New spline methods for solving first and second order ordinary differential equations

Many problems arise from various real life applications may lead to mathematical models which can be expressed as initial value problems (IVPs) and boundary value problems (BVPs) of first and second ordinary differential equations (ODEs).These problems might not have analytical solutions, thus nume...

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Main Author:	Ala'yed, Osama Hasan Salman
Format:	Thesis
Language:	eng eng
Published:	2016
Subjects:	QA Mathematics
Online Access:	https://etd.uum.edu.my/6037/1/s95069_01.pdf https://etd.uum.edu.my/6037/2/s95069_02.pdf
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id	my-uum-etd.6037
record_format	uketd_dc
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng
advisor	Teh, Yuan Ying Saaban, Azizan
topic	QA Mathematics
spellingShingle	QA Mathematics Ala'yed, Osama Hasan Salman New spline methods for solving first and second order ordinary differential equations
description	Many problems arise from various real life applications may lead to mathematical models which can be expressed as initial value problems (IVPs) and boundary value problems (BVPs) of first and second ordinary differential equations (ODEs).These problems might not have analytical solutions, thus numerical methods are needed inapproximating the solutions. When a differential equation is solved numerically, the interval of integration is divided into subintervals.Consequently, numerical solutions at the grid pointscan be determined through numerical computations, whereas the solutions between the grid points still remain unknown. In order to find the approximate solutions between any two grid points, spline methods are introduced. However, most of the existing spline methods are used to approximate the solutions of specific cases of IVPs and BVPs. Therefore, this study develops new spline methods based on polynomial and non-polynomial spline functions for solving general cases of first and second order IVPs and BVPs. The convergence analysis for each new spline method is also discussed. In terms of implementation, the 4-stage fourth order explicit Runge-Kutta method is employed to obtain the solutions at the grid points, while the new spline methods are used to obtain the solutions between the grid points. The performance of the new spline methods are then compared with the existing spline methods in solving12 test problems. Generally, the numerical results indicate that the new spline methods provide better accuracy than their counterparts. Hence, the new spline methods are viable alternatives for solving first and second order IVPs and BVPs.
format	Thesis
qualification_name	Ph.D.
qualification_level	Doctorate
author	Ala'yed, Osama Hasan Salman
author_facet	Ala'yed, Osama Hasan Salman
author_sort	Ala'yed, Osama Hasan Salman
title	New spline methods for solving first and second order ordinary differential equations
title_short	New spline methods for solving first and second order ordinary differential equations
title_full	New spline methods for solving first and second order ordinary differential equations
title_fullStr	New spline methods for solving first and second order ordinary differential equations
title_full_unstemmed	New spline methods for solving first and second order ordinary differential equations
title_sort	new spline methods for solving first and second order ordinary differential equations
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2016
url	https://etd.uum.edu.my/6037/1/s95069_01.pdf https://etd.uum.edu.my/6037/2/s95069_02.pdf
_version_	1747828012161695744
spelling	my-uum-etd.60372021-04-05T01:51:05Z New spline methods for solving first and second order ordinary differential equations 2016 Ala'yed, Osama Hasan Salman Teh, Yuan Ying Saaban, Azizan Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA Mathematics Many problems arise from various real life applications may lead to mathematical models which can be expressed as initial value problems (IVPs) and boundary value problems (BVPs) of first and second ordinary differential equations (ODEs).These problems might not have analytical solutions, thus numerical methods are needed inapproximating the solutions. When a differential equation is solved numerically, the interval of integration is divided into subintervals.Consequently, numerical solutions at the grid pointscan be determined through numerical computations, whereas the solutions between the grid points still remain unknown. In order to find the approximate solutions between any two grid points, spline methods are introduced. However, most of the existing spline methods are used to approximate the solutions of specific cases of IVPs and BVPs. Therefore, this study develops new spline methods based on polynomial and non-polynomial spline functions for solving general cases of first and second order IVPs and BVPs. The convergence analysis for each new spline method is also discussed. In terms of implementation, the 4-stage fourth order explicit Runge-Kutta method is employed to obtain the solutions at the grid points, while the new spline methods are used to obtain the solutions between the grid points. The performance of the new spline methods are then compared with the existing spline methods in solving12 test problems. Generally, the numerical results indicate that the new spline methods provide better accuracy than their counterparts. 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New spline methods for solving first and second order ordinary differential equations

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