Quartic-spline collocation methods for fourth-order two-point boundary value problems.

摘要

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 and the collocation discretization methodology with the midpoints of a uniform partition being the collocation points. The standard quartic-spline collocation method is second order. Two sixth-order quartic-spline collocation methods are developed and analyzed. They are both based on a high order perturbation of the differential equation and boundary conditions operators. 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, and fifth 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. Numerical results verify the orders of convergence predicted by analysis.