Ahmad, Azhar
(2021)
Fourth-Order Spline Methods For
Solving Nonlinear Schrödinger
Equation.
PhD thesis, Perpustakaan Hamzah Sendut.
Abstract
The Nonlinear Schrödinger (NLS) equation is an important and fundamental equation
in Mathematical Physics. In this thesis, fourth-order cubic B-spline collocation
method and fourth-order cubic Exponential B-spline collocation method are developed
in order to solve problems involving the NLS equation. The established Cubic
B-spline Collocation Method and Cubic Exponential B-spline Collocation Method are
of second-order accuracy. The methods developed in this thesis are of fourth-order
accuracy. The time dimension of the NLS equation is discretized using the Finite Difference
Method and the space dimension is discretized based on the particular B-spline
methods used. The Taylor series approach and Besse approaches are used to handle
the nonlinear term of the NLS equation. Since the methods result in an underdetermined
system, the supplementary initial and boundary conditions are used to solve the
system. The developed methods are tested for stability and are found to be unconditionally
stable. Error analysis and convergence analysis are also carried out. The
efficiency of the methods are assessed on three test problems involving solitons and
the approximations are found to be very accurate. Besides that, the numerical order
of convergence is calculated and associated theoretical statements are proved. In conclusion,
the proposed methods in this study worked well and give accurate numerical
results for the NLS equation.
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