Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations

The focus of this thesis is on the derivations of Diagonally Implicit Block Backward Differentiation Formulas (DBBDF) of constant step size. The first part of the thesis discusses on the modification of Fully Implicit Block Backward Differentiation Formulas (FBBDF) to solve first order fuzzy differe...

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Main Author:	Ismail, Nurzeehan
Format:	Thesis
Language:	English
Published:	2014
Subjects:	Differential equations - Numerical solutions Fuzzy sets
Online Access:	http://psasir.upm.edu.my/id/eprint/70591/1/FS%202014%2083%20-%20IR.pdf
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spelling	my-upm-ir.705912019-10-30T00:36:43Z Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations 2014-12 Ismail, Nurzeehan The focus of this thesis is on the derivations of Diagonally Implicit Block Backward Differentiation Formulas (DBBDF) of constant step size. The first part of the thesis discusses on the modification of Fully Implicit Block Backward Differentiation Formulas (FBBDF) to solve first order fuzzy differential equations (FDEs). The subsequent part of the thesis focuses on the derivations of Diagonally Implicit Three-point BBDF of order two and three (DBBDF (3, 2) and DBBDF (3, 3)) for solving first order ordinary differential equations (ODEs) and FDEs. The convergence properties for DBBDF methods and the adequate stability regions for the proposed methods are presented to show that the methods are capable of solving stiff ODEs. The derived methods are then implemented using Newton iteration which is normally used since the methods derived are implicit in nature. Numerical results are presented to verify the efficiency of DBBDF methods for ODEs. The derived methods are then compared with Diagonally Implicit Two-point BBDF of order two, three and four (DBBDF (2, 2), DBBDF (2, 3) and DBBDF (2, 4)). The accuracy of the proposed methods outperformed DBBDF (2, 3) and DBBDF (2, 4) as the step size gets smaller while the computational time of the proposed methods are smaller than the existing methods. Since there are very few block methods used to solve fuzzy differential equations, the derived methods are extended to solve first order fuzzy initial value problems (FIVPs). The fuzzification of DBBDF methods is proposed and the convergence of the corresponding methods when applied to FDEs is also proven. Numerical results of DBBDF methods when solving FDEs are provided and compared with the existing methods. On the whole, this study reveals that the FBBDF and DBBDF methods are capable and efficient for solving first order ordinary and fuzzy differential equations. Differential equations - Numerical solutions Fuzzy sets 2014-12 Thesis http://psasir.upm.edu.my/id/eprint/70591/ http://psasir.upm.edu.my/id/eprint/70591/1/FS%202014%2083%20-%20IR.pdf text en public masters Universiti Putra Malaysia Differential equations - Numerical solutions Fuzzy sets
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Differential equations - Numerical solutions Fuzzy sets
spellingShingle	Differential equations - Numerical solutions Fuzzy sets Ismail, Nurzeehan Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
description	The focus of this thesis is on the derivations of Diagonally Implicit Block Backward Differentiation Formulas (DBBDF) of constant step size. The first part of the thesis discusses on the modification of Fully Implicit Block Backward Differentiation Formulas (FBBDF) to solve first order fuzzy differential equations (FDEs). The subsequent part of the thesis focuses on the derivations of Diagonally Implicit Three-point BBDF of order two and three (DBBDF (3, 2) and DBBDF (3, 3)) for solving first order ordinary differential equations (ODEs) and FDEs. The convergence properties for DBBDF methods and the adequate stability regions for the proposed methods are presented to show that the methods are capable of solving stiff ODEs. The derived methods are then implemented using Newton iteration which is normally used since the methods derived are implicit in nature. Numerical results are presented to verify the efficiency of DBBDF methods for ODEs. The derived methods are then compared with Diagonally Implicit Two-point BBDF of order two, three and four (DBBDF (2, 2), DBBDF (2, 3) and DBBDF (2, 4)). The accuracy of the proposed methods outperformed DBBDF (2, 3) and DBBDF (2, 4) as the step size gets smaller while the computational time of the proposed methods are smaller than the existing methods. Since there are very few block methods used to solve fuzzy differential equations, the derived methods are extended to solve first order fuzzy initial value problems (FIVPs). The fuzzification of DBBDF methods is proposed and the convergence of the corresponding methods when applied to FDEs is also proven. Numerical results of DBBDF methods when solving FDEs are provided and compared with the existing methods. On the whole, this study reveals that the FBBDF and DBBDF methods are capable and efficient for solving first order ordinary and fuzzy differential equations.
format	Thesis
qualification_level	Master's degree
author	Ismail, Nurzeehan
author_facet	Ismail, Nurzeehan
author_sort	Ismail, Nurzeehan
title	Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
title_short	Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
title_full	Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
title_fullStr	Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
title_full_unstemmed	Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
title_sort	three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations
granting_institution	Universiti Putra Malaysia
publishDate	2014
url	http://psasir.upm.edu.my/id/eprint/70591/1/FS%202014%2083%20-%20IR.pdf
_version_	1747812871343964160

Three-point diagonally implicit block methods for solving ordinary and fuzzy differential equations

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