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Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions

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  • Mohammadi, Reza

Abstract

The present paper uses a relatively new approach and methodology to solve one and two dimensional nonlinear Schrödinger equations numerically. We use the horizontal method of lines and θ-method, θ ∈ [1/2, 1] for time discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in space subsequently have been solved by quintic polynomial spline scheme. Convergence of the scheme in maximum norm is established rigorously. The convergence orders are O(k+hx4+hy4) and O(k2+hx4+hy4), where k is the temporal grid size and hx and hy are spatial grid sizes, respectively. Matrix stability analysis shows that the method is conditionally stable. The efficacy of proposed approach has been confirmed with four numerical experiments, where comparison is made with some earlier works. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. The present scheme is very simple, effective and convenient for obtaining numerical solution of Schrödinger equation.

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  • Mohammadi, Reza, 2018. "Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 204-215.
    • Handle: RePEc:eee:chsofr:v:107:y:2018:i:c:p:204-215
    DOI: 10.1016/j.chaos.2018.01.006
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    References listed on IDEAS

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    1. Dehghan, Mehdi, 2006. "Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 16-30.
    2. Zhang, Rongpei & Zhu, Jiang & Yu, Xijun & Li, Mingjun & Loula, Abimael F.D., 2017. "A conservative spectral collocation method for the nonlinear Schrödinger equation in two dimensions," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 194-203.
    3. Shindin, Sergey & Parumasur, Nabendra & Govinder, Saieshan, 2017. "Analysis of a Chebyshev-type pseudo-spectral scheme for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 271-289.
    4. Bashan, Ali & Yagmurlu, Nuri Murat & Ucar, Yusuf & Esen, Alaattin, 2017. "An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method," Chaos, Solitons & Fractals, Elsevier, vol. 100(C), pages 45-56.
    5. Zhang, Jin-Liang & Li, Bao-An & Wang, Ming-Liang, 2009. "The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 858-865.
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    Cited by:

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