Solving the generalized time-space fractional Schrödinger equation by homotopy perturbation Sumudu transform method
DOI:
https://doi.org/10.37376/1571-000-049-009Keywords:
Caputo derivative, Homotopy perturbation method, Sumudu transform method;, Schrödinger equation.Abstract
We suggest and analyze a technique by combining the homotopy perturbation method and the Sumudu transform method. This method is called the homotopy perturbation sumudu transform method. We use this method for solving Generalized Time-space Fractional Schrödinger equation. The fractional derivative is described in Caputo sense. The proposed scheme finds the solution without any discritization, transformation or restrictive assumptions. Several example is given to check the reliability and efficiency of the proposed technique.
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Behzadi, S.S. (2011) Solving Schrödinger equation by using modified variational iteration and homotopy analysis methods. Journal of Applied Analysis and Computation. 1(4):427-437.
Belgacem, F.B.M. and Karaballi, A.A. (2006) Sumudu Transform Fundamental Properties Investigations and Application. Journal of Applied Mathematics and Stochastic Analysis. 1-23.
Chen, Y. and Li, B. (2008) An extended sub equation rational expansion method with symbolic compuation and solutions of the nonlinear Schrödinger equation model. Nonlinear Analysis: Hybrid Systems. 2(2):242-255.
Dalir, M. and Bashour, M. (2010) Applications of fractional calculus. Applied Mathematical Sciences. 4:1021-1032.
Ghazanfari, B. and Ghazanfari, A.G. (2012) Solving fractional nonlinear Schrödinger equation by fractional complex transform method. International Journal of Mathematical Modelling and Computations. 2(4):277-281.
Ghorbani, A. (2009) Beyond Adomian Polynomials: He Polynomials. Chaos, Solitons & Fractals. 39, 1486-1492.
Gupta, P.k. and Singh, M. (2011) Homotopy perturbation method for fractional Fornberg-Whitham equation. Computers and Mathematics with Applications. 61(2):250-254.
Gupta, V.G. and Sharma, B. (2010) Application of Sumudu Transform in Reaction-Diffusion Systems and Nonlinear Waves. Applied Mathematical Sciences. 4, 435-446.
Hao, R., Li, L., Li, Z., Xue, W. and Zhou, G. (2004) A new approach to exact solition solutions and solition interaction for nonlinear Schrödinger equation with variable coefficients. Optics Communications. 236(1-3):79-86.
Herzallah, M.A.E. and Gepreel, K.A. (2012) Approximate solution to the time-space fractional cubicnonlinear Schrödinger equation. Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. 36(11):5678-5685.
Hong, B. and Lu, D. (2014) Modified fractional variational iteration method for solving the generalized time-space fractional Schrödinger equation. The Scientific World Journal. Article ID 964643, P. 6.
Jumarie, G. (2007) Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution. J. App. Math. Computing. 1:31-48.
Karbalaie, A., Montazeri, M.M. and Muhammed, H.H (2014) Exact solution of Time-Fractional partial differential equations using Sumudu transform. Wseast Transactions on Mathematics. 13:142-151.
Kilbas, A.A., Saigo, M. and Saxena, R.K. (2004) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms and Special Functions. 15:31-49.
Laskin, N. (2000) Fractional market dynamics. Physica A. 287( 3-4):482-492.
Li, B. and Chen, Y. (2004) On exact solutions of the nonlinear Schrödinger equations in optical fiber. Chaos, Solitons and Fractals. 21(1):241-247.
Magin, R.L. and Ovadia, M. (2008) Modeling the cardiac tissue electrode in-terface using fractional calculus. J. Vibration and Control , Vol. 14, pp. 1431–1442
Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, NY, USA.
Momani, S. and Odibat, Z. (2007) Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physica Letters A. 365( 5-6):345-350.
Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York, NY, USA.
Rida, S.Z., El-sherbiny, H.M. and Arafa, A.A.M. (2008) On the solution of the fractional nonlinear Schrödinger equation. Physica Letters A. 372(5):553-558.
Singh, J., Kumar, D. and Sushila, Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations. Adv. Theor. Appl. Mech, Vol. 4, No. 4, pp. 165–175.
Sun, H.H., Abdelwahab, A.A and Onaral, B. (1984) Linear approximation of transfer function with a pole of fractional order. IEEE Ttansactions on Automatic Control. 29(5):441-444.
Wang, Q. (2008) Homotopy perturbation method for fractional KdV-Burgers equation. Chaos, Solitons and Fractals. 35(5):843-850.
Wazwaz, A.A. (2008) Study on linear and nonlinear Schrödinger equations by the variational iteration method. Chaos, Solitons and Fractals. 37(4):1136-1142.
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