One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations

One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations

Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinar...

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Main Author: Abdelrahim, Ra'ft Abdelmajid Moh'd
Format: Thesis
Language:eng
eng
Published: 2016
Subjects:
QA273-280 Probabilities
Mathematical statistics
QA299.6-433 Analysis
Online Access:https://etd.uum.edu.my/6177/1/s94643_01.pdf
https://etd.uum.edu.my/6177/2/s94643_02.pdf
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id my-uum-etd.6177
record_format uketd_dc
institution Universiti Utara Malaysia
collection UUM ETD
language eng
eng
advisor Omar, Zurni
topic QA273-280 Probabilities
Mathematical statistics
QA299.6-433 Analysis
spellingShingle QA273-280 Probabilities
Mathematical statistics
QA299.6-433 Analysis
Abdelrahim, Ra'ft Abdelmajid Moh'd
One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
description Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally solved by first converting them into their equivalent systems of first order ordinary differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently, hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step hybrid block methods with generalised off-step point(s) for solving higher order ordinary differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential equations directly.
format Thesis
qualification_name Ph.D.
qualification_level Doctorate
author Abdelrahim, Ra'ft Abdelmajid Moh'd
author_facet Abdelrahim, Ra'ft Abdelmajid Moh'd
author_sort Abdelrahim, Ra'ft Abdelmajid Moh'd
title One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
title_short One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
title_full One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
title_fullStr One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
title_full_unstemmed One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
title_sort one step hybrid block methods with generalised off-step points for solving directly higher 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/6177/1/s94643_01.pdf
https://etd.uum.edu.my/6177/2/s94643_02.pdf
_version_ 1747828032407601152
spelling my-uum-etd.61772021-04-04T07:30:06Z One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations 2016 Abdelrahim, Ra'ft Abdelmajid Moh'd Omar, Zurni Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA273-280 Probabilities. Mathematical statistics QA299.6-433 Analysis Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally solved by first converting them into their equivalent systems of first order ordinary differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently, hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step hybrid block methods with generalised off-step point(s) for solving higher order ordinary differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential equations directly. 2016 Thesis https://etd.uum.edu.my/6177/ https://etd.uum.edu.my/6177/1/s94643_01.pdf text eng public https://etd.uum.edu.my/6177/2/s94643_02.pdf text eng public Ph.D. doctoral Universiti Utara Malaysia Adeniyi, R. & Adeyefa, E. (2013). 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