Fuzzy Linear Equations Research Articles

Numerical solution of fully fuzzy linear systems by fuzzy neural network

In this paper, a new hybrid method based on fuzzy neural network (FNN) for approximate solution of fuzzy linear systems of the form $$Ax=d,$$ where $$A$$ is a square matrix of fuzzy coefficients, $$x$$ and $$d$$ are fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution of an $$n\times n$$ system of fuzzy linear equations that supposedly has a unique fuzzy solution, a simple algorithm from the cost function of the FNN is proposed. Finally, we illustrate our approach by some numerical examples.

A New Approach for Solving Fully Fuzzy Linear Systems

Several authors have proposed different methods to find the solution of fully fuzzy linear systems (FFLSs) that is, fuzzy linear system with fuzzy coefficients involving fuzzy variables. But all the existing methods are based on the assumption that all the fuzzy coefficients and the fuzzy variables are nonnegative fuzzy numbers. In this paper a new method is proposed to solve an FFLS with arbitrary coefficients and arbitrary solution vector, that is, there is no restriction on the elements that have been used in the FFLS. The primary objective of this paper is thus to introduce the concept and a computational method for solving FFLS with no non negative constraint on the parameters. The method incorporates the principles of linear programming in solving an FFLS with arbitrary coefficients and is not only easier to understand but also widens the scope of fuzzy linear equations in scientific applications. To show the advantages of the proposed method over existing methods we solve three FFLSs.

A new method for solving fuzzy linear differential equations

In this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. The proposed method is illustrated with numerical examples.

Numerical Methods for Fuzzy Linear Partial Differential Equations under new Definition for Derivative

In this paper difference methods to solve partial differential (FPDE) such as fuzzy hyperbolic and fuzzy parabolic equations are considered. The existence of the solution and stability of the method are examined in detail. Finally examples are presented to show that the Hausdorff distance between the exact solution and approximate solution tends to zero.

On the Solution of Fully Fuzzy Linear Systems

A linear system is called a fully fuzzy linear system (FFLS) if quantities in this system are all fuzzy numbers. For the FFLS, we investigate its solution and develop a new approximate method for solving the FFLS. Observing the numerical results, we find that our method is accurate than the iterative Jacobi and GaussSeidel methods on approximating the solution of FFLS. Keywords—fully fuzzy linear equations, iterative method, homotopy perturbation method, approximate solutions

Stability analysis of fuzzy linear differential equations

Differential equation is a very important field of study not only in theory but in applications, and stochastic differential equation is also a significant branch of research in theory and applications. Based on the concepts of fuzzy process and fuzzy differential equation, we present some concepts of stability for fuzzy differential equations. Furthermore we propose the sufficient and necessary conditions of stability for fuzzy linear differential equations.

The numerical solution of linear fuzzy Fredholm integral equations of the second kind by using finite and divided differences methods

In recent years, many numerical methods have been proposed for solving fuzzy linear integral equations. In this paper, we use the divided differences and finite differences methods for solving a parametric of the fuzzy Fredholm integral equations of the second kind with arbitrary kernel and present some examples to illustrate this method.

An Approximate Solutions of Fuzzy Linear Fredholm Integral Equations

An Approximate Solutions of Fuzzy Linear Fredholm Integral Equations

A generalization of Hukuhara difference and division for interval and fuzzy arithmetic

We propose a generalization of the Hukuhara difference. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers. Applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown.

Fuzzy linear matrix equation

The main aim of this paper is to discuss Fuzzy Linear Matrix Equations (shown as FLME) of the form AXB = C for finding its fuzzy solutions. In this paper, the parametric form of the fuzzy linear system is used. Necessary and sufficient conditions for the existence of the set of fuzzy solutions are derived, and a numerical procedure for calculating the solutions is designed.

A method for solving nth order fuzzy linear differential equations

In this paper, an analytic method (eigenvalue–eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered. In this method, three cases are introduced, in each case, it is shown that the solution of differential equation is a fuzzy number. In addition, the method is illustrated by solving several numerical examples.

Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix

Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix

Development of reservoir operation policies considering variable agricultural water demands

In this paper, a genetic algorithm (GA) optimization model is developed for reservoir operation optimization considering variations in water demands. In order to incorporate the demand uncertainties in optimal operation policies, different types of linear equations with different combinations of inflow, storage at the beginning of the month, and water demands as independent variables have been considered. The coefficients of optimal operation policies are obtained using classic and fuzzy regression analysis. In the case of fuzzy regression, both symmetric and asymmetric membership functions are used. Efficiency of operation policies are evaluated based on the long-term operation simulation of Zayandeh-Rud Reservoir in central part of Iran. Estimated figures for the four criteria of reliability, resiliency, total vulnerability, and maximum monthly vulnerability and also the statistical criteria of correlation coefficient, coefficient of efficiency, and standard error indicate that the fuzzy linear regression equations with inflow, storage, and demand as independent variables, in which asymmetric membership functions are used for the coefficients of the regression equation, has the best long-term performance in meeting variable demands.

Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix

A method to solve linear fuzzy equations with a symmetric matrix is proposed. Ignoring the symmetry leads to an overestimation of the solution. Our method to find the solution of a system of linear fuzzy equations takes the symmetry of the matrix into account and is based on parametric functions. It is a practical algorithm using parametric functions in which the variables are given by elements of the support of the fuzzy coefficients of the system.

Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix

Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix

Nth-order fuzzy linear differential equations

In this paper a numerical method for solving nth-order linear differential equations with fuzzy initial conditions is considered. The idea is based on the collocation method. The existence theorem of the fuzzy solution is considered. This method is illustrated by solving several examples.

Solving Systems of Linear Fuzzy Equations by Parametric Functions

Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the alpha-levels are calculated. We propose in this paper a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. We show that our algorithm is much more efficient than the method of Buckley and Qu.

A Genetic Algorithm Approach for Solving Fuzzy Linear and Quadratic Equations

Iin this paper a genetic algorithms approach for solving the linear and quadratic fuzzy equations (Ax) over tilde = (B) over tilde and (Ax) over tilde (2) + (Bx) over tilde = (C) over tilde, where (A) over tilde, and (B) over tilde, (C) over tilde and (x) over tilde are fuzzy numbers is proposed by genetic algorithms. Our genetic based method initially starts with a set of random fuzzy solutions. Then in each generation of genetic algorithms, the solution candidates converge more to better fuzzy solution (x) over tilde (b). in this proposed method the final reached (x) over tilde (b) is not only restricted to fuzzy triangular and it can be fuzzy number.

Solving systems of linear fuzzy equations by parametric functions—An improved algorithm

Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the α -levels are calculated. We propose a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. By observing the monotonicity of the parametric functions in each variable, i.e. each fuzzy coefficient in the system, we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions. We show that our algorithm is much more efficient than the method of Buckley and Qu.

Minimal solution of general dual fuzzy linear systems

Fuzzy linear systems of equations, play a major role in several applications in various area such as engineering, physics and economics. In this paper, we investigate the existence of a minimal solution of general dual fuzzy linear equation systems. Two necessary and sufficient conditions for the minimal solution existence are given. Also, some examples in engineering and economic are considered.