time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

on the existence of solutions to periodic boundary value problems for fuzzy linear differential equations in this work we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first order linear fuzzy differential equations by using generalized differentiability and switching points in comparison with some previous works we consider equations whose coefficient may change its sign a finite number of times in the interval of interest we also study the existence of solutions which are crisp or real at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

improved predictor corrector method for solving fuzzy initial value problems in this paper an improved predictor corrector ipc method is proposed to solve the fuzzy initial value problem the ipc method is generated by combining an explicit three step method and an implicit two step method the methods are compared with the methods discussed in t allahviranloo n ahmady e ahmady numerical solution of fuzzy differential equations by predictor corrector method information sciences 177 7 2007 1633 1647 and are shown to be more accurate the convergence and stability of the proposed methods are also presented in detail these methods are illustrated by solving some examples

hukuhara difference of z numbers abstract real world problems in decision analysis economics optimization and other fields are characterized by fuzzy and partially reliable information in this regard prof zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers a and b the first of which describes a linguistic value and the second one is a fuzzy value of probability measure of the first one playing a role of reliability of information construction of mathematical models to handle z number based information is a new challenge which requires development of z number valued analysis including z number valued equations z number valued derivative and integral and other concepts one of the basic concepts in this realm is an inverse of addition operation in the existing literature devoted to set valued arithmetic and fuzzy arithmetic this operation is referred to as hukuhara difference for interval numbers and fuzzy numbers the existence conditions of hukuhara difference are derived unfortunately the concept of hukuhara difference for z numbers is not developed in this work for the first time we suggest a fundamental approach for development of the concept of hukuhara difference of z numbers examples are provided in the paper to show validity of the suggested approach

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

on periodic solutions to first order linear fuzzy differential equations under differential inclusions approach in this work we study some properties of the solutions to first order linear fuzzy differential equations under differential inclusions approach we show that the differential inclusions approach generally adapts better to some properties of the solutions to ordinary differential equations sufficient conditions for the solutions under this approach to be periodic and bounded are presented we also give some examples to show the applicability of these results

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

monotone method for fuzzy differential equations we develop the monotone iterative technique to approximate the extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval to our procedure it is crucial to have at our disposal fuzzy comparison results and several properties of sets of fuzzy functions and fuzzy numbers including a compactness result in spaces of fuzzy functions which are useful from the point of view of applications

existence theorems for solutions to random fuzzy differential equations abstract we examine a random fuzzy initial value problem with two kinds of fuzzy derivatives for both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations the existence of global solutions is also obtained

a new apn function which is not equivalent to a power mapping a new almost perfect nonlinear function apn on f 2 sup 10 which is not equivalent to any of the previously known apn mappings is constructed this is the first example of an apn mapping which is not equivalent to a power mapping

numerical methods forfuzzy differential inclusions in this paper a numerical method for solving fuzzy differential inclusions is considered fuzzy reachable set can be approximated by proposed method with complete error analysis which is discussed in detail the method is illustrated by solving some linear and nonlinear fuzzy initial value problems

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on random fuzzy differential equations we consider a cauchy problem in a random fuzzy setting under the condition of lipschitzean right hand side the existence and uniqueness of the solution is proven also the continuous dependence on the right hand side and initial condition is shown some kind of boundedness of the solution is established

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

towards the theory of fuzzy differential equations comparative analysis of two ideologically distinct approaches in the theory of fuzzy differential equations is given domains of applications advantages and shortcomings of those approaches are described new definitions of solutions to fuzzy differential equations are 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 retarded fuzzy functional differential equations and nonabsolute fuzzy integrals abstract we prove a generalized convergence theorem for fuzzy henstock integrals based on the sequence of fuzzy number valued functions which are weak generalized uniformly absolute continuous u a c g to widen the applications of this convergence theorem we provide some existence theorems for the generalized solution for retarded fuzzy functional differential equations and the dependence of the solutions on a parameter under the assumption of the strong generalized derivative of the solution our results generalize existence results in a kaleva integral setting to a fuzzy henstock integral setting

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

eulerian lagrangian analysis of transport conditioned on hydraulic data 2 effects of log transmissivity and hydraulic head measurements in paper 1 of this series we described an analytical numerical method to predict deterministically solute transport under uncertainty the method is based on a unified eulerian lagrangian theory which allows conditioning of the predictions on hydraulic measurements conditioning on measured concentrations is also possible as demonstrated by neuman et al 1993 in this paper we condition velocity on log transmissivity and or hydraulic head data via cokriging we then combine an early time analytical solution with a pseudo fickian galerkin finite element scheme for later time to obtain conditional predictions of concentration and lower bounds on its variance and coefficient of variation the pseudo fickian scheme involves a conditional dispersion tensor which depends on information and varies in space time hence the predicted plumes travel along curved trajectories and attain irregular non gaussian shapes we also compute a measure of uncertainty for the original source location of a solute particle whose position at some later time is known from sampling spatial maps of this particle origin covariance provide vivid images of preferential flow paths and exclusion zones identified by the available data we illustrate these concepts and results on instantaneous point and area sources

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

comparation between some approaches to solve fuzzy differential equations in this paper we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via zadeh s extension principle we get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented we also give some properties and we show the relationships with others interpretation finally we propose a numerical procedure for obtaining the fuzzy solution

numerical solutions for linear system of first order fuzzy differential equations with fuzzy constant coefficients in this paper a novel iterative method is proposed to obtain approximate analytical solutions for the linear systems of first order fuzzy differential equations fdes with fuzzy constant coefficients fccs while avoiding the complexities of eigen value computations a theorem for the convergence and the validity of the approach is also presented in detail numerical experiments and comparisons with exact solutions reveal that the proposed method is capable of generating accurate results

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented