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Lashkarian, Elham; Hejazi, S. Reza

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods. (English) Zbl 1391.76655

Math. Methods Appl. Sci. 41, No. 7, 2664-2672 (2018).
Summary: In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann-Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi-Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.

Cited in 12 Documents

MSC:

76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
34K37 Functional-differential equations with fractional derivatives

Keywords:

Erdelyi-Kober operators; fractional derivatives; invariant subspace; Lie point symmetry; Mittag-Leffler function; Schrödinger equation
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References:

[1] GalaktionovVA, SvirshchevskiiSR. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. London: Chapman and Hall/CRC; 2007. · Zbl 1153.35001
[2] GalaktionovVA. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 125; 1995; Cambridge University Press. 225‐246. · Zbl 0824.35128
[3] SvirshchevskiiSR. Invariant linear spaces and exact solutions of nonlinear evolution equations. J Nonlinear Math Phys. 1996;3:164‐169. · Zbl 1044.35511
[4] GazizovR, KasatkinA, LukashchukSY. Symmetry properties of fractional diffusion equations. Physica Scripta. 2009;T136:014016.
[5] BuckwarE, LuchkoY. Invariance of a partial differential equation of fractional order under the lie group of scaling transformations. J Math Anal Appl. 1998;1:81‐97. · Zbl 0932.58038
[6] DjordjevicVD, AtanackovicTM. Similarity solutions to non‐linear heat conduction and Burgers/Korteweg‐deVries fractional equations. J Comput Appl Math. 2008;2:701‐714. · Zbl 1157.35470
[7] PodlubnyI, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. San Diego, CA: Academic.
[8] KnopsRJ, StuartCA. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch Ration Mech Anal. 1984;186(13):233‐249. 2013; 66(5):576-584. · Zbl 0589.73017
[9] KiryakovaV. Generalised fractional calculus and applications. Pitman Research Notes in Mathematics. Proceedings of the Edinburgh Mathematical Society. Vol. 38. Cambridge University Press; 1994:189‐190. · Zbl 0882.26003
[10] lashkarianE, Reza HejaziS. Group analysis of the time fractional generalized diffusion equation. Physica A: Stat Mech Appl. 2017;479:572‐579. · Zbl 1495.35194
[11] lashkarianE, Reza HejaziS. Conservation laws of (3+α)‐dimensional time‐fractional diffusion equation. Comput Math Appl. 10.1016/j.camwa.2017.10.001, (in press).
[12] LiuH. Complete group classifications and symmetry reductions of the fractional fifth‐order KDV types of equations. Stud Appl Math. 2013;4:317‐330. · Zbl 1277.35305
[13] OuhadanA, ElkinaniEH. Exact solutions of time fractional Kolmogorov equation by using Lie symmetry analysis. J Fractional Calculus Appl. 2014;1:97‐104. · Zbl 1488.34003
[14] SahadevanR, BakkyarajT. Invariant analysis of time fractional generalized Burgers and Korteweg‐de Vries equations. J Math Anal Appl. 2012;2:341‐347. · Zbl 1245.35142
[15] GodlewskiE, RaviartP‐A. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118. Berlin, Germany: Springer; 1996. · Zbl 0860.65075
[16] LeVequeRJ. Numerical Methods for Conservation Laws. 2nd ed., Lectures in Mathematics ETH Zurich. Basel, Switzerland: Birkhauser; 1992. · Zbl 0847.65053
[17] WangGW, XuTZ. Symmetry properties and explicit solutions of the nonlinear time fractional KDV equation. Boundary Value Prob. 2013;1:232. · Zbl 1293.22006
[18] WangGW, LiuXQ, ZhangYY. Lie symmetry analysis to the time fractional generalized fifth‐order KDV equation. Commun Nonlinear Sci Numer Simul. 2013;9:2321‐2326. · Zbl 1304.35624
[19] WangGW, XuTZ, FengT. Lie symmetry analysis and explicit solutions of the time fractional fifth‐order KDV equation. PLOS ONE. 2014;9(2):e88336.
[20] GazizovRR, KasatkinA. Construction of exact solutions for fractional order differential equations by the invariant subspace method. Comput Math Appl. 2013;66:576‐584. · Zbl 1348.34012
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