Solution Of Fuzzy Differential Equations Research Articles

On stability of motion with respect to two measures under uncertainty

General conditions for the stability with respect to two measures of the solution of fuzzy differential equations are established. The approach used is based on reducing a fuzzy differential equation to a hybrid system. The stability conditions are formulated in terms of restrictions on the evolution operator of the system of linear approximation

Note on “Numerical solutions of fuzzy differential equations by predictor–corrector method”

In the present note it is shown that the examples presented in a recent paper by Allahviranloo et al., are incorrect. Namely, the “exact solutions” proposed by the authors are not solutions of the given fuzzy differential equations (FDEs). The correct exact solutions are also presented here, together with some results for characterizing solutions of FDEs under Hukuhara differentiability by an equivalent system of ODEs. In this way a new direction for the numerical solutions of FDEs is proposed.

Comparison results for fuzzy differential equations

In this paper, we present several comparison results for the solutions of fuzzy differential equations, by using the Hukuhara derivative. These results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations. Besides, some side-results about the Hukuhara difference and the partial orderings are given. We illustrate the applicability of the new results by showing several examples.

Periodic boundary value problems for impulsive fuzzy differential equations

The peculiarity of the Hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations. In this work, we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method. Fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic ‘linear’ differential problems is also provided.

On new solutions of fuzzy differential equations

We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made.

Numerical solution of fuzzy differential equations by predictor–corrector method

In this paper three numerical methods to solve “The fuzzy ordinary differential equations” are discussed. These methods are Adams–Bashforth, Adams–Moulton and predictor–corrector. Predictor–corrector is obtained by combining Adams–Bashforth and Adams–Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α - cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α - cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

A Note on the Asymptotic Stability of Fuzzy Differential Equations

We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations.

Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations

The usual concept of differentiability of fuzzy-number-valued functions, has the following shortcoming: if c is a fuzzy number and g : [ a , b ] → R is an usual real-valued function differentiable on x 0 ∈ ( a , b ) with g ′ ( x 0 ) ⩽ 0 , then f ( x ) = c ⊙ g ( x ) is not differentiable on x 0 . In this paper we introduce and study generalized concepts of differentiability (of any order n ∈ N ), which solves this shortcoming. Newton–Leibnitz-type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied. Also, some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c ⊙ g ( x ) , are given.

Stability of a periodic solution for fuzzy differential equations

In this paper, we consider the fuzzy differential equations $$\frac{{dx(t)}}{{dt}} = F(t,x(t)) and x(t_0 ) = x_0 \in E^n $$ whereF(t,x(t)) is a continuous fuzzy mapping on [0, ∞)×E n . The purpose of this paper is to prove that the solution ϕ(t) of the fuzzy differential equations is equiasymptotically stable in the large and uniformly asymptotically stable in the large.

Existence and interrelation between set and fuzzy differential equations

An existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general. Then, the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it, is studied.

Numerically solution of fuzzy differential equations by Adomian method

In this paper a method for numerical approximation of fuzzy first-order initial value problem is presented. Buckly and Feruing [Fuzzy Sets Syst. 110 (2000) 43] have solved fuzzy first-order initial value problem, they solved problem in crisp case and then extend the solution to fuzzy case. In this paper, we use Adomian method to find the approximate solution in crisp case and extend it to fuzzy case. Using this method, we solve problems that the classical methods could not be applied for them in crisp case.

Numerical Solution of Fuzzy Differential Equation

In this paper numerical algorithms for solving fuzzy ordinary differential equations are considered. A scheme based on the 2nd Taylor method in detail is discussed and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy cauchy problems.

Numerical Solutions of Fuzzy Differential Equations by Taylor Method

Abstract In this paper, numerical algorithms for solving “fuzzy ordinary differential equations” are considered. A scheme based on the Taylor method of order p is discussed in detail and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy Cauchy problems.

Carathéodory solutions of fuzzy differential equations

In this paper, the existence of solutions for fuzzy differential equations with right-hand side satisfying a Carathéodory condition on a class of fuzzy sets is established.

Numerical solutions of fuzzy differential equations

Numerical algorithms for solving ‘fuzzy ordinary differential equations’ (FODE) are considered. A scheme based on the classical Euler method is discussed in detail, and this is followed by a complete error analysis. The algorithm is illustrated by solving several linear and nonlinear fuzzy Cauchy problems.

Existence and uniqueness theorem for a solution of fuzzy differential equations

By using the method of successive approximation, we prove the existence and uniqueness of a solution of the fuzzy differential equation x′(t) = f(t, x(t)), x(t0) = x0. We also consider an ϵ‐approximate solution of the above fuzzy differential equation.

Frequency domain methods for the solutions of N-order fuzzy differential equations

The solutions of the fuzzy differential equations in Refs. [8–10, 12, 14, 16] have so far been obtained by integration or fuzzy integration in the time domain. It is often more convenient, particularly, in the fuzzy random vibration problems [12] and the fuzzy stochastic dynamic systems [14] to obtain the solutions by integration or fuzzy integration in the frequency domain. This is accomplished by the use to generalized fuzzy harmonic analysis [12, 14]. In the second paper of a series of reports on fuzzy differenrial equations, we continue studying the nth-order fuzzy differential equation X ∼ (n)(t) + a n−1(t) X ∼ n−1(t) + ⋯ + a 0(t) X ∼ (t) = F(t) , where X ( n) ( t), X ( n−1) ( t), … X (1)( t) are nth, ( n−1)th…, 1st same-oorder (or reverse-order) derived functions of an unknown fuzzy set-valued function X( t), respectively; X( t) is known fuzzy set-valued function; a i ( t), i = 0,1, …, n − 1, are deterministic functions of parameter t. The solving processes of frequency domain for nth-order fuzzy differential equations are put forward. One example is considered in order to demonstrate the rationality and validity of the methods. The work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations [8], fuzzy dynamical systems [2], fuzzy random vibration [12], fuzzy stochastic dynamical systems [14, 18–20] and fuzzy stochastic systems [21–23].

Asymptotic behavior of solutions of fuzzy differential equations

We study the asymptotic behaviour of the solutions of the fuzzy differential equations. The method of successive approximation is used to establish our results.

Existence of the solutions of fuzzy differential equations with parameters

In this paper, we study the existence of the solutions of fuzzy differential equations with parameters by the topological degree method.