A comparison of numerical methods for solving the bratu and bratu-type problems

The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique solution is obtained when A = Ac where Ac = 3.513830719 is the critical value. The First Bratu-Type problem corresponds A = _7[2 while the...

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Main Author:	Md.Kasmani, Ruhaila
Format:	Thesis
Language:	English
Published:	2005
Subjects:	QA Mathematics QA299.6-433 Analysis
Online Access:	http://eprints.uthm.edu.my/7437/1/24p%20RUHAILA%20MD.KASMANI.pdf
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id	my-uthm-ep.7437
record_format	uketd_dc
spelling	my-uthm-ep.74372022-07-24T03:46:09Z A comparison of numerical methods for solving the bratu and bratu-type problems 2005-03 Md.Kasmani, Ruhaila QA Mathematics QA299.6-433 Analysis The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique solution is obtained when A = Ac where Ac = 3.513830719 is the critical value. The First Bratu-Type problem corresponds A = _7[2 while the Second Bratu-Type problem is ul/(x) + 7[ 2e-u(x) = o. The exact solution of the First Bratu-Type problem blows up at x = 0.5 while the Second Bratu-Type problem is continuous. The present work seeks to compare various numerical methods for solving the Bratu and Bratu-Type problems. The numerical methods are the standard Adomian decomposition method, the modified Adomian decomposition method, the shooting method and the finite difference method. These methods are implemented using Maple. Convergence is achieved by applying the four methods when 0 < A ::; 2, however the shooting method is the most effective method as it gives the smallest maximum absolute error. ·When A = Ac, none of these methods give the convergence solutions. Due to the nature of the solution of the First Bratu-Type problem, only the shooting method and the modified Adomian decomposition method can give the convergence values to the exact solution. The finite difference method is proved to be the most effective method for the Second Bratu-Type problem compared to other methods. Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposi�tion method, modified Adomian decomposition method, shooting method, finite difference method. 2005-03 Thesis http://eprints.uthm.edu.my/7437/ http://eprints.uthm.edu.my/7437/1/24p%20RUHAILA%20MD.KASMANI.pdf text en public mphil masters Universiti Teknologi Malaysia Fakulti Sains
institution	Universiti Tun Hussein Onn Malaysia
collection	UTHM Institutional Repository
language	English
topic	QA Mathematics QA299.6-433 Analysis
spellingShingle	QA Mathematics QA299.6-433 Analysis Md.Kasmani, Ruhaila A comparison of numerical methods for solving the bratu and bratu-type problems
description	The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique solution is obtained when A = Ac where Ac = 3.513830719 is the critical value. The First Bratu-Type problem corresponds A = _7[2 while the Second Bratu-Type problem is ul/(x) + 7[ 2e-u(x) = o. The exact solution of the First Bratu-Type problem blows up at x = 0.5 while the Second Bratu-Type problem is continuous. The present work seeks to compare various numerical methods for solving the Bratu and Bratu-Type problems. The numerical methods are the standard Adomian decomposition method, the modified Adomian decomposition method, the shooting method and the finite difference method. These methods are implemented using Maple. Convergence is achieved by applying the four methods when 0 < A ::; 2, however the shooting method is the most effective method as it gives the smallest maximum absolute error. ·When A = Ac, none of these methods give the convergence solutions. Due to the nature of the solution of the First Bratu-Type problem, only the shooting method and the modified Adomian decomposition method can give the convergence values to the exact solution. The finite difference method is proved to be the most effective method for the Second Bratu-Type problem compared to other methods. Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposi�tion method, modified Adomian decomposition method, shooting method, finite difference method.
format	Thesis
qualification_name	Master of Philosophy (M.Phil.)
qualification_level	Master's degree
author	Md.Kasmani, Ruhaila
author_facet	Md.Kasmani, Ruhaila
author_sort	Md.Kasmani, Ruhaila
title	A comparison of numerical methods for solving the bratu and bratu-type problems
title_short	A comparison of numerical methods for solving the bratu and bratu-type problems
title_full	A comparison of numerical methods for solving the bratu and bratu-type problems
title_fullStr	A comparison of numerical methods for solving the bratu and bratu-type problems
title_full_unstemmed	A comparison of numerical methods for solving the bratu and bratu-type problems
title_sort	comparison of numerical methods for solving the bratu and bratu-type problems
granting_institution	Universiti Teknologi Malaysia
granting_department	Fakulti Sains
publishDate	2005
url	http://eprints.uthm.edu.my/7437/1/24p%20RUHAILA%20MD.KASMANI.pdf
_version_	1747831151160983552

A comparison of numerical methods for solving the bratu and bratu-type problems

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