Abstract. The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving (nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.
T e r r y L . He,rdmant V i r g i n i a P o l y t e c h n i c I n s t i t u t e a n d S t a t e U n i v e r s i t y Blacksburg, VA 24061 A complete dynamic model i s f o r m u l a t e d f o r a system i n w h i c h t h e e l a s t i c m o t i o n s o f a s t r u c t u r e a r e c o u p l e d w i t h t h e m o t i o n s o f t h e s u r r o u n d i n g f l u i d . W h i l e c e r t a i n a s p e c t s o f t h e p r o b l e m a r e w e l l -s t u d i e d , the emphasis here is on development of a well-posed s t a t e -s p a c e f o r m u l a t i o n . Such models have proven conc e p t u a l a n d c o m p u t a t i o n a l v a l u e i n p r o b l e m s o f o p t i m a l c o n t r o l a n d p a r a m e t e r i d e n t i f i c a t i o n . I. I n t r o d u c t i o n A c t i v e f l u t t e r c o n t r o l i s a p o t e n t i a l l y i m p o r t a n t technology i n the development and design of fuele f f i c i e n t a n d / o r h i g h l y m a n e u v e r a b l e a i r c r a f t . The c o n c e p t u a l f o r m u l a t i o n o f c o n t r o l a n d i d e n t i f i c a t i o n p r o b l e m s , t h e i r n u m e r i c a l s o l u t i o n , a s w e l l a s t h e e v a l u a t i o n o f p o t e n t i a l d e s i g n s , a l l b e n e f i t f r o m m a t h e m a t i c a l m o d e l s t h a t f a i t h f u l l y p r e d i c t t h e d y n a m i c b e h a v i o r o f t h e a e r o e l a s t i c s y s t e m . I n p a r t i c u l a r , t h e h i g h f r e q u e n c y o s c i l l a t i o n s e n c o u n t e r e d i n some a p p l ic a t i o n s r e q u i r e a p p r o p r i a t e m o d e l i n g o f u n s t e a d y a e r odynamics. Whereas t h e a c t u a l f l o w may be described as three-dimensional, unsteady, viscous and compressible [3], i t seems t h a t u s e f u l m a t h e m a t i c a l m o d e l s o f s u c h s y s t e m s a r e b e y o n d c u r r e n t c a p a b i l i t i e s . H o w e v e r , as a f i r s t s t e p i n t h e d e v e l o p m e n t o f t h e s e c o m p l e x models one should investigate models for somewhat simpler systems. I n t h i s p a p e r we develop a s t a t e space model f o r m o t i o n o f a n a i r f o i l i n t w o -d i m e n s i o n a l , u n s t e a d y f l o w o f a n i n v i s c i d , i n c o m p r e s s i b l e f l u i d [3,6]. This problem has been addressed by Balakrishnan and Edwards [1,2,5]. Our model i s s i m i l a r t o t h a t proposed by Balakrishnan [l]; h o w e v e r , t h e r e a l i z a t i o n we d e s c r i b e d i f f e r s i n a t l e a s t t w o i m p o r t a n t a s p e c t s . The " s t a t e " i n o u r model i s d i r e c t l y r e l a t e d t o c e r t a i n p h y s i c a l q u a n t i t i e s ( i .e. p i t c h , p l u n g e , c i r c u l a t i o n ) a n d s e c o n d l y , t h e i n d i c i a 1 model considered i n [l] may b e r e c o n s t r u c t e d b y t h e p r o p e r c h o i c e o f i n i t i a l c o n d it i o n s . I n S e c t i o n 2 we summarize the basic equations of incompressible, two-dimensional flow and derive an e v o l u t i o n e q u a t ...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.