Solving ordinary differential equations using block multistep method

An eficient code based on block multistep method is developed for solving first and higher order initial value problems (IVPs) of ordinary differential equations (ODEs) using variable step size strategy. Unlike previous methods, the Gauss-Seidel approach is used for the implementation of the propose...

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Main Author:	Mehrkanoon, Siamak
Format:	Thesis
Language:	English English
Published:	2011
Subjects:	Initial value problems Differential equations - Numerical solutions Numerical analysis
Online Access:	http://psasir.upm.edu.my/id/eprint/27697/1/FS%202011%2098R.pdf
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id	my-upm-ir.27697
record_format	uketd_dc
spelling	my-upm-ir.276972014-04-10T06:40:28Z Solving ordinary differential equations using block multistep method 2011-03 Mehrkanoon, Siamak An eficient code based on block multistep method is developed for solving first and higher order initial value problems (IVPs) of ordinary differential equations (ODEs) using variable step size strategy. Unlike previous methods, the Gauss-Seidel approach is used for the implementation of the proposed method instead of Jacobi iteration. Therefore by applying the current technique, the desired order of accuracy can be achieved with less number of function evaluation, since the latest available information is used to advance the numerical solution through a given step size. The higher order IVPs are solved directly without reducing it into the first order IVPs. In addition, the stability of the proposed methods are discussed. Furthermore, the parallel version of the proposed methods are constructed in order to solve large system of first and second order of IVPs. The parallelism across the system is considered for the implementation of the parallel block methods. Less computational time is required when the parallel versions are applied compared to sequential methods. The parallel implementation is supported by Message Passing Interface (MPI). Both sequential and parallel algorithms were carried out on Sunfire V1280 with eight homogeneous processors located at Institute of Mathematical Research (INSPEM), University Putra Malaysia. Subsequently, a variable block method is proposed to solve first and higher order IVPs of ODEs using variable step size and order strategy. In previous researches, in order to increase the order of the method, it supposed to use the information of more back points. In the proposed technique, attempts have been made to promote the order of the method via involving the advanced points. By utilizing the present code we are able to not only increase the order of the method but also to reach the end of the given interval faster. In conclusion, the developed new codes deserve to be used for solving system of first and higher order IVPs of ODEs. Initial value problems Differential equations - Numerical solutions Numerical analysis 2011-03 Thesis http://psasir.upm.edu.my/id/eprint/27697/ http://psasir.upm.edu.my/id/eprint/27697/1/FS%202011%2098R.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Initial value problems Differential equations - Numerical solutions Numerical analysis Faculty of Science English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Initial value problems Differential equations - Numerical solutions Numerical analysis
spellingShingle	Initial value problems Differential equations - Numerical solutions Numerical analysis Mehrkanoon, Siamak Solving ordinary differential equations using block multistep method
description	An eficient code based on block multistep method is developed for solving first and higher order initial value problems (IVPs) of ordinary differential equations (ODEs) using variable step size strategy. Unlike previous methods, the Gauss-Seidel approach is used for the implementation of the proposed method instead of Jacobi iteration. Therefore by applying the current technique, the desired order of accuracy can be achieved with less number of function evaluation, since the latest available information is used to advance the numerical solution through a given step size. The higher order IVPs are solved directly without reducing it into the first order IVPs. In addition, the stability of the proposed methods are discussed. Furthermore, the parallel version of the proposed methods are constructed in order to solve large system of first and second order of IVPs. The parallelism across the system is considered for the implementation of the parallel block methods. Less computational time is required when the parallel versions are applied compared to sequential methods. The parallel implementation is supported by Message Passing Interface (MPI). Both sequential and parallel algorithms were carried out on Sunfire V1280 with eight homogeneous processors located at Institute of Mathematical Research (INSPEM), University Putra Malaysia. Subsequently, a variable block method is proposed to solve first and higher order IVPs of ODEs using variable step size and order strategy. In previous researches, in order to increase the order of the method, it supposed to use the information of more back points. In the proposed technique, attempts have been made to promote the order of the method via involving the advanced points. By utilizing the present code we are able to not only increase the order of the method but also to reach the end of the given interval faster. In conclusion, the developed new codes deserve to be used for solving system of first and higher order IVPs of ODEs.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Mehrkanoon, Siamak
author_facet	Mehrkanoon, Siamak
author_sort	Mehrkanoon, Siamak
title	Solving ordinary differential equations using block multistep method
title_short	Solving ordinary differential equations using block multistep method
title_full	Solving ordinary differential equations using block multistep method
title_fullStr	Solving ordinary differential equations using block multistep method
title_full_unstemmed	Solving ordinary differential equations using block multistep method
title_sort	solving ordinary differential equations using block multistep method
granting_institution	Universiti Putra Malaysia
granting_department	Faculty of Science
publishDate	2011
url	http://psasir.upm.edu.my/id/eprint/27697/1/FS%202011%2098R.pdf
_version_	1747811595340218368

Solving ordinary differential equations using block multistep method

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