我們以附有延拓法之徑向基底函數配置法求解葛羅斯-皮塔夫斯基方程。方程式藉由以逆多元二次徑向基底函數在均勻網格點上之線性組合內插方法離散化。我們利用方程式線性部分之精確解作為延拓法的初始值，而在非線性部分中的散射長度常數及角動量常數分別當作連續法之獨立參數。在數值實驗中可觀察到，此方法在修正過程中的收斂區域很廣，即使當通過奇異點時通常也能收斂到數值解。

We compute the numerical solution of Gross-Pitaevskii equation by a continuation algorithm embedded in the radial basis function collocation method. The equation is interpolated by a linear combination of the inverse multiquadric functions under uniform grid points. The exact solution of the linear portion of the equation is taken as the initial value, and the other nonlinear parts, scattering term and rotating term, are independently extended by continuation algorithm. The numerical experiments show that under such formulation, the convergent regions are quite large in the correction process so that it usually converges to a numerical solution even when passing a singular point.

論文審定書 ------------------------------------------------------------------------- i
摘要 ---------------------------------------------------------------------------------- ii
Abstract ----------------------------------------------------------------------------- iii
1. Introduction --------------------------------------------------------------------- 1
2. Preliminary ---------------------------------------------------------------------- 3
2.1 Collocation method ------------------------------------------------------- 3
2.2 Radial basis function ----------------------------------------------------- 4
3. Discretization ------------------------------------------------------------------- 7
3.1 Collocation points --------------------------------------------------------- 7
3.2 The nonlinear Schrodinger equation ------------------------------ 8
3.3 The mass conservation constraint ------------------------------------ 10
4. Numerical methods ----------------------------------------------------------- 12
4.1 Linear Schrodinger equation ---------------------------------------- 12
4.2 Continuation method with parameter μ ------------------------------ 14
4.3 Continuation method with parameter ω ----------------------------- 15
5. Numerical results -------------------------------------------------------------- 19
6. Conclusions --------------------------------------------------------------------- 24
7. Appendix ------------------------------------------------------------------------- 25
References -------------------------------------------------------------------------- 27

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