Abstract
Wentzel–Kramer–Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case, the wave function is constructed such that the phase factor is the same as the Hamilton’s principle function S. To demonstrate our proposed approach, two examples are investigated in detail.
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Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach, Linghorne (1993)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Connecticut (2006)
Tenreiro Machado, J.A.: A probabilistic interpretation of the fractional-order differentiation. Fractional Calc. Appl. Anal. 8, 73–80 (2003)
Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space–time fractional diffusion equation. Fractional Calc. Appl. Anal. 4(2), 153 (2001)
Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)
Tofighi, A.: The intrinsic damping of the fractional oscillator. Physica A 329, 29–34 (2006)
Stanislavsky, A.A.: Fractional oscillator. Phys. Rev. E 70, 051103 (2004)
Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivatives. Czech. J. Phys. 1, 1348–1354 (2001)
Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52, 1247–1253 (2002)
Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)
Riewe, F.: Non-conservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)
Rabei, E.M., Alhalholy T.S. and Rousan, A.A.: Potential of arbitrary forces with fractional derivatives. Int. J. Mod. Phys. A 19:17&18, 3083–3092 (2004)
Rabei, E.M., Ajlouni, A., Ghassib, H.B.: Quantisation of Brownian motion. Int. J. Theor. Phys. 45, 1613–1623 (2005)
Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007)
Rabei, E.M., Ababneh, B.S.: Hamilton–Jacobi fractional sequential mechanics. Preprint, hep-th/07040519
Rabei, E.M., Hasan, E.H., Ghassib, H.B.: Hamilton–Jacobi treatment of constrained systems with second-order Lagrangians. Int. J. Theor. Phys. 43(4), 1073–1096 (2004)
Rabei, E.M., Nawafleh, K.I., Abdelrahman, Y.S., Omari, H.Y.R.: Hamilton–Jacobi treatment of Lagrangians with linear velocities. Mod. Phys. Lett. A 18, 1591–1596 (2003)
Rabei, E.M., Eyad, H., Hasan, E.H., Ghassib, H.B., Muslih, S.I.: Quantisation of second-order constrained Lagrangian systems using the WKB approximation. Int. J. Geom. Methods Mod. Phys. 2, 1–20 (2005)
Nawafleh, K.I., Rabei, E.M., Ghassib, H.B.: Hamilton–Jacobi treatment of constrained systems. Int. J. Mod. Phys. A 19, 347–354 (2004)
Rabei, E.M., Nawafleh, K.I., Ghassib, H.B.: Quantisation of constrained systems using the WKB approximation. Phys. Rev. A 66, 24101–24011 (2002)
Griffiths, D.J.: Introduction to Quantum Mechanics. Prentice-Hall, New Jersey (1995)
Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading (1980)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
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Rabei, E.M., Altarazi, I.M.A., Muslih, S.I. et al. Fractional WKB approximation. Nonlinear Dyn 57, 171–175 (2009). https://doi.org/10.1007/s11071-008-9430-7
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DOI: https://doi.org/10.1007/s11071-008-9430-7