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Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

1
School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
2
Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fluids 2019, 4(2), 108; https://doi.org/10.3390/fluids4020108
Received: 17 April 2019 / Revised: 25 May 2019 / Accepted: 7 June 2019 / Published: 12 June 2019
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics)
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Abstract

The non-linear Schrödinger (NLS) equation has often been used as a model equation in the study of quantum states of physical systems. Numerical solution of NLS equation is obtained using cubic B-spline Galerkin method. We have applied the Crank–Nicolson scheme for time discretization and the cubic B-spline basis function for space discretization. Three numerical problems, including single soliton, interaction of two solitons and birth of standing soliton, are demonstrated to evaluate to the performance and accuracy of the method. The error norms and conservation laws are determined and found to be in good agreement with the published results. The obtained results show that the approach is feasible and accurate. The proposed method has almost second order convergence. The linear stability of the method is performed using the Von Neumann method. View Full-Text
Keywords: non-linear Schrödinger equation; cubic B-spline basis functions; Galerkin method non-linear Schrödinger equation; cubic B-spline basis functions; Galerkin method
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Iqbal, A.; Abd Hamid, N.N.; Md. Ismail, A.I. Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method. Fluids 2019, 4, 108.

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