### Title page for etd-0810110-200158

URN etd-0810110-200158 Hsiu-Chen Hsieh No Public. This thesis had been viewed 5225 times. Download 0 times. Applied Mathematics 2009 2 Master English Collocation Fourier methods for Elliptic andEigenvalue Problems 2010-06-03 97 eigenvalue problems Bose-Einstein condensation periodic potential stability analysis energy level spectral method elliptic problems Error analysis In spectral methods for numerical PDEs, when the solutions are periodical, the Fourierfunctions may be used. However, when the solutions are non-periodical, the Legendre andChebyshev polynomials are recommended, reported in many papers and books. Thereseems to exist few reports for the study of non-periodical solutions by spectral Fouriermethods under the Dirichlet conditions and other boundary conditions. In this paper, wewill explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM)for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thusfor saving a great deal of the CPU time. The collocation Fourier methods (CFM) canbe regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule.Furthermore, the error bounds are derived for both the CFM and the SFM. When thereexist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is asaccurate as the spectral method using Legendre and Chebyshev polynomials. However,once there exists the truncation errors of the trapezoidal rule, the errors of the ellipticsolutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is themesh length of uniform collocation grids, which are just equivalent to those by the linearelements and the finite difference method (FDM). The O(h^2) and even the superconvergenceO(h4) are found numerically. The traditional condition number of the CFMis O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methodsusing the Legendre and Chebyshev polynomials. Also the effective condition number isonly O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems,to support the analysis made. The simplicity of algorithms and the promising numericalcomputation with O(h^4) may grant the CFM to be competent in application in numericalphysics, chemistry, engineering, etc., see [7]. Tzon-Tzer Lu - chair Ming-Gong Lee - co-chair Chien-Sen Huang - co-chair Zi-Cai Li - advisor Hung-Tsai Huang - advisor indicate not accessible 2010-08-10

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