on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

a foundation on semigroups of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations abstract in this article we study the strongly continuous semigroups of fuzzy valued operators defined on the space of triangular fuzzy numbers new notions of fuzzy infinitesimal generators and fuzzy resolvent operators of fuzzy semigroups are established in the sense of the generalized hukuhara difference a hille yosida like theorem in the framework of fuzzy metric space is investigated we demonstrate the efficiency of theoretical results by studying the existence of mild solutions of fuzzy nonlinear fractional evolution equations

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

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

on a new set valued stochastic integral with respect to semimartingales and its applications abstract we present a new approach to a concept of a set valued stochastic integral with respect to semimartingales such an integral called set valued stochastic up trajectory integral is compatible with the decomposition of the semimartingale some properties of this integral are stated we show applicability of the new integral in set valued stochastic integral equations driven by multidimensional semimartingales the uniqueness theorem is presented then we extend the notion of the set valued stochastic up trajectory integral to definition of a fuzzy stochastic up trajectory integral with respect to semimartingales a result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up trajectory integral driven by the multidimensional semimartingale is stated

existence results for extremal solutions of interval fractional functional integro differential equations abstract in this study we introduce interval fractional functional integro differential equations iffides under the caputo generalized hukuhara fractional differentiability we establish some necessary comparison results and use the monotone iterative technique combined with the method of upper and lower solutions to investigate the existence of extremal solutions for iffides an example is presented to illustrate the results

numerical approach for solution to an uncertain fractional differential equation abstract uncertain fractional differential equation ufde is of importance tool for the description of uncertain dynamic systems generally we may not obtain its analytic solutions in most cases this paper focuses on proposing a numerical method for solving ufde involving caputo derivative first the concept of path to an ufde with initial value conditions is introduced which is a solution of the corresponding fractional differential equation fde involving with the same initial value conditions then the relations between its solution and associate path are investigated besides a formula is derived for calculating expected value of a monotonic function with respect to solutions of ufdes based on the established relations numerical algorithms are designed finally some numerical experiments of nonlinear ufdes are given to demonstrate the effectiveness of the numerical algorithms

on fuzzy solutions for partial differential equations in this study we investigate heat wave and poisson equations as classical models of partial differential equations pdes with uncertain parameters considering the parameters as fuzzy numbers the fuzzy solution is built from fuzzification of the deterministic solution the continuity of the zadeh extension is used to obtain qualitative properties on regular a cuts of the fuzzy solution we prove the stability with respect to the initial boundary data and show that as time goes to zero the diameter of the fuzzy solution converges to zero and as a consequence to the cylindrical surface determined by the curve of the degree of membership numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method we theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time point for the heat and poisson equations and in a particular case for the wave equation the deterministic solution and the defuzzified surface intercept are numerically compared using the euclidean distance

tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose the oil palm frond a lignocellulosic material is a high yielding energy crop that can be utilized as a promising source of xylose it holds the potential as a feedstock for bioethanol production due to being free and inexpensive in terms of collection storage and cropping practices the aim of the paper is to calculate the concentration and yield of xylose from the acid hydrolysis of the oil palm frond through a fuzzy fractional kinetic model the approximate solution of the derived fuzzy fractional model is achieved by using a tau method based on the fuzzy operational matrix of the generalized laguerre polynomials the results validate the effectiveness and applicability of the proposed solution method for solving this type of fuzzy kinetic model a new fractional kinetic equation under uncertainty was addressed to depict the chemical reaction arising in palm oil frond an efficient numerical simulation based on a tau method was derived to solve the proposed kinetic equation different cases were solved to demonstrate the validity and efficiency of the proposed technique

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

existence of global solutions to nonlinear fuzzy volterra integro differential equations abstract in this paper we introduce new solutions for fuzzy differential equations as mixed solutions and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro differential operators of volterra type one example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions

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 fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

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

some properties of strong solutions to stochastic fuzzy differential equations we consider stochastic fuzzy differential equations driven by m dimensional brownian motion such equations can be useful in modeling of hybrid dynamic systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously under a boundedness condition which is weaker than linear growth condition and the lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations solutions which are fuzzy stochastic processes and their uniqueness are considered to be in a strong sense an estimation of error of the picard approximate solution is established we give a boundedness type result for the solution defined on finite time interval also the stabilities of solution on initial condition and coefficients of the equation are shown the existence and uniqueness of a solution defined on infinite time interval is proven finally some applications of fuzzy stochastic differential equations are considered all the results presented in this paper apply to set valued stochastic differential equations

an anisotropic error indicator based on zienkiewicz zhu error estimator application to elliptic and parabolic problems the anisotropic error indicator presented in m picasso comm numer methods engrg 19 2003 pp 13 23 in the frame of the laplace equation is extended to elliptic and parabolic problems our error indicator is derived using the anisotropic interpolation estimates of l formaggia and s perotto numer math 89 2001 pp 641 667 l formaggia and s perotto numer math 2002 doi 10 1007 s002110200415 together with a zienkiewicz zhu error estimator to approach the error gradient a numerical study of the effectivity index is proposed for elliptic diffusion convection and parabolic problems an adaptive algorithm is implemented aimed at controlling the relative estimated error

fuzzy and set valued stochastic differential equations with local lipschitz condition we are concerned with the fuzzy stochastic differential equations driven by multidimensional brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises under the uniform lipschitz condition we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations next we show assuming the lipschitz condition is satisfied only locally that these equations have a unique solution the fact that the solution is bounded is also proved we conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set valued stochastic differential equations with local lipschitz condition

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

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

on random fuzzy functional differential equations in this paper we prove the existence and uniqueness result for random fuzzy functional differential equations for the existence and uniqueness we use the method of successive approximations some kind of boundedness of the solution is established

adaptive t s fuzzy control design for fractional order systems with parametric uncertainty and input constraint abstract thanks to the superiority of fractional differential equations in modeling high technology dynamical real world systems the design and control of fractional order systems have been widely investigated in recent decades on the other hand supposing that dynamic equations of the fractional order systems usually include various parameters and nonlinear terms takagi sugeno t s fuzzy models with if then rules can used to characterize the systems dynamics also the control community knows that there are usually some nonlinearities such as dead zone backlash hysteresis and saturation in control actuators which can decrease the ideal prescribed performance of the fractional systems the saturation nonlinearity is perhaps the most commonly existed one in control applications of fractional nonlinear systems that should be compensated by a proper method so this article proposes a novel adaptive t s fuzzy variable structure control technique for nonlinear fractional order systems in spite of the saturating input and control fluctuations the parameters of the fractional system as well as those of the saturation function are presumed to be unknown accordingly update rules are extracted to estimate such fluctuations then simple adaptive linear like control rules are developed according to the t s fuzzy control theory the stability of the origin equilibrium point of the final closed loop system is assured by lyapunov stability theory at last comparative computer simulations are presented to highlight the accuracy and applicability of the proposed adaptive fuzzy saturated control methodology

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

on a system of fuzzy fractional differential inclusions with projection operators abstract the aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators which captures the desired features of both fuzzy differential inclusions and fractional order projective dynamical systems within the same framework the existence of solutions for the open situation is proved by using continuous selection theorem moreover the existence of solutions for the closed situation is showed by employing lipschitzean selection theorem and fixed point theorem respectively furthermore an application and a numerical example are also given

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

an a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes a new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes all considerations carry over to anisotropic triangular meshes with minor changes only the lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities in order to prove the upper error bound it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration to measure this correspondence a so called matching function is defined and its discussion shows it to be a useful tool with its help anisotropic interpolation estimates and subsequently the upper error bound are proven additionally it is pointed out how to treat robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes a numerical example supports the anisotropic error analysis

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

some results on boundary value problems for fuzzy differential equations with functional dependence abstract we calculate the exact solution for a class of boundary value problems for first order linear fuzzy differential equations with impulses under hukuhara differentiability by finding green s function some conclusions can be derived for impulsive fuzzy differential equations including a term with functional dependence for which existence and uniqueness results are proved

fuzzy stochastic differential systems abstract basing on the work of feng fuzzy sets and systems 102 1999 271 280 on mean square calculus for fuzzy stochastic processes we discuss in this paper the general theory of fuzzy stochastic differential systems including the existence and uniqueness of a solution the continuity of the solution with respect to the initial value and the stability of systems when there are perturbations of the coefficients and the initial conditions the explicit representation of solutions for the first order linear fuzzy stochastic systems are established

hyers ulam stability of fuzzy fractional volterra integral equations with the kernel ψ function via successive approximation method abstract in this paper we investigative the hyers ulam stability and the hyers ulam rassias stability of fuzzy fractional volterra integral equations ffvies involving the kernel function we propose conditions for the existence uniqueness and stability of ffvie by employing the method of successive approximation to visualize the theoretical results some examples and numerical simulations are given

anisotropic adaptive simulation of transient flows using discontinuous galerkin methods an anisotropic adaptive analysis procedure based on a discontinuous galerkin finite element discretization and local mesh modification of simplex elements is presented the procedure is applied to transient two and three dimensional problems governed by euler s equation a smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified the mesh metric field in smooth portions of the domain is controlled by a hessian matrix constructed using a variational procedure to calculate the second derivatives the transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain copyright c 2004 john wiley sons ltd

on set differential equations in banach spaces a second type hukuhara differentiability approach in this paper we consider set differential equations in banach spaces and with the second type hukuhara derivative we show that a sequence of successive approximations converges uniformly to a unique local solution of a cauchy problem some comparison results are formulated and a continuity of solution with respect to equation data is shown the existence of a global solution is also established

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

a posteriori error estimates based on the polynomial preserving recovery superconvergence of order o h 1 rho for some rho 0 is established for the gradient recovered with the polynomial preserving recovery ppr when the mesh is mildly structured consequently the ppr recovered gradient can be used in building an asymptotically exact a posteriori error estimator

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

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

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

interval cauchy problem with a second type hukuhara derivative in this paper we consider interval differential equations such the equations can be appropriate in modeling of dynamical systems under presence of uncertainty of parameters we study an interval initial value problem with a second type hukuhara derivative by an example of real world application we indicate the advantages of the usage of such a kind of interval valued derivative a continuous dependence of the solution on initial value and right hand side of the equation is shown the existence of approximate local solutions is proven and then it is used in the derivation of existence of at least one local solution to interval cauchy problem with second type hukuhara derivative the compactness of solutions set is also stated finally the explicit formulae for local solutions to linear interval differential equations are provided

some remarks on fuzzy differential equations via differential inclusions abstract in this paper we discuss the formulation and procedure for solving fuzzy differential equations via differential inclusions we give several examples showing the correct and incorrect procedure for solving fuzzy differential equations we show the connection between fuzzy differential equations and fuzzy differential inclusions finally we give some remarks on numerical algorithms for solving fuzzy differential equations via differential inclusions

adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives an anisotropic error estimator involving only first order derivatives is proposed for the laplace problem and continuous piecewise linear finite elements upper and lower bounds are presented the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching an anisotropic adaptive algorithm is then proposed with aim to equidistribute the error gradient in the directions of maximum and minimum stretching numerical results in two and three space dimensions show that the effectivity index is aspect ratio independent on such adapted meshes

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

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

the cauchy problem of fuzzy differential equations under generalized differentiability the generalization of the concept of h differentiability can be of great help in the dynamic study of fuzzy differential equations in this paper the concept of generalized differentiability is described from a new perspective on the basis of this concept the class of differentiable fuzzy set valued mappings is enlarged the cauchy problem for fuzzy differential equations is investigated in this enlarged setting as a result some new solutions are obtained the lengths of the support sets of these solutions may be non monotonic several examples are also shown

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

a note on initial value problems for fractional fuzzy differential equations abstract in this note we present some remarks on solutions of fractional fuzzy differential equation in general a fractional fuzzy differential equation and a fractional fuzzy integral equation are not equivalent in this paper we give an appropriate condition so that this equivalence is valid some examples are provided to illustrate the theory

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

approximation schemes for fuzzy stochastic integral equations we consider fuzzy stochastic integral equations with stochastic lebesgue trajectory integrals and fuzzy stochastic ito trajectory integrals some methods of construction of approximate solutions to such the equations are examined we study the picard type approximations the caratheodory type approximations and the maruyama type approximations of solutions in considered framework the solutions and approximate solutions are mappings with values in the space of fuzzy sets with basis of square integrable random vectors under lipschitz and linear growth conditions each sequence of considered approximate solutions converges to the exact unique solution of the given fuzzy stochastic integral equation for each type of approximate solutions we show that the sequence of approximations is uniformly bounded and obtain some bounds for a distance between nth approximation and exact solution

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985