UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Numerical solution of boundary value problems in ordinary differential equations Usmani, Riaz Ahmad

Abstract

In the numerical solution of the two point "boundary value problem, [ equation omitted ] (1) the usual method is to approximate the problem by a finite difference analogue of the form [ equation omitted ] (2) with k = 2, and the truncation error T.E. = O(h⁴) or O(h⁶), where h is the step-size. Varga (1962) has obtained error bounds for the former when the problem (1) is linear and of class M . In this thesis, more accurate finite difference methods are considered. These can be obtained in essentially two different ways, either by increasing the value k in difference equations (2), or by introducing higher order derivatives. Several methods of both types have been derived. Also, it is shown how the initial value problem y' = ϕ(x,y) can be formulated as a two point boundary value problem and solved using the latter approach. Error bounds have been derived for all of these methods for linear problems of class M . In particular, more accurate bounds have been derived than those obtained by Varga (1962) and Aziz and Hubbard (1964). Some error estimates are suggested for the case where [ equation omitted ], but these are not accurate bounds, especially when [ equation omitted ] not a constant. In the case of non-linear differential equations, sufficient conditions are derived for the convergence of the solution of the system of equations (2) by a generalized Newton's method. Some numerical results are included and the observed errors compared with theoretical error bounds.

Item Media

Item Citations and Data

License

For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

Usage Statistics