Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations

This thesis focuses on solving first order stiff Ordinary Differential Equations (ODEs) using 2-point and 3-point block methods. The 2-point block method will compute the solutions yn+1 and yn+2 at points xn+1 and xn+2 simultaneously in a block at each step. Thus, the derivations of 2-point block me...

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Main Author:	Mohamad Noor, Nursyazwani
Format:	Thesis
Language:	English
Published:	2018
Subjects:	Differential equations Stiff computation (Differential equations)
Online Access:	http://psasir.upm.edu.my/id/eprint/83301/1/FS%202019%2066%20ir.pdf
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spelling	my-upm-ir.833012022-01-07T01:43:41Z Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations 2018-12 Mohamad Noor, Nursyazwani This thesis focuses on solving first order stiff Ordinary Differential Equations (ODEs) using 2-point and 3-point block methods. The 2-point block method will compute the solutions yn+1 and yn+2 at points xn+1 and xn+2 simultaneously in a block at each step. Thus, the derivations of 2-point block methods of third and fifth order are presented. Order and error constant of the methods are determined. Newton’s Method is used to implement in the 2-point block methods. The numerical results for each method are presented and compared with the existing methods. Furthermore, the stability properties of all 2-point block methods are analysed to ensure that the methods are A(a)-stable. Hence, its suitable for solving stiff problems. Convergence characteristics of the methods are also investigated. The 2-point block method with fifth order is then extended to 3-point block method with same order. Advantage of the 3-point block method is the solutions will be approximated at three points concurrently which are xn+1, xn+2 and xn+3. Thus, the derivation of 3-point block method using Taylor’s series expansion is presented. Order and error constant of the method are verified. Stability and convergence properties of the method are investigated by determining the zero-stable, stability region, A(a)-stable and consistency. The 3-point block method is implemented by using Newton’s iteration to measure its efficiency. Numerical results of the method are presented and performance of the method are compared with the existing methods. An application problem of SIR model is solved by using the proposed methods. The numerical results are presented in tables for s, i and r groups for each method. Based on the analysis, the proposed methods can be an alternative solver for solving the application problem. Differential equations Stiff computation (Differential equations) 2018-12 Thesis http://psasir.upm.edu.my/id/eprint/83301/ http://psasir.upm.edu.my/id/eprint/83301/1/FS%202019%2066%20ir.pdf text en public masters Universiti Putra Malaysia Differential equations Stiff computation (Differential equations) Ibrahim, Zarina Bibi
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
advisor	Ibrahim, Zarina Bibi
topic	Differential equations Stiff computation (Differential equations)
spellingShingle	Differential equations Stiff computation (Differential equations) Mohamad Noor, Nursyazwani Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
description	This thesis focuses on solving first order stiff Ordinary Differential Equations (ODEs) using 2-point and 3-point block methods. The 2-point block method will compute the solutions yn+1 and yn+2 at points xn+1 and xn+2 simultaneously in a block at each step. Thus, the derivations of 2-point block methods of third and fifth order are presented. Order and error constant of the methods are determined. Newton’s Method is used to implement in the 2-point block methods. The numerical results for each method are presented and compared with the existing methods. Furthermore, the stability properties of all 2-point block methods are analysed to ensure that the methods are A(a)-stable. Hence, its suitable for solving stiff problems. Convergence characteristics of the methods are also investigated. The 2-point block method with fifth order is then extended to 3-point block method with same order. Advantage of the 3-point block method is the solutions will be approximated at three points concurrently which are xn+1, xn+2 and xn+3. Thus, the derivation of 3-point block method using Taylor’s series expansion is presented. Order and error constant of the method are verified. Stability and convergence properties of the method are investigated by determining the zero-stable, stability region, A(a)-stable and consistency. The 3-point block method is implemented by using Newton’s iteration to measure its efficiency. Numerical results of the method are presented and performance of the method are compared with the existing methods. An application problem of SIR model is solved by using the proposed methods. The numerical results are presented in tables for s, i and r groups for each method. Based on the analysis, the proposed methods can be an alternative solver for solving the application problem.
format	Thesis
qualification_level	Master's degree
author	Mohamad Noor, Nursyazwani
author_facet	Mohamad Noor, Nursyazwani
author_sort	Mohamad Noor, Nursyazwani
title	Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
title_short	Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
title_full	Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
title_fullStr	Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
title_full_unstemmed	Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
title_sort	extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
granting_institution	Universiti Putra Malaysia
publishDate	2018
url	http://psasir.upm.edu.my/id/eprint/83301/1/FS%202019%2066%20ir.pdf
_version_	1747813376197656576

Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations

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