The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: FDTD; Linear interpolation; MEMS; NEMS; Photonic crystals; Wavelength conversion; Frequency conversion; Doppler. Includes bibliographical references (leaves 107-114).

Stuttgart, Univ., Diss., 2006.

Thesis (M. S.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Russell Mersereau; Committee Member: Anthony J. Yezzi; Committee Member: Yucel Altunbasak.

Thesis (MSc)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.

Thesis (M.S.)--California Polytechnic State University, 2008. / Major professor: Zoë Wood, Ph.D. "Presented to the faculty of California Polytechnic State University, San Luis Obispo." "In partial fulfillment of the requirements for the degree [of] Master of Science in Computer Science." "June 2008." Includes bibliographical references (leaves 44-45). Also available online. Also available on microfiche (1 sheet).

Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Chin-Hui Lee; Committee Member: Anthony Yezzi; Committee Member: Evans Harrell; Committee Member: Fred Juang; Committee Member: James H. McClellan. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Stuttgart, Univ., Diss., 2000.

Mannheim, Univ., Diss., 2005.

Zugl.: München, Techn. Univ., Diss., 2009