UM ETheses Collection (澳門大學電子學位論文庫)
 Title

Some efficient numerical methods for Schrodingertype equations and financial derivatives
 English Abstract

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The first part of this thesis is concerned with onedimensional coupled nonlinear Schrodinger (CNLS) equations, which can model a beam propagation inside crys ¨ tals or photorefractives as well as water wave interactions. First, the use of the orthogonal spline collocation (OSC) method for the semidiscretization scheme of the onedimensional CNLS equations is presented. In particular, conservation properties, including mass conservation, energy conservation, and momentum conservation, are shown in both theory and practice. Later, numerical tests with various initial conditions of the CNLS equations like single soliton, interaction of two solitons, interaction of arbitrary initial conditions, and periodic boundary conditions are reported. In the second part, we consider a discretetime OSC scheme for solving Schrodinger ¨ equation with wave operator. The scheme was proposed recently by Wang et al. (J. Comput. Appl. Math., 235 (2011), 1993–2005.) and was shown to have high order convergence rate when a parameter θ in the scheme was chosen in [ 1 4 , +∞). In this chapter, we show that the result can be extended to include θ ∈ (0, 1 4 ). A numerical example is given to justify the theoretical result. In the third part, we understand OSC method for solving the nonlinear sineGordon equation. The nonlinear sineGordon equation might be viewed as the most simplest form of wave equations. This scheme uses Hermite basis functions to approximate the solution at Gaussian points. The convergence rate with order O(h 4 + τ 2 ) and stability of the scheme are proved. The numerical results are presented, and are compared with analytical solutions to confirm the accuracy of the presented scheme. In the last part, we investigate option price arising from financial derivatives. An efficient option pricing method based on Fouriercosine series expansion, proposed by Fang and Oosterlee [40], and was used in twodimensional cases by Ruijter and Oosterlee [76]. In this part, we consider a modification of the Fourier sinesine expansion, whereby in [−1, 1] the sin(πnx) functions are replaced by sin(π(n − 1 2 )x), n ≥ 1, for pricing rainbow options. Feasible approaches to obtain the pricing formulae are presented. Practical applications to basket options and correlation options under difiii ferent models are also given. They indicate satisfactory convergence and efficiency as expected
 Issue date

2013.
 Author

Meng, Qing Jiang
 Faculty

Faculty of Science and Technology
 Department

Department of Mathematics
 Degree

Ph.D.
 Subject

Schrödinger equation
Nonlinear theories
Numerical analysis
Finance  Mathematical models
 Supervisor

Jin, Xiao Qing
Ding, Deng
Sun, Hai Wei
 Files In This Item
 Location
 1/F Zone C
 Library URL
 991004703849706306