Quartic-spline Collocation Methods for Fourth-order Two-point Boundary Value Problems Abstract Quartic-spline Collocation Methods for Fourth-order Two-point Boundary Value Problems

Quartic-Spline Collocation Methods for Fourth-Order Two-Point Boundary Value Problems Ying Zhu Master of Science Graduate Department of Computer Science University of Toronto 2001 This thesis presents numerical methods for the solution of general linear fourth-order boundary value problems in one dimension. The methods are based on quartic splines, that is, piecewise quartic polynomials with C3 continuity, and the collocation discretization methodology with the midpoints of a uniform partition being the collocation points. The standard quartic-spline collocation method is shown to be second order. Two sixthorder quartic-spline collocation methods are developed and analyzed. They are both based on a high order perturbation of the di erential equation and boundary conditions operators. The one-step method forces the approximation to satisfy a perturbed problem, while the three-step method proceeds in three steps and perturbs the right sides of the equations appropriately. The error analysis follows the Green's function approach and shows that both methods exhibit optimal order of convergence, that is, they are locally sixth order on the gridpoints and midpoints of the uniform partition, and fth order globally. The properties of the matrices arising from a restricted class of problems are studied. Analytic formulae for the eigenvalues and eigenvectors are developed, and related to those arising from quadratic-spline collocation matrices. Numerical results verify the orders of convergence predicted by the analysis. ii Dedication To my parents, Luo Ji Dai and Zhu Guo Ping, who are also my best friends.