Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation - Ahmad, Azhar (2021)
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.