Abstract.
The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrödinger equation (T-FSE) and the space-fractional Schrödinger equation (S-FSE). The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, the presented approach is also applied to solve the time-fractional coupled Schrödinger system (T-FCSS). In order to demonstrate the validity and accuracy of the numerical scheme proposed, several numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.
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References
G. Bohannan, J. Vibr. Control 14, 1487 (2008)
H.G. Sun, W. Chen, H. Wei, Y.Q. Chen, Eur. Phys. J. ST 193, 185 (2011)
S. Chen, F. Liu, Appl. Math. Modell. 33, 256 (2009)
H.G. Sun, W. Chen, C. Li, Y.Q. Chen, Phys. A Stat. Mech. Appl. 389, 2719 (2010)
S. Das, Functional Fractional Calculus for System Identification and Controls (Springer, NewYork, 2008)
S.S. Ray, Appl. Math. Comput. 218, 5239 (2012)
M. Dehghan, J.M. Heris, A. Saadatmandi, Numer. Methods Partial Differ. Equ. 26, 448 (2010)
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Appl. Math. Comput. 234, 267 (2014)
P. Zhuang, Y.T. Gu, F. Liu, I. Turner, P.K. Yarlagadda, Int. J. Numer. Methods Eng. 88, 1346 (2011)
H. Zhou, W. Tian, W. Deng, J. Sci. Comput. 56, 45 (2013)
Q. Yang, I. Turner, F. Liu, M. Ilis, SIAM J. Sci. Comput. 33, 1159 (2011)
R.K. Pandey, O.P. Singh, V.K. Baranwal, Comput. Phys. Commun. 182, 1134 (2011)
J. Liu, G. Hou, Appl. Math. Comput. 217, 7001 (2011)
Y. Jiang, J. Ma, J. Comput. Appl. Math. 235, 3285 (2011)
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 35, 5662 (2011)
A.H. Bhrawy, A.S. Alofi, S.S. Ezz-Eldien, Appl. Math. Lett. 24, 2146 (2011)
A.H. Bhrawy, M.M. Al-Shomrani, Adv. Differ. Equ. 2012, 1 (2012)
E.H. Doha, A.H. Bhrawy, D. Baleanu, S.S. Ezz-Eldien, Appl. Math. Comput. 219, 8042 (2013)
A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011)
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 36, 4931 (2012)
A.H. Bhrawy, E.H. Doha, D. Baleanu, S.S. Ezz-Eldien, J. Comput. Phys., DOI:10.1016/j.jcp.2014.03.039
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Cent. Eur. J. Phys. 11, 1494 (2013)
E.H. Doha, A.H. Bhrawy, M.A. Abdelkawy, R.A.V. Gorder, J. Comput. Phys. 261, 244 (2014)
M.R. Eslahchi, M. Dehghan, M. Parvizi, J. Comput. Appl. Math. 257, 105 (2014)
A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 62, 1135 (2011)
M. Dehghan, A. Taleei, Comput. Phys. Commun. 181, 43 (2010)
Z. Gao, S. Xie, Appl. Numer. Math. 61, 593 (2011)
X. Guo, M. Xu, J. Math. Phys. 47, 82 (2006)
A. Iomin, Chaos Solitons Fractals 44, 348 (2011)
N. Laskin, Phys. Rev. E 66, 056108 (2002)
N. Laskin, Phy. Rev. E 62, 3135 (2000)
N. Laskin, Phys. Lett. A 268, 298 (2000)
M. Naber, J. Math. Phys. 45, 3339 (2004)
S.Z. Rida, H.M. El-Sherbiny, A.A.M. Arafa, Phys. Lett. A 372, 553 (2008)
N.A. Khan, M. Jamil, A. Ara, ISRN Math. Phys. 2012, 197068 (2012)
A. Yildirim, Int. J. Nonlinear Sci. Numer. Simulat. 10, 445 (2009)
K. Aruna, A.S.V.R. Kanth, Natl. Acad. Sci. Lett. 36, 201 (2013)
P. Felmer, Proc. R. Soc. Edinburgh 142, 1237 (2012)
M.A.E. Herzallah, K.A. Gepreel, Appl. Math. Modell. 36, 5678 (2012)
J. Hu, J. Xin, H. Lu, Comput. Math. Appl. 62, 1510 (2011)
S.I. Muslih, O.P. Agrawal, D. Baleanu, Int. J. Theor. Phys. 49, 1746 (2010)
P. Wang, C. Huang, J. Comput. Phys., DOI:10.1016/j.jcp.2014.03.037
L. Wei, Y. He, X. Zhang, S. Wang, Finite Elements Anal. Des. 59, 28 (2012)
A. Mohebbi, M. Abbaszadeh, M. Dehghan, Eng. Anal. Bound. Elements 37, 475 (2013)
C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995)
S. Leble, B. Reichel, Eur. Phys. J. ST 173, 5 (2009)
A. Aydin, B. Karasözen, Int. J. Comput. Math. 86, 864 (2009)
M.S. Ismail, Appl. Math. Comput. 196, 273 (2008)
T. Wang, B. Guo, L. Zhang, Appl. Math. Comput. 217, 1604 (2010)
X.W. Zhou, L. Wang, Comput. Math. Appl. 61, 2035 (2011)
L. Wei, X. Zhang, S. Kumar, A. Yildirim, Comput. Math. Appl. 64, 2603 (2012)
D. Wang, A. Xiao, W. Yang, J. Comput. Phys. 242, 670 (2013)
D. Wang, A. Xiao, W. Yang, J. Comput. Phys. 272, 644 (2014)
M. Gasca, T. Sauer, J. Comput. Appl. Math. 122, 23 (2000)
J. De Villiers, Mathematics of Approximation (Atlantis Press, 2012)
M.R. Eslahchi, M. Dehghan, M. Parvizi, J. Comput. Appl. Math. 257, 105 (2014)
Y. Yang, Y. Huang, Adv. Math. Phys 2013, 821327 (2013)
Y. Yang, Y. Chen, Y. Huang, Acta Math. Sci. 34B, 673 (2014)
Q. Xu, J.S. Hesthaven, J. Comput. Phys. 257, 241 (2014)
X. Ma, C. Huang, Appl. Math. Modell. 38, 1434 (2014)
X. Li, T. Tang, Front. Math. China 1, 69 (2012)
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Bhrawy, A.H., Doha, E.H., Ezz-Eldien, S.S. et al. A new Jacobi spectral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems. Eur. Phys. J. Plus 129, 260 (2014). https://doi.org/10.1140/epjp/i2014-14260-6
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DOI: https://doi.org/10.1140/epjp/i2014-14260-6