Abstract
In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called \(\mathsf {WKB}\) method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the \(\mathsf {WKB}\) to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the \(\mathsf {WKB}\) method in the scope of the fractional differential equation. By means of this extension, the \(\mathsf {WKB}\) analysis based on the Borel resummation, for fractional differential operators of \(\mathsf {WKB}\) type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified \(\mathsf {WKB}\).
Similar content being viewed by others
References
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, North-Holland (2006)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore (2012)
Sayevand, K., Pichaghchi, K.: Successive approximation: a survey on stable manifold of fractional differential systems. Fract. Calculus Appl. Anal. 18(3), 621–641 (2015)
Sayevand, K.: Analytical treatment of Volterra integro-differential equations of fractional order. Appl. Math. Model. 39(15), 4330–4336 (2015)
Voros, A.: The return of the quartic oscillator. The complex WKB method. Ann. Inst. H. Poincaré A 39, 211–338 (1983)
Voros, A.: From exact-WKB towards singular quantum perturbation theory. 40, 973–990 (2004)
Iwaki, K., Nakanishi, T.: Exact WKB analysis and cluster algebras. arXiv:1401.7094v4 [math.CA] (2014)
Nayefeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1993)
Rabei, E.M.: Quantization of Christ-Lee model using the WKB approximation. Int. J. Theor. Phys. 42(9), 20–97 (2003)
Rabei, E.M., Nawafleh, K.I., Ghassib, H.B.: Quantization of constrained systems using the WKB approximation. Phys. Rev. A 66, 024–101 (2002)
Rabei, E.M., Nawafleh, K.I., Ghassib, H.B.: Quantization of reparameterizatized systems using the WKB method. Tur. J. Phys. 29(1), 151–162 (2005)
Rabei, E.M., Hassan, E.H., Ghassib, H.B.: Quantization of second-order constrained Lagrangians systems using the WKB approximation. Int. J. Geom. Methods Mod. Phys. 2(3), 485–504 (2005)
Rabei, E.M., Hassan, E.H., Ghassib, H.B.: Quantization of higher-order constrained Lagrangian systems using the WKB approximation. Int. J. Theor. Phys. 43(11), 22–85 (2004)
Rabei, E.M., Altarazi, I.M.A., Muslih, S.I., Baleanu, D.: Fractional WKB approximation. Nonlinear Dyn. 57(1–2), 171–175 (2009)
Rabei, E.M., Muslih, S.I., Baleanu, D.: Quantization of fractional systems using WKB approximation. Commun. Nonlinear Sci. Numer. Simulat. 15, 807–811 (2010)
Yang, X.J.: Advanced Local Fractional Calculus and its Applications. World Science Publisher, New York (2012)
Yang, X.J.: Local fractional integral transforms. Prog. Nonlinear Sci. 4, 1–225 (2011)
Hu, M.S., Baleanu, D., Yang, X.J.: One-phase problems for discontinuous heat transfer in fractal media. Math. Probl. Eng. 2013, 358–473 (2013)
Yang, X.J.: A short note on local fractional calculus of function of one variable. J. Appl. Libr. Inf. Sci. 1(1), 1–13 (2012)
Zhong, W.P., Gao, F., Shen, X.M.: Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral. Adv. Math. Res. 461, 306–310 (2012)
Yang, X.J.: Local fractional variational iteration method and its algorithms. Adv. Comput. Math. Appl. 1(3), 139–145 (2012)
He, J.H.: A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53, 3698–3718 (2014)
Kolwankar, K.M.: Local fractional calculus: a review. arXiv preprint arXiv:1307.0739 (2013)
Babakhani, A., Daftardar-Gejji, V.: On calculus of local fractional derivatives. J. Math. Anal. Appl. 270, 66–79 (2002)
Adda, F.B., Cresson, J.: About non-differentiable functions. J. Math. Anal. Appl. 263, 721–737 (2001)
Aoki, T.: Calcul exponentiel des operateurs microdifferentiels dordre inni. I. Ann. Inst. Fourier (Grenoble) 33, 227–250 (1983)
Aoki, T.: Symbols and formal symbols of pseudodifferential operators. Adv. Stud. Pure Math. 4, 181–208 (1984)
Aoki, T.: Calcul exponentiel des operateurs microdifferentiels dordre inni, II. Ann. Inst. Fourier (Grenoble) 36, 143–165 (1986)
Aoki, T., Kawai, T., Koike, T., Takei, Y.: On the exact WKB analysis of operators admitting innitely many phases. Adv. Math. 181, 165–189 (2004)
Guillemin, V.W., Kashiwara, M., Kawai, T.: Seminar on micro-local analysis. Princeton University Press and University of Tokyo Press, Princeton, New Jeresy (1979)
Aoki, T.: Quantized contact transformations and pseudodifferential operators of innite-order. Publ. Res. Inst. Math. Sci. Kyoto Univ. 26, 505–519 (1990)
Malgrange, B.: L’involutivite des caracteristiques des systemes differentiels et micro-differentiels. Sem. Bourbaki 522, 277–289 (1977/78)
Jarad, F., Abdeljawad, T., Gundogdu, E., Baleanu, D.: On the Mittag-Leffler stability of q-fractional nonlinear dynamical system. Proc. Rom. Acad. Ser. A 12, 309–314 (2011)
Momani, S., Hadid, S.: Lyapunov stability solutions of fractional integro differential equations. Int. J. Math. Math. Sci. 47, 2503–2507 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sayevand, K., Pichaghchi, K. A modification of \(\mathsf {WKB}\) method for fractional differential operators of Schrödinger’s type. Anal.Math.Phys. 7, 291–318 (2017). https://doi.org/10.1007/s13324-016-0143-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13324-016-0143-7
Keywords
- \(\mathsf {WKB}\) approximation
- Exact \(\mathsf {WKB}\) analysis
- Schrödinger equation
- Local fractional derivative
- Mittag-Leffler stability