2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations
| Format: | General Document |
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| collectionurl | https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 |
| copyright | Copyright©PWB2025 |
| country | Malaysia |
| date | 2018-08-01 |
| format | General Document |
| id | 16214 |
| institution | UniSZA |
| originalfilename | 16214_750de0a184f199d.pdf |
| person | Ibrahim Sulaiman Mohammed |
| recordtype | oai_dc |
| resourceurl | https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16214 |
| sourcemedia | Server storage Scanned document |
| spelling | 16214 https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16214 https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 General Document Malaysia Library Staff (Top Management) Library Staff (Management) Library Staff (Support) Terengganu Faculty of Informatics & Computing English application/pdf 1.5 20 Server storage Scanned document Universiti Sultan Zainal Abidin UniSZA Private Access UNIVERSITI SULTAN ZAINAL ABIDIN SAMBox 2.3.4; modified using iTextSharp™ 5.5.10 ©2000-2016 iText Group NV (AGPL-version) Copyright©PWB2025 Ibrahim Sulaiman Mohammed Iterative Methods 2018-08-01 16214_750de0a184f199d.pdf Iterative Methods (Mathematics) Fuzzy Mathematics Dual Fuzzy Systems 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations One of the most significant problems in fuzzy set theory is solving fuzzy equations. Numerous researchers have applied different numerical methods to solve this problem, which focused on equations with nonsingular Jacobian at the solution point. Numerical investigation indicates that most of these methods are computationally expensive due to the storage of Jacobian or approximate Jacobian at every iteration. Furthermore, the algorithms may not be defined when the Jacobian is singular. Yet, the researchers assumed this could be overcome by suitably chosen alternative methods. Besides that, singular problems are inherently more difficult to solve than nonsingular problems, and even the linear convergence in a reasonably stable fashion is considered quite an achievement. Therefore, new kinds of numerical methods are proposed to obtain the solution of fuzzy and singular fuzzy non-linear equations. Numerical algorithms are developed, and the solutions are verified by comparing the results with the analytical solutions. The conjugate gradient (CG) method has been developed under the exact line search for the solution of fuzzy, dual fuzzy and singular fuzzy non-linear equations. The CG is one of the simplest and most efficient numerical methods for solving non-linear problems. Diagonal updating Shamanskii method for solving fuzzy, dual fuzzy, and singular fuzzy non-linear equations has been successfully derived. This method tackled the problem of singularity by approximating the Jacobian into a diagonal matrix. Also, modified Levenberg-Marquardt method was employed to find the solution of fuzzy, dual fuzzy, and singular fuzzy non-linear equations. This method possesses the excellent local convergence exhibited by the Newton's method. The Barzilai-Borwein method is also employed to obtain the solution of fuzzy non-linear equations. This method is less sensitive to ill-conditioning with a rapid convergence rate. Additionally, the Regula Falsi method, also known as the false position method, is proposed numerically. This method is derivative-free and converges linearly. Results which were obtained using several benchmark problems have shown that the proposed methods are reliable and very effective compared to the classical methods for solving fuzzy non-linear equations. The proposed methods achieve 100% success and never fail to converge throughout the numerical experiments due to the derivative-free approach that was employed. The global convergence proof of the proposed methods is also provided. This study presents several efficient numerical algorithms for solving fuzzy non-linear equations. Their robustness and ability to solve problems with singular Jacobian at the solution point make them as several of the good alternatives for solving fuzzy non-linear equations. Dissertations, Academic Thesis |
| spellingShingle | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| state | Terengganu |
| subject | Iterative Methods (Mathematics) Dissertations, Academic |
| summary | One of the most significant problems in fuzzy set theory is solving fuzzy equations. Numerous researchers have applied different numerical methods to solve this problem, which focused on equations with nonsingular Jacobian at the solution point. Numerical investigation indicates that most of these methods are computationally expensive due to the storage of Jacobian or approximate Jacobian at every iteration. Furthermore, the algorithms may not be defined when the Jacobian is singular. Yet, the researchers assumed this could be overcome by suitably chosen alternative methods. Besides that, singular problems are inherently more difficult to solve than nonsingular problems, and even the linear convergence in a reasonably stable fashion is considered quite an achievement. Therefore, new kinds of numerical methods are proposed to obtain the solution of fuzzy and singular fuzzy non-linear equations. Numerical algorithms are developed, and the solutions are verified by comparing the results with the analytical solutions. The conjugate gradient (CG) method has been developed under the exact line search for the solution of fuzzy, dual fuzzy and singular fuzzy non-linear equations. The CG is one of the simplest and most efficient numerical methods for solving non-linear problems. Diagonal updating Shamanskii method for solving fuzzy, dual fuzzy, and singular fuzzy non-linear equations has been successfully derived. This method tackled the problem of singularity by approximating the Jacobian into a diagonal matrix. Also, modified Levenberg-Marquardt method was employed to find the solution of fuzzy, dual fuzzy, and singular fuzzy non-linear equations. This method possesses the excellent local convergence exhibited by the Newton's method. The Barzilai-Borwein method is also employed to obtain the solution of fuzzy non-linear equations. This method is less sensitive to ill-conditioning with a rapid convergence rate. Additionally, the Regula Falsi method, also known as the false position method, is proposed numerically. This method is derivative-free and converges linearly. Results which were obtained using several benchmark problems have shown that the proposed methods are reliable and very effective compared to the classical methods for solving fuzzy non-linear equations. The proposed methods achieve 100% success and never fail to converge throughout the numerical experiments due to the derivative-free approach that was employed. The global convergence proof of the proposed methods is also provided. This study presents several efficient numerical algorithms for solving fuzzy non-linear equations. Their robustness and ability to solve problems with singular Jacobian at the solution point make them as several of the good alternatives for solving fuzzy non-linear equations. |
| title | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| title_full | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| title_fullStr | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| title_full_unstemmed | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| title_short | 2018_New Iterative Methods for Solving Fuzzy and Dual Fuzzy Nonlinear Equations |
| title_sort | 2018_new iterative methods for solving fuzzy and dual fuzzy nonlinear equations |