calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

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

ito type stochastic fuzzy differential equations with delay abstract in the paper we give some foundations for the studies of stochastic fuzzy delayed differential equations we prove the existence and uniqueness of solutions to such the equations to obtain our result we assume that the coefficients of the equation satisfy the lipschitz condition together with linear growth condition we estimate the distance between approximate solution and exact solution also the stability of solution with respect to the initial history is shown an application of stochastic fuzzy delayed differential equations in the modeling of population growth is indicated

existence of solutions to fuzzy differential equations with generalized hukuhara derivative via contractive like mapping principles abstract we study the existence and uniqueness of solution for fuzzy initial value problems in the setting of a generalized hukuhara derivative and by using some recent results of fixed point of weakly contractive mappings on partially ordered sets

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

generalized fuzzy differentiability with lu parametric representation in the present paper we use a new generalization of the hukuhara difference and derivative for fuzzy valued functions and we study several properties of the new concepts in the setting of the lu parametric representation of fuzzy numbers assessed both from theoretical and computational points of view

two point boundary value problems of undamped uncertain dynamical systems in this paper we study two point boundary value problems of a class of uncertain dynamical systems i e undamped uncertain dynamical systems we introduce the concept of large solutions and small solutions for two point boundary value problems of undamped uncertain dynamical systems and under some conditions establish the existence and uniqueness of large solutions and small solutions and settle the relationship between large solutions and small solutions

on the solutions to first order linear fuzzy differential equations in this paper we study different formulations of first order linear fuzzy differential equations using the concept of generalized differentiability we present sufficient conditions for the existence of solutions and obtain the general expression of these solutions which exhibit different behavior some examples are given to illustrate our results

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

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

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

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

hypergraph partitioning for the parallel computing of fuzzy differential equations in our previous work fuzzy differential equations have been used to describe service composition and the solutions have been used for the performance analysis this paper presents a method to compute these fuzzy differential equations in parallel we first map the underlying petri net that generates fuzzy differential equations to a hypergraph and then partition the hypergraph to minimize inter processor communication while maintaining a good load balance based on the partition result we divide the fuzzy differential equations into several blocks finally we design parallel computing algorithm to compute these equations software hmetis is used to partition the hypergraph and software sundials is used to support the parallel computing of fuzzy differential equations online shop problem from ibm web site has been used to demonstrate the benefit of our method

fuzzy differential equations an approach via fuzzification of the derivative operator abstract in this paper we study fuzzy differential equations fdes in terms of derivative for fuzzy functions in a different way from the traditional hukuhara derivative defined for set valued functions the derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions we discuss the relation of this approach to fuzzy differential inclusions fdis and hukuhara and strongly generalized derivatives a theorem of existence of a solution is studied with hypothesis similar to those assumed for fdis some examples are explored in order to illustrate the theory

fuzzy differential equations with interactive derivative in this paper we introduce and study new concept of differentiability for fuzzy set valued functions this derivative considers possible local interactivity in the process studied several properties of differentiability and integrability are investigated for the new concept and they are compared to similar fuzzy differentiabilities like hukuhara differentiability and generalized hukuhara differentiability furthermore we establish theorems as the fundamental theorem of calculus ultimately we exhibit some results for fuzzy initial value problem and an application

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

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

minimizing the laplacian eigenvalues for trees with given domination number abstract in this paper we study the laplacian spectral radius of trees on n vertices with domination number where n k k 2 is an integer and determine the extremal trees that attain the minimal laplacian spectral radius when 2 3 4

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

existence and uniqueness results for fuzzy linear differential algebraic equations abstract we discuss the existence results for a fuzzy initial value problem of linear differential algebraic equations and provide an explicit representation for the solution a few illustrative examples are given

periodic problems of first order uncertain dynamical systems it is well known that there are no periodic solutions to fuzzy differential equations in the sense of hukuhara derivatives i e h derivatives so fuzzy differential equations in the sense of h derivatives cannot be used to describe periodic phenomena in the real world to overcome this disadvantage this paper studies periodic problems of first order fuzzy differential equations in the sense of differential inclusions i e periodic problems of first order uncertain dynamical systems and obtains the existence of periodic solutions for first order uncertain dynamical systems

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

revisiting fuzzy differential equations abstract we revisit the original formulation of fuzzy differential equations fdes and provide examples to show that the theory of fdes is richer than the corresponding theory of ordinary differential equations odes and therefore need not be rejected the alternative formulation in terms of differential inclusions does not always have advantage over odes as claimed we also discuss a stability result via lyapunov functions in this setup

a class of linear differential dynamical systems with fuzzy matrices abstract this paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices we use a complex number representation of the level sets of the fuzzy system and obtain the solution by employing such representation it is applicable to practical computations and has also some implications for the theory of fuzzy differential equations we then present some properties of the 2 dimensional dynamical systems and their phase portraits some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear we finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems

fuzzy delay differential equations under generalized differentiability abstract we interpret a fuzzy delay differential equation using the concept of generalized differentiability in this setting we prove the existence of two fuzzy solutions each one corresponding to a different type of differentiability uniqueness is understood in the sense that the solutions considered do not have switching points the applicability of the theoretical results is illustrated with some real world examples

analytical and numerical solutions of fuzzy differential equations in this paper we study analytical and numerical solutions of fuzzy differential equations based on the extension principle for linear fuzzy differential equations we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative in order to approximate the solutions of linear and non linear fuzzy differential equations we propose a new fuzzification of the classical euler method and then incorporate an unconstrained optimisation technique this combination offers a powerful tool to tackle uncertainty in any numerical method an efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain several illustrative examples are given

on the topological equivalence of some fuzzy flows near hyperbolic equilibria the purpose of this work is to study fuzzy dynamical systems associated with deterministic systems the grobman hartman theorem states that near hyperbolic equilibria there exists a homeomorphism between the nonlinear system s trajectories and the linearized correspondent system s trajectories that is these systems are topologically equivalent a theorem similar to grobman hartman theorem to fuzzy flows is the main result in this article for fuzzy flows obtained from each system it states that the nonlinear and the linearized are topologically equivalent

initial value problem for fuzzy differential equations under dissipative conditions a new concept of inner product on the fuzzy space e n d is introduced studied and used to prove several theorems stating the existence uniqueness and boundedness of solutions of fuzzy differential equations a stability result is also proved in the same context

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

on linear fuzzy differential equations by differential inclusions approach abstract in this paper we study first order linear fuzzy differential equations under differential inclusions and strongly generalized differentiability approaches we present some new results on the relation between their solutions finally some examples are given to illustrate our results

set valued and fuzzy stochastic differential equations driven by semimartingales abstract in the paper we present set valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties then we study the existence of solutions to set valued and fuzzy set valued stochastic differential equations driven by semimartingales the stability of solutions is also established

grid generation for internal flow configurations abstract this paper is concerned with analysis and applications of geometry grid generation methodologies for internal flow configurations weighted transfinite interpolation technique is combined with cubic splines and bezier curves to accomplish well distributed zonal grids the concept of surface distribution mesh and volume distribution mesh is presented and its natural application to adaptive grids is developed grid refinement algorithms are developed using applications of elliptic grid systems various computational examples of practical interest are presented

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

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

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

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

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

an existence and uniqueness result for fuzzy goursat partial differential equation abstract in this paper we study a fuzzy partial differential equation the so called goursat problem under generalized differentiability concept we give necessary and sufficient conditions for the existence of a unique solution we also provide some sufficient conditions for the coincidence of the second order mixed derivatives of a fuzzy function which is useful for the study of higher order fuzzy partial differential equations