Finite-difference solutions of tenth-order boundary-value problems

Abstract

In this thesis finite difference methods are used to obtain numerical solutions for a class of high-order ordinary differential equations with applications to eigenvalue problems. Two families of numerical methods are developed for tenth-order boundary-value problems and global extrapolations on two and three grids are considered for the special problem. Special nonlinear tenth-order boundary-value problems are solved using a family of direct finite difference methods which are adapted to solve a general linear and nonlinear boundary-value problem. These methods convert the ordinary differential equation into a set of algebraic equations. If the original ordinary differential equations are linear, the finite difference equations will give linear algebraic equations. If the ordinary differential equation are nonlinear, the resulting finite difference equations will be nonlinear algebraic equations. These nonlinear equations are first linearized by Newton's method. The methods developed are of orders two, four, six, eight, ten and twelve. The error analyses are discussed. A generalized form is given to solve a class of high-order boundary-value problems by converting the differential equation to a system of first-order equations. The method based on using a Pade rational approximant to the exponential function for general boundary-value problems is applied to a tenth-order eigenvalue problem associated with instability in a Benard layer and numerical results are compared with asymtotic estimates appearing in the literature. This method may be implernented on a parallel computer. The method is extended to a twelfth-order eigenvalue problern in an appendix. The algorithms developed are tested on a variety of problems from the literature. The REDUCE package is used to obtain the parameters in the numerical methods and all computations are carried out on a Sun Workstation at Brunel University using Fortran 77 with double precision arithmetic.