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

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

numerical solution method for the dbar equation in the plane a fast method for solving equations of the form v tv is presented where v and t are complex valued functions of two real variables the multigrid method of vainikko int soc anal appl comput 5 2000 is adapted to the problem with a fft implementation convergence with rate o h is proved for the method applied to equations of the form above one grid and two grid versions of the method are implemented and their effectiveness is demonstrated on an application arising in electrical impedance tomography eit

existence theorem to the cauchy problem of fuzzy differential equations under compactness type conditions kaleva fuzzy sets and systems 35 1990 389 396 has concluded that the peano theorem does not hold for fuzzy differential equations on en d this paper deals with the cauchy problem of fuzzy differential equations the existence theorems under compactness type conditions are obtained

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

decomposing a matrix into circulant and diagonal factors abstract let k be a field of characteristic 2 it is proven that for k f 3 f 5 each square matrix with entries in k can be written as a product of circulant and diagonal matrices with entries in k

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

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

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 and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

initial value problems for higher order fuzzy differential equations we consider nth order fuzzy differential equations with initial value conditions we prove the existence and uniqueness of solution for nonlinearities satisfying a lipschitz condition we apply the obtained results to the particular case of linear fuzzy problems

a class of linear differential dynamical systems with fuzzy initial condition this paper investigates linear first order fuzzy differential dynamical systems where the initial condition is described by a fuzzy number we use a complex number representation of the a level sets of the fuzzy system and prove theorems that provide the solutions under such representation which is applicable to practical computations and also has some implications for theory then the paper shows some properties of the two dimensional dynamical systems and their phase portraits are described by means of examples there may be a significant difference between the solutions according to whether the matrix is nonnegative or not finally the paper points out future research on the fuzzy dynamical systems

on fuzzy boundary value problems the statement that a two point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by lakshmikantham et al and o regan et al recently bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two point boundary value problems have no solutions under a new structure and certain conditions we show that a two point boundary value problem is equivalent to a fuzzy integral equation we also prove the existence of solutions to the two point boundary value problem in some sense this is an amendment to results of lakshmikantham et al and o regan et al and it is an answer to one of bede s problems

on the validity of the peano theorem for fuzzy differential equations abstract kaleva 6 attempted to derive the validity of the classical peano theorem over a locally compact subset of en d based on this attempt kloeden 8 concluded that the peano theorem is also valid over en dp this paper challenges both attempts by counterexamples

algorithmic design of diffractive optical systems for information processing abstract diffractive optical systems are used for information processing we pose the problem of diffractive computability which is equivalent to the factorization of a matrix into diagonal matrices and circulant matrices two instances of this problem will be answered which 2 m 2 m matrices can in principle be factored into diagonal and circulant factors which are the matrices that are optically realizable using up to three arbitrary diffractive elements we give a fast algorithm to solve the nonlinear equations involved in designing optical setups of the latter type

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

euler polygonal method for metric dynamical systems following some ideas developed for equations with a nonlinear differential we consider the existence and uniqueness of solution for the metric dynamical system f x 0 and we study the approximation under appropriate hypotheses of the unique solution using the euler polygonal approximation method in relation with function f

how real is your matrix for scalars there is essentially just one way to define reality real part and to measure nonreality in this paper various ways of defining respective concepts for complex entried matrices are considered in connection with this products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised multiplicative structures have applications for instance in diffractive optics preconditioning and fast fourier expansions

numerical solutions of fuzzy differential equations abstract 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

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian 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

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

approximating ideal diffractive optical systems in numerical wave optics involving diffractive optical elements one is required at some point to form a multiplicative matrix factorization this paper is concerned with such representations and related approximations of ideal diffractive optical systems various simple diffractive optical elements are used the basic factoring problem being highly nonlinear a minor parallelism in terms of beam splitters and beam combiners makes the task computationally much more tractable and versatile this gives rise to several matrix nearness problems of geometric nature

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

modular functions and the uniform distribution of cm points 1 zd i d 2 if d 0 mod 4 1 i d 2 if d 3 mod 4 the j function has the remarkable property that j zd is an algebraic integer of degree h d the class number of k q zd in fact k j zd is the hilbert class field of k 4 the first few values of j zd are j z3 0 j z4 12 j z7 153 j z8 20 j z11 323 j z15 191025 85995 5 2 j z19 963 j z20 632000 282880 5