin (2010).
  • [34] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, London (1993).
  • [35] Silva, M.F., MacHado, J.A.T.: Fractional order PD joint control of legged robot. J. Vib. Control, 12 (12), 1483-1501 (2006).
  • [36] Sommacal, L., Melchior, P., Oustaloup, A., Cabelguen, J.-M., Ijspeert, A. J.: Fractional multi-models of the frog gastrocnemius muscle. J. Vib. Control 14 (9-10), 1415-1430 (2008).
  • [37] Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simulat. 18, 2945–2948 (2013).
  • [38] Veeresha, P., Prakasha, D.G., Baskonus, H.M.: New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos 29, 013119 (2019), doi.org/10.1063/1.5074099.
  • [39] Yano, K., Kon, M.: Structures on Manifolds. World Sci. 1985.
  • [40] Yajima, T., Nagahama, H.: Differential geometry of viscoelastic models with fractional-order derivatives. J. Phys. A: Math. Theor. 43 (38) (2010), 385207; doi: 10.1088/1751-8113/43/38/385207.
  • [41] Yajima, T., Nagahama, H.: Geometric structures of fractional dynamical systems in non-Riemannian space: Applications to mechanical and electromechanical systems. Ann. Phys. (Berlin) 530 (5) (2018), 1700391; doi: 10.1002/andp.201700391.
  • [42] 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) (2012), 065201; doi:10.1088/17518113/45/6/065201.
  • [43] 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).
  • [44] Ye, H., Ding, Y.: Nonlinear dynamics and chaos in a fractional-order HIV model. Mathematical Problems in Engineering, 2009, Article ID 378614, 12 pages, 2009. https://doi.org/10.1155/2009/378614.

Differential Geometry of Curves in Euclidean 3-Space with Fractional Order

Year 2021, Volume: 14 Issue: 1, 132 - 144, 15.04.2021

Abstract

In this paper, for a given curve in the Euclidean 3-space $\mathbb{R}^{3}$ we introduce new invariants such as arc-length, curvature and torsion with
fractional-order and provide certain relations between these and the standart invariants. We obtain the Frenet-Serret formulas in $\mathbb{R}^{3}$
and then construct the ways of determining a curve in $\mathbb{R}^{2}$ and $% \mathbb{R}^{3}$ in terms of the new invariants. Several examples are also
given by figures.

References

  • [1] Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 (3), 201–210 (1983).
  • [2] Baleanu, D., Fernandez, A.: On fractional operators and their classifications. Mathematics 2019, 7, 830; doi:10.3390/math7090830.
  • [3] 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).
  • [4] Baleanu, D., Vacaru, S.I.: Fractional almost Kähler-Lagrange geometry. Nonlinear Dyn. 64 (4), 365–373 (2011).
  • [5] Baleanu, D., Diethelm, K., Scalas, E., Trujillo J.J.: Fractional Calculus: Models and Numerical Methods.World Scientific, New Jersey (2012).
  • [6] Baleanu, D., Trujillo, J.J.: A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. Comm. Nonlin. Sci. Numer. Simul. 15 (5), 1111–1115 (2010).
  • [7] Bas, E., Ozarslan, R.: Real world applications of fractional models by Atangana–Baleanu fractional derivative. Chaos Solit. Fractals 116, 121–125 (2018).
  • [8] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 13 (5), 529-539 (1967).
  • [9] do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976).
  • [10] Dokuyucu, M.A., Celik, E., Bulut, H., Baskonus, H.M.: Cancer treatment model with the Caputo-Fabrizio fractional derivative. Eur. Phys. J. Plus 133 (92) (2018). doi:10.1140/epjp/i2018-11950-y.
  • [11] De Espíndola, J., Bavastri, C., De Oliveira Lopes, E.: Design of optimum systems of viscoelastic vibration absorbers for a given material based on the fractional calculus model. J. Vib. Control 14 (9–10), 1607-1630 (2008).
  • [12] El-Nabulsi, R.A.: Fractional derivatives generalization of Einstein’s field equations. Ind. J. Phys. 87 (2), 195-200 (2013).
  • [13] El-Nabulsi, R.A.: Modifications at large distances from fractional and fractal arguments. Fractals 18 (02), 185-190 (2010).
  • [14] El-Nabulsi, R.A.: Fractional nonlocal Newton’s law of motion and emergence of Bagley-Torvik equation. J. Peridyn Nonlocal Model. 2, 50-58 (2020).
  • [15] El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. Royal Society A 476, 20190729, (2020).
  • [16] Gozutok, U., Coban, H.A., Sagiroglu, Y.: Frenet frame with respect to conformable derivative. Filomat 33 (6), 1541-1550 (2019).
  • [17] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. Second Ed. CRC Press, (1998).
  • [18] Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91 (3) (2003), 034101; doi: 10.1103/PhysRevLett.91.034101.
  • [19] Haba, T.C., Ablart, G., Camps, T., Olivie, F.: Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos, Solit. Fractals 24 (2), 479-490 (2005).
  • [20] Herrmann, R.: Towards a geometric interpretation of generalized fractional integrals-Erdelyi-Kober type integrals on RN as an example. Fract. Calc. Appl. Anal. 17 (2), 361-370 (2014).
  • [21] Heymans, N.: Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state. J. Vib. Control 14 (9–10), 1587–1596 (2008).
  • [22] Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk. J. Math. 28 153-163, (2004).
  • [23] Kilbas, A., Srivastava, H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Math. Studies. North-Holland, New York (2006).
  • [24] Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51 (2), 299–307 (1984).
  • [25] Kula, L., Ekmekci, N., Yayli, Y., Ilarslan, K.: Characterizations of slant helices in Euclidean 3-space. Turk. J. Math. 34, 261–273 (2010).
  • [26] Lancret, M.A.: Mémoire sur les courbes à double courbure. Mémoires présentés à l’Institut 1, 416-454 (1806).
  • [27] Lazopoulos, K.A., Lazopoulos, A.K.: Fractional differential geometry of curves & surfaces. Progr. Fract. Differ. Appl. 2 (3), 169–186 (2016).
  • [28] Lazopoulos, K.A., Lazopoulos, A.K.: Fractional vector calculus and fractional continuum mechanics. Progr. Fract. Differ. Appl. 2 (2), 85–104 (2016).
  • [29] Lazopoulos, K.A., Lazopoulos, A.K.: On the mathematical formulation of fractional derivatives. Progr. Fract. Differ. Appl. 5 (4), 261–267 (2019).
  • [30] Magin, R. L.: Fractional Calculus in Bioengineering. Begell House, Connecticut, UK, (2006).
  • [31] Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Franklin Inst. 335 B(6), 1077-1086 (1998).
  • [32] Ozarslan, R., Ercan, A., Bas, E.: Novel fractional models compatible with real world problems. Fractal Fract. 3 (15), (2019) doi:10.3390/fractalfract3020015.
  • [33] Pressley, A.: Elementary Differential Geometry, Springer Verlag, Berlin (2010).
  • [34] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, London (1993).
  • [35] Silva, M.F., MacHado, J.A.T.: Fractional order PD joint control of legged robot. J. Vib. Control, 12 (12), 1483-1501 (2006).
  • [36] Sommacal, L., Melchior, P., Oustaloup, A., Cabelguen, J.-M., Ijspeert, A. J.: Fractional multi-models of the frog gastrocnemius muscle. J. Vib. Control 14 (9-10), 1415-1430 (2008).
  • [37] Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simulat. 18, 2945–2948 (2013).
  • [38] Veeresha, P., Prakasha, D.G., Baskonus, H.M.: New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos 29, 013119 (2019), doi.org/10.1063/1.5074099.
  • [39] Yano, K., Kon, M.: Structures on Manifolds. World Sci. 1985.
  • [40] Yajima, T., Nagahama, H.: Differential geometry of viscoelastic models with fractional-order derivatives. J. Phys. A: Math. Theor. 43 (38) (2010), 385207; doi: 10.1088/1751-8113/43/38/385207.
  • [41] Yajima, T., Nagahama, H.: Geometric structures of fractional dynamical systems in non-Riemannian space: Applications to mechanical and electromechanical systems. Ann. Phys. (Berlin) 530 (5) (2018), 1700391; doi: 10.1002/andp.201700391.
  • [42] 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) (2012), 065201; doi:10.1088/17518113/45/6/065201.
  • [43] 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).
  • [44] Ye, H., Ding, Y.: Nonlinear dynamics and chaos in a fractional-order HIV model. Mathematical Problems in Engineering, 2009, Article ID 378614, 12 pages, 2009. https://doi.org/10.1155/2009/378614.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Muhittin Evren Aydın 0000-0001-9337-8165

Mehmet Bektaş 0000-0002-5797-4944

Alper Öğremiş 0000-0001-5008-2655

Asıf Yokuş 0000-0001-7568-2251

Publication Date April 15, 2021
Acceptance Date October 11, 2020
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Aydın, M. E., Bektaş, M., Öğremiş, A., Yokuş, A. (2021). Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. International Electronic Journal of Geometry, 14(1), 132-144.
AMA Aydın ME, Bektaş M, Öğremiş A, Yokuş A. Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. Int. Electron. J. Geom. April 2021;14(1):132-144.
Chicago Aydın, Muhittin Evren, Mehmet Bektaş, Alper Öğremiş, and Asıf Yokuş. “Differential Geometry of Curves in Euclidean 3-Space With Fractional Order”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 132-44.
EndNote Aydın ME, Bektaş M, Öğremiş A, Yokuş A (April 1, 2021) Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. International Electronic Journal of Geometry 14 1 132–144.
IEEE M. E. Aydın, M. Bektaş, A. Öğremiş, and A. Yokuş, “Differential Geometry of Curves in Euclidean 3-Space with Fractional Order”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 132–144, 2021.
ISNAD Aydın, Muhittin Evren et al. “Differential Geometry of Curves in Euclidean 3-Space With Fractional Order”. International Electronic Journal of Geometry 14/1 (April 2021), 132-144.
JAMA Aydın ME, Bektaş M, Öğremiş A, Yokuş A. Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. Int. Electron. J. Geom. 2021;14:132–144.
MLA Aydın, Muhittin Evren et al. “Differential Geometry of Curves in Euclidean 3-Space With Fractional Order”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 132-44.
Vancouver Aydın ME, Bektaş M, Öğremiş A, Yokuş A. Differential Geometry of Curves in Euclidean 3-Space with Fractional Order. Int. Electron. J. Geom. 2021;14(1):132-44.