Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations

Differential equations are useful in modelling various concepts and scenarios. However, when considering the initial or boundary conditions for these differential equation models, the use of fuzzy numbers is more realistic and flexible, as the parameters may vary within a certain range. Such situati...

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Main Author:	Hussain, Kashif
Format:	Thesis
Language:	eng eng eng
Published:	2023
Subjects:	QA Mathematics QA76.76 Fuzzy System.
Online Access:	https://etd.uum.edu.my/10892/1/permission%20to%20deposit-embargo%2036%20months-s904029.pdf https://etd.uum.edu.my/10892/2/s904029_01.pdf https://etd.uum.edu.my/10892/3/s904029_02.pdf
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spelling	my-uum-etd.108922024-01-15T00:37:21Z Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations 2023 Hussain, Kashif Ahmad, Nazihah Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Art & Sciences QA Mathematics QA76.76 Fuzzy System. Differential equations are useful in modelling various concepts and scenarios. However, when considering the initial or boundary conditions for these differential equation models, the use of fuzzy numbers is more realistic and flexible, as the parameters may vary within a certain range. Such situations are referred as unpredictable circumstances which introduce the concept of uncertainty. Thus, fuzzy derivatives and fuzzy differential equations (FDEs) are used to deal with these situations. Numerical methods are used to provide approximate solutions to differential equations, such as FDEs, in the absence of exact solutions. Conventionally, higher-order fuzzy ordinary differential equations (FODEs) are solved by first reducing the equations to their equivalent systems of first-order FODEs. Then, suitable existing numerical methods for first-order FODEs are employed to solve the resulting systems. This approach results in having more equations and thus increases the computational burden which jeopardises the accuracy of the solution. For this reason, several existing studies have developed numerical methods for the direct solution of FODEs. However, the results obtained possess accuracy that could be improved. Other issues encountered in existing studies that consider the numerical solution of FODEs include the methods being non self-starting, not solving both fuzzy initial value problems (FIVPs) and fuzzy boundary value problems (FBVPs), and the absence of a generalised form for the developed numerical methods. Therefore, to overcome these problems, this study developed more accurate higher derivative self -starting block methods with generalised steplength for the direct numerical solution of FODEs. A linear block approach using Taylor series expansion is adopted for the derivation of the block methods with the presence of two higher derivatives terms. Investigations on the order, error constant, zero-stability, consistency, convergence, and region of absolute stability of the new block methods with the presence of two higher derivatives were also conducted, and the developed new block methods satisfied all convergence properties in the fuzzy form successfully. To investigate the accuracy, the developed block methods were employed to solve both FIVPs and FBVPs considered in the literature. The numerical results confirmed the superiority of the new methods over the existing methods when solving the same test problems in terms of absolute errors. In conclusion, this study has successfully developed self -starting and efficient block methods in solving FODEs directly. 2023 Thesis https://etd.uum.edu.my/10892/ https://etd.uum.edu.my/10892/1/permission%20to%20deposit-embargo%2036%20months-s904029.pdf text eng 2026-10-17 staffonly https://etd.uum.edu.my/10892/2/s904029_01.pdf text eng 2026-10-17 staffonly https://etd.uum.edu.my/10892/3/s904029_02.pdf text eng staffonly other doctoral Universiti Utara Malaysia
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng eng
advisor	Ahmad, Nazihah
topic	QA Mathematics QA76.76 Fuzzy System.
spellingShingle	QA Mathematics QA76.76 Fuzzy System. Hussain, Kashif Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
description	Differential equations are useful in modelling various concepts and scenarios. However, when considering the initial or boundary conditions for these differential equation models, the use of fuzzy numbers is more realistic and flexible, as the parameters may vary within a certain range. Such situations are referred as unpredictable circumstances which introduce the concept of uncertainty. Thus, fuzzy derivatives and fuzzy differential equations (FDEs) are used to deal with these situations. Numerical methods are used to provide approximate solutions to differential equations, such as FDEs, in the absence of exact solutions. Conventionally, higher-order fuzzy ordinary differential equations (FODEs) are solved by first reducing the equations to their equivalent systems of first-order FODEs. Then, suitable existing numerical methods for first-order FODEs are employed to solve the resulting systems. This approach results in having more equations and thus increases the computational burden which jeopardises the accuracy of the solution. For this reason, several existing studies have developed numerical methods for the direct solution of FODEs. However, the results obtained possess accuracy that could be improved. Other issues encountered in existing studies that consider the numerical solution of FODEs include the methods being non self-starting, not solving both fuzzy initial value problems (FIVPs) and fuzzy boundary value problems (FBVPs), and the absence of a generalised form for the developed numerical methods. Therefore, to overcome these problems, this study developed more accurate higher derivative self -starting block methods with generalised steplength for the direct numerical solution of FODEs. A linear block approach using Taylor series expansion is adopted for the derivation of the block methods with the presence of two higher derivatives terms. Investigations on the order, error constant, zero-stability, consistency, convergence, and region of absolute stability of the new block methods with the presence of two higher derivatives were also conducted, and the developed new block methods satisfied all convergence properties in the fuzzy form successfully. To investigate the accuracy, the developed block methods were employed to solve both FIVPs and FBVPs considered in the literature. The numerical results confirmed the superiority of the new methods over the existing methods when solving the same test problems in terms of absolute errors. In conclusion, this study has successfully developed self -starting and efficient block methods in solving FODEs directly.
format	Thesis
qualification_name	other
qualification_level	Doctorate
author	Hussain, Kashif
author_facet	Hussain, Kashif
author_sort	Hussain, Kashif
title	Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
title_short	Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
title_full	Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
title_fullStr	Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
title_full_unstemmed	Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
title_sort	higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2023
url	https://etd.uum.edu.my/10892/1/permission%20to%20deposit-embargo%2036%20months-s904029.pdf https://etd.uum.edu.my/10892/2/s904029_01.pdf https://etd.uum.edu.my/10892/3/s904029_02.pdf
_version_	1794023783373209600

Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations

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