Thesis and Dissertationhttp://dspace.napata.edu.sd/handle/123456789/2612024-10-14T19:47:02Z2024-10-14T19:47:02Z11COLLOCATION ALGORITHMS AND ERROR ANALYSIS FOR APPROXIMATE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONSAwad Hagali Ahmedhttp://dspace.napata.edu.sd/handle/123456789/2622023-01-22T10:48:43Z1981-01-01T00:00:00Zdc.title: COLLOCATION ALGORITHMS AND ERROR ANALYSIS FOR APPROXIMATE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
dc.contributor.author: Awad Hagali Ahmed
dc.description.abstract: This thesis is mainly concerned with an error analysis of
numerical methods for two point boundary value problems. in particular
for the method of collocation using polynomial and certain piecewise
polynomial bases.
As in previous work on strict error bounds an operator
theoretical approach is taken. The setting for the theory and the
principal results for later use are firstly considered. Then two types
of 'a posteriori' error bounds are developed. These"bounds are made
computable by relating the inverse of the approximating operator to the
inverse of certain matrices formed in the actual application of the
approximation method.
The application of this theory to the numerical solution of
linear two point boundary value problems is then considered. It is
demonstrated how the differential equation can be split to fit into the
setting required by the theory. It is also demonstrated how the global
and the piecewise collocation method can be expressed in terms of a
projection method applied to the operator equation. The conditions
required by the theory are expressed in terms of continuity requirements
on the coefficients of the differential equation and in terms of the
distribution of the collocation points. In examining these bounds on
a variety of problems. it is noticed that with some problems the
conditions for applicability may not hold except for more points than one
actually required to obtain a satisfactory solution. To improve the
applicability. the theory is reconsidered with a different splitting
of the differential equation. The method of collocation is expressed
accordingly in terms of a new projection operator which is proved to
have some nice properties in practice. This new approach is then compared
with the original one and it is shown to be superior on various problems.
By examining the inverse differential operator and the residual
improved error bounds and estimates are shown to be obtainable. These
estimates are tested in a large variety of examples and some graphs
are presented to describe their behaviour in more detail. Finally
these estimates are used to develop various adaptive mesh selection
algorithms for solving two point boundary value problems. These
strategies are tested and compared in several representative examples
and some conclusions are drawn.
The thesis concludes with a brief review of the work with an
indication of possible improvements and extensions.
1981-01-01T00:00:00Z