Numerical Solution of Two Point Boundary Value Problems

Abstract

AcknowledgementsThe a u t h o r wishes t o thank P r o f e s s o r Herbert B, K e l l e r f o r h i s p a t i e n t guidence and i n v a l u a b l e a s s i s t a n c e i n t h e p r e p a r a t i o n of t h i s t h e s i s . H i s encouragement and enthusiasm were e s s e n t i a l t o t h e completion of t h i s work.The o p p o r t u n i t y t o c a r r y o u t t h i s r e s e a r c h was provided by C a l i f o r n i a I n s t i t u t e of Technology Graduate Teaching A s s i s t a n t s h i p s f o r which t h e a u t h o r i s g r a t e f u l . H e a l s o wishes t o e x p r e s s h i s a p p r e c i a t i o n t o t h e e n t i r e Department of Applied Mathematics, both f a c u l t y and s t u d e n t s , f o r providing an i n v i g o r a t i n g c l i m a t e i n which t o pursue t h i s work.The a u t h o r wishes t o thank M r s . J a n e t Smith and Mrs. Mary Beth Briggs f o r t h e i r e x p e r t typing of a d e t a i l e d and t e d i o u s manuscript. ABSTRACTThe approximation of two-point boundary-value problenls by g e n e r a l f i n i t e d i f f e r e n c e schemes i s t r e a t e d . A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e s t a b i l i t y of t h e l i n e a r d i s c r e t e boundary-value problem i s derived i n terms of t h e a s s o c i a t e d d i s c r e t e i n i t i a l -v a l u e problem. P a r a l l e l s h o o t i n g methods a r e shown t o be e q u i v a l e n t t o t h e d i s c r e t e boundary-value problem. One-step d i f f e r e n c e schemes a r e considered i n d e t a i l and a c l a s s of computationally e f f i c i e n t schemes of a r b i t r a r i l y h i g h o r d e r of accuracy i s e x h i b i t e d . S u f f i c i e n t c o n d i t i o n s a r e found t o i n s u r et h e convergence of d i s c r e t e f i n i t e d i f f e r e n c e approximations t o n o n l i n e a r boundary-value problems w i t h i s o l a t e d s o l u t i o n s . ~e w t o n ' s method i s considered a s a procedure f o r s o l v i n g t h e r e s u l t i n g n o n l i n e a r a l g e b r a i c equations. A new, e f f i c i e n t f a c t o r i z a t i o n scheme f o r block t r i d i a g o n a l m a t r i c e s i s d e r i v e d . The t h e o r y