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Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations

Many problems in computing, sciences, and engineering’s applications represented by the initial value problems (IVPs) and boundary value problems (BVPs) of ordinary differential equations (ODEs) cannot be solved explicitly. Analytical approximation methods in the form of polynomial or piecewise poly...

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Main Author: Bhatti, Abdul Hadi
Format: Thesis
Language:English
English
English
Published: 2024
Subjects:
QA Mathematics
Online Access:https://etd.uum.edu.my/11484/1/permission%20to%20deposit-embargo%2036%20months-s902916.pdf
https://etd.uum.edu.my/11484/2/s902916_01.pdf
https://etd.uum.edu.my/11484/3/s902916_02.pdf
https://etd.uum.edu.my/11484/
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spelling my.uum.etd.114842025-01-05T07:23:48Z https://etd.uum.edu.my/11484/ Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations Bhatti, Abdul Hadi QA Mathematics Many problems in computing, sciences, and engineering’s applications represented by the initial value problems (IVPs) and boundary value problems (BVPs) of ordinary differential equations (ODEs) cannot be solved explicitly. Analytical approximation methods in the form of polynomial or piecewise polynomial functions have been developed to address such problems. Among existing studies, Bézier curve and Ball curve consisting of the same geometrical structure are commonly used to approximate analytical solutions. However, Bézier curve required higher degree to find their best control points compared to Ball curve, which increased computational burden, thus affecting the accuracy. Moreover, the least squares method (LSM) is incorporated within the Bézier curve algorithm to minimize errors during the determination of control points. Therefore, by taking advantages of this strength, this study intends to incorporate Ball curve with LSM. The main objective of the study is to develop approximation methods base on the Ball curve using LSMfor both linear and nonlinear IVPs and BVPs of ODEs. The best control points of Ball curve are determined by minimising the sum of the squared residuals of the control functions. Following this, the convergence analysis of the proposed methods is conducted theoretically and numerically. Three Ball curves with LSM have been developed to solve eleven (11) numerical examples of various order of ODEs: DP Ball curves (DPBC), Said Ball curves (SBC), and Wang Ball curves (WBC). Numerical results show that all three methods performed better compared to the existing methods in terms of accuracy, with SBC as the best among the three. Furthermore, uniqueness and existence theorem are also proven for each method to verify their corresponding numerical result. In conclusion, Ball curves with LSM provided better accuracy for solving IVPs and BVPs of ODEs, with potential application in animation, computer aided graphic design, fluid dynamic, and network. 2024 Thesis NonPeerReviewed text en https://etd.uum.edu.my/11484/1/permission%20to%20deposit-embargo%2036%20months-s902916.pdf text en https://etd.uum.edu.my/11484/2/s902916_01.pdf text en https://etd.uum.edu.my/11484/3/s902916_02.pdf Bhatti, Abdul Hadi (2024) Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations. Doctoral thesis, Universiti Utara Malaysia.
institution Universiti Utara Malaysia
building UUM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Utara Malaysia
content_source UUM Electronic Theses
url_provider http://etd.uum.edu.my/
language English
English
English
topic QA Mathematics
spellingShingle QA Mathematics
Bhatti, Abdul Hadi
Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
description Many problems in computing, sciences, and engineering’s applications represented by the initial value problems (IVPs) and boundary value problems (BVPs) of ordinary differential equations (ODEs) cannot be solved explicitly. Analytical approximation methods in the form of polynomial or piecewise polynomial functions have been developed to address such problems. Among existing studies, Bézier curve and Ball curve consisting of the same geometrical structure are commonly used to approximate analytical solutions. However, Bézier curve required higher degree to find their best control points compared to Ball curve, which increased computational burden, thus affecting the accuracy. Moreover, the least squares method (LSM) is incorporated within the Bézier curve algorithm to minimize errors during the determination of control points. Therefore, by taking advantages of this strength, this study intends to incorporate Ball curve with LSM. The main objective of the study is to develop approximation methods base on the Ball curve using LSMfor both linear and nonlinear IVPs and BVPs of ODEs. The best control points of Ball curve are determined by minimising the sum of the squared residuals of the control functions. Following this, the convergence analysis of the proposed methods is conducted theoretically and numerically. Three Ball curves with LSM have been developed to solve eleven (11) numerical examples of various order of ODEs: DP Ball curves (DPBC), Said Ball curves (SBC), and Wang Ball curves (WBC). Numerical results show that all three methods performed better compared to the existing methods in terms of accuracy, with SBC as the best among the three. Furthermore, uniqueness and existence theorem are also proven for each method to verify their corresponding numerical result. In conclusion, Ball curves with LSM provided better accuracy for solving IVPs and BVPs of ODEs, with potential application in animation, computer aided graphic design, fluid dynamic, and network.
format Thesis
author Bhatti, Abdul Hadi
author_facet Bhatti, Abdul Hadi
author_sort Bhatti, Abdul Hadi
title Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
title_short Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
title_full Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
title_fullStr Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
title_full_unstemmed Analysis of least squares methods based on Ball curves representation in solving ordinary differential equations
title_sort analysis of least squares methods based on ball curves representation in solving ordinary differential equations
publishDate 2024
url https://etd.uum.edu.my/11484/1/permission%20to%20deposit-embargo%2036%20months-s902916.pdf
https://etd.uum.edu.my/11484/2/s902916_01.pdf
https://etd.uum.edu.my/11484/3/s902916_02.pdf
https://etd.uum.edu.my/11484/
_version_ 1821005243947679744
score 13.244413

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