Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations

This thesis describes the development of sequential and parallel methods for solving delay differential equations. A new sequential code for the numerical solution of delay differential equations is considered. The variable order variable stepsize formulae based on the Adams-Bashforth-Moulton met...

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Main Author:	Ishak, Fuziyah
Format:	Thesis
Language:	English English
Published:	2009
Subjects:	Mathematical analysis Mathematical statistic
Online Access:	http://psasir.upm.edu.my/id/eprint/7556/1/ABS_----__IPM_2009_9.pdf
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spelling	my-upm-ir.75562013-05-27T07:35:35Z Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations 2009 Ishak, Fuziyah This thesis describes the development of sequential and parallel methods for solving delay differential equations. A new sequential code for the numerical solution of delay differential equations is considered. The variable order variable stepsize formulae based on the Adams-Bashforth-Moulton methods are represented in divided difference form. Derivative discontinuities are detected by local error estimate at the grid points. Large magnitude of the local error estimate indicates the presence of derivative discontinuity. Stepsize is then reduced and eventually the discontinuity point is included in the grid. The formulae representation proves to be efficient when compared with the existing method in modified divided difference form. We also consider the development of two-point block methods on sequential and parallel computers. Formulae for three two-point block methods for solving delay differential equations are derived. The implicit block methods are implemented using variable stepsize variable order technique. The formulae for two-point diagonally and triangu larly implicit block methods using predictor-corrector application are represented in divided difference form. Meanwhile, the predictor-corrector formulae for two-point fully implicit block method are calculated beforehand and stored at the beginning of the code. All of the block methods rely on the local error estimates to detect derivative discontinuities. In all of the developed methods, regions of absolute stability are presented and compared. Comparison among the developed methods indicates that all of the methods achieve the desired accuracy. Block methods are efficient when compared with the sequential non-block method as the total steps taken can be reduced. The new block methods are then used for the parallel implementation in solving large system of delay differential equations. The parallel programs using Message Passing Interface are run on Sun Fire V1280 using two processors. Numerical results indicate that parallel implementation increases the performance of the block methods. Mathematical analysis Mathematical statistic 2009 Thesis http://psasir.upm.edu.my/id/eprint/7556/ http://psasir.upm.edu.my/id/eprint/7556/1/ABS_----__IPM_2009_9.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Mathematical analysis Mathematical statistic Institute : Institute for Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Mathematical analysis Mathematical statistic
spellingShingle	Mathematical analysis Mathematical statistic Ishak, Fuziyah Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
description	This thesis describes the development of sequential and parallel methods for solving delay differential equations. A new sequential code for the numerical solution of delay differential equations is considered. The variable order variable stepsize formulae based on the Adams-Bashforth-Moulton methods are represented in divided difference form. Derivative discontinuities are detected by local error estimate at the grid points. Large magnitude of the local error estimate indicates the presence of derivative discontinuity. Stepsize is then reduced and eventually the discontinuity point is included in the grid. The formulae representation proves to be efficient when compared with the existing method in modified divided difference form. We also consider the development of two-point block methods on sequential and parallel computers. Formulae for three two-point block methods for solving delay differential equations are derived. The implicit block methods are implemented using variable stepsize variable order technique. The formulae for two-point diagonally and triangu larly implicit block methods using predictor-corrector application are represented in divided difference form. Meanwhile, the predictor-corrector formulae for two-point fully implicit block method are calculated beforehand and stored at the beginning of the code. All of the block methods rely on the local error estimates to detect derivative discontinuities. In all of the developed methods, regions of absolute stability are presented and compared. Comparison among the developed methods indicates that all of the methods achieve the desired accuracy. Block methods are efficient when compared with the sequential non-block method as the total steps taken can be reduced. The new block methods are then used for the parallel implementation in solving large system of delay differential equations. The parallel programs using Message Passing Interface are run on Sun Fire V1280 using two processors. Numerical results indicate that parallel implementation increases the performance of the block methods.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Ishak, Fuziyah
author_facet	Ishak, Fuziyah
author_sort	Ishak, Fuziyah
title	Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
title_short	Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
title_full	Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
title_fullStr	Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
title_full_unstemmed	Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations
title_sort	sequential and parallel methods for numerical solutions of delay differential equations
granting_institution	Universiti Putra Malaysia
granting_department	Institute : Institute for Mathematical Research
publishDate	2009
url	http://psasir.upm.edu.my/id/eprint/7556/1/ABS_----__IPM_2009_9.pdf
_version_	1747810700117409792

Sequential and Parallel Methods for Numerical Solutions of Delay Differential Equations

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