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Some efficient numerical methods for Schrodinger-type equations and financial derivatives

English Abstract

The first part of this thesis is concerned with one-dimensional 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 semi-discretization scheme of the one-dimensional 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 discrete-time 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 sine-Gordon equation. The nonlinear sine-Gordon 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 Fourier-cosine series expansion, proposed by Fang and Oosterlee [40], and was used in two-dimensional cases by Ruijter and Oosterlee [76]. In this part, we consider a modification of the Fourier sine-sine 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

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Meng, Qing Jiang


Faculty of Science and Technology


Department of Mathematics




Schrödinger equation

Nonlinear theories

Numerical analysis

Finance -- Mathematical models


Jin, Xiao Qing

Ding, Deng

Sun, Hai Wei

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