UBC Theses and Dissertations
Numerical solution of boundary value problems in ordinary differential equations Usmani, Riaz Ahmad
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.
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