Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations
Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing...
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| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/3/446 |
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| author | Ahmed Abdel Aziz Elsayed Bassem Saassouh Nazihah Ahmad Ghassan Malkawi |
| author_facet | Ahmed Abdel Aziz Elsayed Bassem Saassouh Nazihah Ahmad Ghassan Malkawi |
| author_sort | Ahmed Abdel Aziz Elsayed |
| collection | DOAJ |
| description | Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing analytical methods for solving FFSME are based on Vec-operator and Kronecker product. Nevertheless, these methods are only applicable for nonnegative fuzzy numbers, which limits the applications of the existing methods. Thus, this paper proposes a new numerical method for solving arbitrary Trapezoidal FFSME (TrFFSME), which includes near-zero trapezoidal fuzzy numbers to overcome this limitation. The TrFFSME is converted to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication operations. Then the non-linear system is solved using a newly developed two-stage algorithm. In the first stage algorithm, initial values are determined. Subsequently, the second stage algorithm obtains all possible finite fuzzy solutions. A numerical example is solved to illustrate the proposed method. Besides, this proposed method can solve other forms of fuzzy matrix equations and produces finite fuzzy and non-fuzzy solutions compared to the existing methods. |
| first_indexed | 2024-03-09T12:25:57Z |
| format | Article |
| id | doaj.art-98776a7119274ab693bc4009e02108c9 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:25:57Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-98776a7119274ab693bc4009e02108c92023-11-30T22:34:44ZengMDPI AGSymmetry2073-89942022-02-0114344610.3390/sym14030446Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix EquationsAhmed Abdel Aziz Elsayed0Bassem Saassouh1Nazihah Ahmad2Ghassan Malkawi3Department of Mathematics, Institute of Applied Technology, Mohamed Bin Zayed City 33884, United Arab EmiratesAcademic Support Department, Abu Dhabi Polytechnic College, Abu Dhabi 111499, United Arab EmiratesSchool of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kedah, MalaysiaFaculty of Engineering, Math and Natural Science Division, Higher Colleges of Technology (HCT), Al Ain Campus, Abu Dhabi 17155, United Arab EmiratesSylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing analytical methods for solving FFSME are based on Vec-operator and Kronecker product. Nevertheless, these methods are only applicable for nonnegative fuzzy numbers, which limits the applications of the existing methods. Thus, this paper proposes a new numerical method for solving arbitrary Trapezoidal FFSME (TrFFSME), which includes near-zero trapezoidal fuzzy numbers to overcome this limitation. The TrFFSME is converted to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication operations. Then the non-linear system is solved using a newly developed two-stage algorithm. In the first stage algorithm, initial values are determined. Subsequently, the second stage algorithm obtains all possible finite fuzzy solutions. A numerical example is solved to illustrate the proposed method. Besides, this proposed method can solve other forms of fuzzy matrix equations and produces finite fuzzy and non-fuzzy solutions compared to the existing methods.https://www.mdpi.com/2073-8994/14/3/446fully fuzzy sylvester matrix equationsfuzzy matrix equationnumerical fuzzy solutiontrapezoidal fuzzy multiplication |
| spellingShingle | Ahmed Abdel Aziz Elsayed Bassem Saassouh Nazihah Ahmad Ghassan Malkawi Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations Symmetry fully fuzzy sylvester matrix equations fuzzy matrix equation numerical fuzzy solution trapezoidal fuzzy multiplication |
| title | Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations |
| title_full | Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations |
| title_fullStr | Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations |
| title_full_unstemmed | Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations |
| title_short | Two-Stage Algorithm for Solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations |
| title_sort | two stage algorithm for solving arbitrary trapezoidal fully fuzzy sylvester matrix equations |
| topic | fully fuzzy sylvester matrix equations fuzzy matrix equation numerical fuzzy solution trapezoidal fuzzy multiplication |
| url | https://www.mdpi.com/2073-8994/14/3/446 |
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