Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas

References

[1] Om.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, Issue 1 (2002), 368–379; 10.1016/S0022-247X(02)00180-4.Search in Google Scholar

[2] T.M. Atanacković, S. Konjik, Lj. Oparnica, S. Pilipović, Generalized Hamilton’s principle with fractional derivatives. J. Phys. A: Math. Theor. 43, No 25 (2010), # 255203; 10.1088/1751-8113/43/25/255203.Search in Google Scholar

[3] R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dynam. 14, No 2 (1991), 304–311; 10.2514/3.20641.Search in Google Scholar

[4] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, No 3 (1983), 201–210; 10.1122/1.549724.Search in Google Scholar

[5] R.L. Bagley, P.J. Torvik, Fractional calculus - A different approach to the analysis of viscoelastically damped structures. AIAA J. 21, No 5 (1983), 741–748; 10.2514/3.8142.Search in Google Scholar

[6] R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, No 1 (1986), 133–155; 10.1122/1.549887.Search in Google Scholar

[7] D. Baleanu, J.J. Trujillo, A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. Comm. Nonlin. Sci. Numer. Simul. 15, No 5 (2010), 1111–1115; 10.1016/j.cnsns.2009.05.023.Search in Google Scholar

[8] D. Baleanu, S.I. Vacaru, Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics. Cent. Eur. J. Phys. 9, No 5 (2011), 1267–1279; 10.2478/s11534-011-0040-5.Search in Google Scholar

[9] D. Baleanu, S.I. Vacaru, Fractional almost Kähler-Lagrange geometry. Nonlinear Dyn. 64, No 4 (2011), 365–373; 10.1007/s11071-010-9867-3.Search in Google Scholar

[10] D. Baleanu, S.I. Vacaru, Fractional curve flows and solitonic hierarchies in gravity and geometric mechanics. J. Math. Phys. 52, No 5 (2011), # 053514; 10.1063/1.3589964.Search in Google Scholar

[11] D. Baleanu, S.I. Vacaru, Fedosov quantization of fractional Lagrange spaces. Int. J. Theor. Phys. 50, No 1 (2011), 233–243; 10.1007/s10773-010-0514-z.Search in Google Scholar

[12] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific, New Jersey (2012).10.1142/8180Search in Google Scholar

[13] D. Baleanu, T. Maaraba (Abdeljawad), F. Jarad, Fractional variational principles with delay. J. Phys. A: Math. Theor. 41, No 31 (2008), # 315403; 10.1088/1751-8113/41/31/315403.Search in Google Scholar

[14] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 13, No 5 (1967), 529–539; 10.1111/j.1365-246X.1967.tb02303.x.Search in Google Scholar

[15] M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pageoph91, No 1 (1971), 134–147, 10.1007/BF00879562.Search in Google Scholar

[16] A. Gjurchinovski, T. Sandev, V. Urumov, Delayed feedback control of fractional-order chaotic systems. J. Phys. A: Math. Theor. 43, No 44 (2010), # 445102; 10.1088/1751-8113/43/44/445102.Search in Google Scholar

[17] I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, No 3 (2003), # 034101; 10.1103/PhysRevLett.91.034101.Search in Google Scholar PubMed

[18] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, New Jersey (2011).10.1142/8072Search in Google Scholar

[19] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[20] R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, No 2 (1984), 299–307; 10.1115/1.3167616.Search in Google Scholar

[21] K.A. Lazopoulos, A.K. Lazopoulos, Fractional differential geometry of curves & surfaces. Progr. Fract. Differ. Appl. 2, No 3 (2016), 169–186; 10.18576/pfda/020302.Search in Google Scholar

[22] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Dover Publications, New York (2006).Search in Google Scholar

[23] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar

[24] D.J. Struik, Lectures on Classical Differential Geometry. Dover Publications, New York (1988).Search in Google Scholar

[25] V.E. Tarasov, Fractional generalization of gradient and Hamiltonian systems. J. Phys. A: Math. Gen. 38, No 26 (2005), 5929–5943; 10.1088/0305-4470/38/26/007.Search in Google Scholar

[26] V.E. Tarasov, Fractional generalization of gradient systems. Lett. Math. Phys. 73, No 1 (2005), 49–58; 10.1007/s11005-005-8444-z.Search in Google Scholar

[27] S.I. Vacaru, Fractional dynamics from Einstein gravity, general solutions, and black holes. Int. J. Theor. Phys. 51, No 5 (2012), 1338–1359; 10.1007/s10773-011-1010-9.Search in Google Scholar

[28] S.I. Vacaru, Fractional nonholonomic Ricci flows. Chaos Soliton. Fract. 45, Nos 9-10 (2012), 1266–1276; 10.1016/j.chaos.2012.06.011.Search in Google Scholar

[29] T. Yajima, H. Nagahama, Differential geometry of viscoelastic models with fractional-order derivatives. J. Phys. A: Math. Theor. 43, No 38 (2010), # 385207; 10.1088/1751-8113/43/38/385207.Search in Google Scholar

[30] T. Yajima, H. Nagahama, Geometric structures of fractional dynamical systems in non-Riemannian space: Applications to mechanical and electromechanical systems. Ann. Phys. (Berlin)530, No 5 (2018), # 1700391; 10.1002/andp.201700391.Search in Google Scholar

[31] T. Yajima, K. Yamasaki, Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows. J. Phys. A: Math. Theor. 45, No 6 (2012), # 065201; 10.1088/1751-8113/45/6/065201.Search in Google Scholar