scholarly journals LU Decomposition Scheme for Solving m-Polar Fuzzy System of Linear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-19 ◽  
Author(s):  
Ali N. A. Koam ◽  
Muhammad Akram ◽  
Ghulam Muhammad ◽  
Nawab Hussain
Keyword(s):  
Fuzzy System ◽  
Matrix Theory ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Least Square ◽  
System Of Linear Equations ◽  
Fuzzy Matrix ◽  
Decomposition Scheme ◽  
Symmetric Positive Definite ◽  
Generalized Inverse Matrix

This paper presents a new scheme for solving m-polar fuzzy system of linear equations (m-PFSLEs) by using LU decomposition method. We assume the coefficient matrix of the system is symmetric positive definite, and we discuss this point in detail with some numerical examples. Furthermore, we investigate the inconsistent m×nm-polar fuzzy matrix equation (m-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of m-polar fuzzy LSS of the inconsistent m-PFME. In the end, we present a numerical example to illustrate our approach.

LU Decomposition method to solve bipolar fuzzy linear systems

2020 ◽  
Vol 39 (3) ◽  
pp. 3329-3349 ◽  
Author(s):  
Muhammad Akram ◽  
Ghulam Muhammad ◽  
Tofigh Allahviranloo ◽  
Nawab Hussain
Keyword(s):  
Decomposition Method ◽  
Matrix Theory ◽  
Coefficient Matrix ◽  
Least Square ◽  
Lu Decomposition ◽  
Fuzzy Matrix ◽  
Linear System Of Equations ◽  
Fuzzy Environment ◽  
Generalized Inverse Matrix ◽  
Fuzzy Linear Systems

The aim of this work is to solve the linear system of equations using LU decomposition method in bipolar fuzzy environment. We assume a special case when the coefficient matrix of the system is symmetric positive definite. We discuss this point in detail by giving some numerical examples. Moreover, we investigate m × n inconsistent bipolar fuzzy matrix equation and find the least square solution of the inconsistent bipolar fuzzy matrix using the generalized inverse matrix theory. The existence of the strong bipolar fuzzy least square solution of the inconsistent bipolar fuzzy matrix is discussed. In the end, a numerical example is presented to illustrate our proposed method.

scholarly journals A Novel Technique to Solve the Fuzzy System of Equations

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 850
Author(s):  
Nasser Mikaeilvand ◽  
Zahra Noeiaghdam ◽  
Samad Noeiaghdam ◽  
Juan J. Nieto
Keyword(s):  
Linear System ◽  
Fuzzy Number ◽  
Fuzzy System ◽  
Linear Equations ◽  
Second Step ◽  
Negative Solution ◽  
System Of Linear Equations ◽  
Vector Solution ◽  
Novel Technique ◽  
Fuzzy Linear System

The aim of this research is to apply a novel technique based on the embedding method to solve the n × n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created n × n crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created n × n crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors.

Certain efficient iterative methods for bipolar fuzzy system of linear equations

2020 ◽  
Vol 39 (3) ◽  
pp. 3971-3985 ◽  
Author(s):  
Muhammad Saqib ◽  
Muhammad Akram ◽  
Shahida Bashir
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Iterative Methods ◽  
Fuzzy Set ◽  
Fuzzy System ◽  
Linear Equations ◽  
Performance Validity ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Iterative Schemes

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n ×  n BFSLEs into 4n ×  4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs.

Bipolar fuzzy system of linear equations with polynomial parametric form

2019 ◽  
Vol 37 (6) ◽  
pp. 8275-8287 ◽  
Author(s):  
Muhammad Akram ◽  
Ghulam Muhammad ◽  
Nawab Hussain
Keyword(s):  
Fuzzy System ◽  
Linear Equations ◽  
Parametric Form ◽  
System Of Linear Equations

Solutions of Fuzzy System of Linear Equations

2020 ◽  
pp. 26-33
Author(s):  
Laxminarayan Sahoo
Keyword(s):  
Linear System ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Fuzzy Number ◽  
Fuzzy System ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Interval System ◽  
Solution Methodology

This chapter deals with solution methodology of fuzzy system of linear equations (FSLEs). In fuzzy set theory, finding solutions of FLSEs has long been a well-known problem to the researchers. In this chapter, the fuzzy number has been converted into interval number, and the authors have solved the interval system of linear equation for finding the fuzzy valued solution. Here, a fuzzy valued linear system has been introduced and a numerical example has been solved and presented for illustration of purpose.

Uncertain Static and Dynamic Analysis of Imprecisely Defined Structural Systems

2014 ◽  
pp. 357-382
Author(s):  
S. Chakraverty ◽  
Diptiranjan Behera
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Fuzzy System ◽  
Eigenvalue Problems ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Static And Dynamic Analysis ◽  
Fuzzy Finite Element ◽  
Element Method

This chapter presents the static and dynamic analysis of structures with uncertain parameters using fuzzy finite element method. Uncertainties presents in the parameters are modelled through convex normalised fuzzy sets. Fuzzy finite element method converts the structures into fuzzy system of linear equations and fuzzy eigenvalue problem for static and dynamic problems respectively. As such method to solve fuzzy system of linear equations, fully fuzzy system of linear equations and fuzzy eigenvalue problems are presented. These methods are applied to various structural problems to find out the fuzzy static and dynamic responses of the structures. Also the chapter analyses the numerical solution of uncertain fractionally damped spring-mass system. Uncertainties considered in the initial condition of the system.

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