Residual power series method for timefractional Schrödinger equations
Authors
Yu Zhang
 College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450000, China.
Amit Kumar
 Department of Mathematics, National Institute of Technology, Jamshedpur831014, Jharkhand, India.
Sunil Kumar
 Department of Mathematics, National Institute of Technology, Jamshedpur831014, Jharkhand, India.
Dumitru Baleanu
 Department of Mathematics, Cankya University, Ogretmenler Cad. 14, Balgat06530, Ankara, Turkey.
 Institute of Space Sciences, MagureleBucharest, Romania.
XiaoJun Yang
 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China.
Abstract
In this paper, the residual power series method (RPSM) is effectively applied to find the exact solutions
of fractionalorder time dependent Schrödinger equations. The competency of the method is examined by
applying it to the several numerical examples. Mainly, we find that our solutions obtained by the proposed
method are completely compatible with the solutions available in the literature. The obtained results
interpret that the proposed method is very effective and simple for handling different types of fractional
differential equations (FDEs).
Keywords
 residual power series
 fractional power series
 Fractional Schrödinger equation
 exact solution.
MSC
References

[1]
S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360 (2006), 109113

[2]
O. Abu Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 3152

[3]
A. A. Kilbas , H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North Holland Mathematics Studies, Elsevier Science B.V.,, Amsterdam (2006)

[4]
S. Kumar, A numerical study for the solution of time fractional nonlinear shallow water equation in oceans, Z. Naturforschung A, 68 (2013), 547553

[5]
S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 31543163

[6]
S. Kumar, A. Kumar, D. Baleanu, Two analytical methods for timefractional nonlinear coupled Boussinesq Burger's equations arise in propagation of shallow water waves, Nonlinear Dynam., 85 (2016), 699715

[7]
A. Kumar, S. Kumar, M. Singh, Residual power series method for fractional SharmaTassoOlever equation, Commun. Numer. Anal., 2016 (2016), 10 pages

[8]
S. Kumar, M. M. Rashidi, New analytical method for gas dynamic equation arising in shock fronts, Comput. Phys. Commun., 185 (2014), 19471954

[9]
S. Momani, Z. Odibat, Analytical solution of a timefractional NavierStokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488494

[10]
M. M. Mousaa, S. F. Ragab, Application of the homotopy perturbation method to linear and nonlinear schrödinger equations, Z. Naturforschung A, 63 (2008), 140144

[11]
Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 2734

[12]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, San Diego (1999)

[13]
A. M. Wazwaz, A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos Solitons Fractals, 37 (2008), 11361142

[14]
X.J. Yang, D. Baleanu, Fractal heat conduction problem solved by local fractional variation iteration method, Therm. Sci., 17 (2013), 625628

[15]
X.J. Yang, D. Baleanu, M. P. Lazarevic, M. S. Cajic, Fractal boundary value problems for integral and differential equations with local fractional operators, Therm. Sci., 19 (2013), 959966

[16]
X.J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)

[17]
X.J. Yang, D. Baleanu, W.P. Zhong, Approximate solutions for diffusion equations on Cantor spacetime, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 14 (2013), 127133

[18]
X.J. Yang, H. M. Srivastava, C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (2015), 752761

[19]
X.J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact travelingwave solutions for local fractional Kortewegde Vries equation, Chaos, 26 (2016), 5 pages