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

on solutions to fuzzy stochastic differential equations with local martingales abstract this paper deals with a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the lipschitzian condition such equations can be useful in modeling hybrid systems where the phenomena are simultaneously subjected to two kinds of uncertainties randomness and fuzziness the solutions of the fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to the fsdes under the lipschitzian condition is first proven moreover some asymptotic properties of the solutions to the fsdes are investigated finally an illustrating example on the interest term model is provided

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

coalgebras in functional programming and type theory this is a survey article on the use of coalgebras in functional programming and type theory it presents the basic theory underlying the implementation of coinductive types families and predicates it gives an overview of the application of corecursive methods to the study of general recursion formal power series tabulations of functions on inductive data it also sketches some advanced topics in the study of the solutions to non guarded corecursive equations and the design of non standard type theory

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

hukuhara differentiability of interval valued functions and interval differential equations on time scales abstract using the concept of the generalized hukuhara difference in this paper we introduce and study the differentiability and the integrability for the interval valued functions on time scales some illustrative examples to interval differential equations on time scales are presented

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

differential equations with fuzzy parameters via differential inclusions abstract we give a definition of solutions of ordinary differential equations in r n containing parameters which are described by changing in time fuzzy sets they are defined as fuzzy subsets of the space of absolutely continuous functions we introduce a hypograph metric in the space of fuzzy sets and prove a theorem on continuous dependence of fuzzy solutions on parameters and initial conditions with respect to that metric

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

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

the interrelation between stochastic differential inclusions and set valued stochastic differential equations abstract in this paper we connect the well established theory of stochastic differential inclusions with a new theory of set valued stochastic differential equations solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty bounded convex and closed subsets of the space l 2 consisting of square integrable random vectors we show that for the solution x to a set valued stochastic differential equation corresponding to a stochastic differential inclusion there exists a solution x for this inclusion that is a l 2 continuous selection of x this result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions as well as to consider the viability problem for stochastic differential inclusions

a stacking theorem approach for fuzzy differential equations abstract this paper presents new existence results for fuzzy differential equations our analysis combines the stacking theorem with new solution set results for ordinary differential equations

analysis of first order errors in shock calculations in two space dimensions numerical computations show that solutions of hyperbolic conservation laws obtained by second or higher order shock capturing methods in many cases are only first order accurate downstream of shocks see e g m h carpenter and j h casper aiaa j 37 1999 pp 1072 1079 we use matched asymptotic expansions to analyze the degeneration in order of accuracy for stationary solutions of hyperbolic conservation laws in two space dimensions

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

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

mean square integral and differential of fuzzy stochastic processes abstract this paper defines the concepts of integral and differential in mean square associated with a class of fuzzy stochastic processes and studies their integrability and differentiability properties the definition and the mean square calculus properties of a gaussian fuzzy stochastic process are also discussed

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

a critical analysis of the modified equation technique of warming and hyett abstract warming and hyett developed a modified equation technique in which the behavior of a difference scheme is evaluated by using the coefficients of a certain modified equation specifically they discovered a connection between these coefficients and the multiplication factor obtained from the von neumann analysis since the dissipation and dispersion off error components are determined by the multiplication factor the former properties can be studied using the coefficients of the modified equation the work of warming and hyett represents a key step in the development of the method of modified equations through this work it became clear that modified equations should be derived from the difference scheme rather than from the original differential equation however in order to prove the above connection warming and hyett incorrectly interpreted their modified equations as the actual partial differential equations solved by the difference schemes the main purpose of the current study is to investigate rigorously the above connection without using their interpretation the result of this investigation shows that the above connection is only partially valid for multilevel schemes in the von neumann analysis the multiplication factor associated with a wave number generally has l 1 roots for an l level scheme it is shown that the coefficients of the modified equation provide information for only the principal root

adiabatic perturbations for compactons under dissipation and numerically induced dissipation compacton propagation under dissipation shows amplitude damping and the generation of tails the numerical simulation of compactons by means of dissipative schemes also show the same behaviors the truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis for dissipative schemes or when artificial dissipation is added the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically induced tails in this paper such methods are applied to the k 2 2 rosenau hyman equation showing a very good agreement between perturbative and numerical results

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

on a superconvergent lattice boltzmann boundary scheme in a seminal paper 20 ginzburg and adler 1994 analyzed the bounce back boundary conditions for the lattice boltzmann scheme and showed that it could be made exact to second order for the poiseuille flow if some expressions depending upon the parameters of the method were satisfied thus defining so called magic parameters using the taylor expansion method that one of us developed we analyze a series of simple situations 1d and 2d for diffusion and for linear fluid problems using bounce back and anti bounce back numerical boundary conditions the result is that magic parameters depend upon the detailed choice of the moments and of their equilibrium values they may also depend upon the way the flow is driven

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

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

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

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

the validity of the modified equation for nonlinear shock waves abstract the modified model equivalent equation is an important tool in designing and analyzing nonlinear difference schemes in this note the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case

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

the principle of the difference of difference quotients as a key to the self adaptive solution of nonlinear partial differential equations abstract a unified method for the numerical solution of nonlinear ordinary and partial differential equations elliptic and parabolic is developed the discretization error is estimated from the difference of difference quotients of a family of difference formulae the difference grid and the difference star of arbitrary order are self adapted and selected so that an optimum method results in which all error terms are well balanced to a prescribed relative tolerance

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

viability theory and fuzzy differential equations new results are presented which guarantee the existence of viable solutions to differential inclusions we show that a viable solution exists if i a tangency condition is satisfied and ii there exists a maximal solution to an appropriate differential equation in addition we discuss the topological structure of the solution set these results will then be used with the stacking theorem to establish new existence criteria for fuzzy differential equations if there exists a maximal solution for an appropriate set of differential equations

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

kripke semantics for higher order type theory applied to constraint logic programming languages abstract we define a kripke semantics for intuitionistic higher order logic with constraints formulated within church s theory of types via the addition of a new constraint base type we then define an executable fragment hohh c of the logic a higher order logic programming language with typed abstraction implication and universal quantification in goals and constraints and give a modified model theory for this fragment both formal systems are shown sound and complete for their respective semantics we also solve the impredicativity problem in prolog semantics namely how to give a definition of truth without appealing to induction on subformula structure in the last section we give a simple semantics based conservative extension proof that the language hohh c satisfies a uniformity property along the lines of 39

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

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

on the method of modified equations iii numerical techniques based on the second equivalent equation for the euler forward difference method new numerical successive correction techniques for ordinary differential equations based on the euler forward explicit method and the first modified or equivalent equation are presented these techniques are similar to iterative updating deferred methods and are based on the application of asymptotic methods to modified equations which do not require initial conditions for the high order derivatives in the truncation terms and which yield stable numerical methods it is shown that depending on the discretization of the high order derivatives in the high order correction equations different methods of as high order of consistency as required can be developed in this paper backward and centered formulas are used but the resulting numerical methods are not self starting it is shown that if the starting procedure is not adequate the numerical order of the method can be smaller than the theoretical one in order to avoid this loss of numerical order a method for consistently starting the asymptotic successive correction technique based on the use of fictitious times is presented and applied to autonomous and non autonomous ordinary differential equations and compared with the results of second and fourth order runge kutta methods it is shown that the fourth order runge kutta method is more accurate than the successive correction techniques for large time steps due to the higher order derivatives in the successive correction but for sufficiently small time steps these techniques have almost the same accuracy as the fourth order runge kutta method

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

on the method of modified equations v asymptotic analysis of and direct correction and asymptotic successive correction techniques for the implicit midpoint method the equivalent second equivalent and simply modified equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step size as small parameter yields exactly the same results for linear problems with constant coefficients they are also equivalent to the original finite difference scheme straightforward regular multiple scales and summed up asymptotic techniques are used for the analysis of the implicit midpoint rule difference method and the accuracy of the resulting asymptotic expansion is assessed for several first order non linear autonomous ordinary differential equations it is shown that when the resulting asymptotic expansion is uniformly valid the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up the modified equation is also studied as a method for the development of new numerical schemes based on both direct correction and asymptotic successive correction techniques applied to the three kinds of modified equations the linear stability of these techniques is analyzed and their results are compared with those of runge kutta schemes for several autonomous and non autonomous first order ordinary differential equations

an investigation of the internal structure of shock profiles for shock capturing schemes the theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions so called continuum shock profiles however the previous papers have focused on linear methods we have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes including riemann solver based flux limiter methods high resolution central schemes and eno type methods in some cases no continuum profiles exists however in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle the possibility of such behavior was anticipated in bultelle grassin and serre 1998 but no specific examples or other evidence of this behavior have previously been given in other cases our results indicate that continuum shock profiles exist but are very complicated we also study phase space orbits with regard to post shock oscillations

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

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

set valued stochastic integral equations driven by martingales abstract we consider a notion of set valued stochastic lebesgue stieltjes trajectory integral and a notion of set valued stochastic trajectory integral with respect to martingale then we use these integrals in a formulation of set valued stochastic integral equations the existence and uniqueness of the solution to such the equations is proven as a generalization of set valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales

self similar radiation from numerical rosenau hyman compactons the numerical simulation of compactons solitary waves with compact support is characterized by the presence of spurious phenomena as numerically induced radiation which is illustrated here using four numerical methods applied to the rosenau hyman k p p equation both forward and backward radiations are emitted from the compacton presenting a self similar shape which has been illustrated graphically by the proper scaling a grid refinement study shows that the amplitude of the radiations decreases as the grid size does confirming its numerical origin the front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters the amplitude of the radiations decreases exponentially in time being characterized by a nearly constant scaling exponent an ansatz for both the backward and forward radiations corresponding to a self similar function characterized by the scaling exponent is suggested by the present numerical results

on the algebraic structure of declarative programming languages we develop an algebraic framework logic programming doctrines for the syntax proof theory operational semantics and model theory of horn clause logic programming based on indexed premonoidal categories our aim is to provide a uniform framework for logic programming and its extensions capable of incorporating constraints abstract data types features imported from other programming language paradigms and a mathematical description of the state space in a declarative manner we define a new way to embed information about data into logic programming derivations by building a sketch like description of data structures directly into an indexed category of proofs we give an algebraic axiomatization of bottom up semantics in this general setting describing categorical models as fixed points of a continuous operator

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 the method of modified equations vi asymptotic analysis of and asymptotic successive corrections techniques for two point boundary value problems in ode s the modified equation technique is extended to two point boundary value problems and a second order accurate implicit centered finite difference scheme for non homogeneous second order ordinary differential equations with linear boundary conditions is analyzed the first second and third modified equations or equivalent second equivalent and simply modified equations respectively for this scheme and its boundary conditions are presented it is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results for a linear problem multiple scales and summed up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme asymptotic successive corrections techniques are also applied to the three kinds of modified equations to obtain higher order schemes higher order boundary conditions are easily treated in the asymptotic successive corrections technique although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries the methods introduced in this paper are applied to homogeneous and non homogeneous second order linear and non linear ordinary differential equations and yield very accurate results

on the method of modified equations i asymptotic analysis of the euler forward difference method the method of modified equations is studied as a technique for the analysis of finite difference equations the non uniqueness of the modified equation of a difference method is stressed and three kinds of modified equations are introduced the first modified or equivalent equation is the natural pseudo differential operator associated to the original numerical method linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modified or second equivalent equation and the third modified or simply modified equation respectively for linear problems with constant coefficients the three kinds of modified equations are equivalent among them and to the original difference scheme for nonlinear problems the three kinds of modified equations are asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step as small parameter yields exactly the same results in this paper both regular and multiple scales asymptotic techniques are used for the analysis of the euler forward difference method and the resulting asymptotic expansions are verified for several nonlinear autonomous ordinary differential equations it is shown that when the resulting asymptotic expansion is uniformly valid the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up even for large step sizes

a theory of observables for logic programs we define a semantic framework to reason about properties of abstractions of sld derivations the framework allows us to address problems such as the relation between the top down operational semantics and the bottom up denotational semantics the existence of a denotation for a set of definite clauses and their properties compositionality w r t various syntactic operators correctness minimality and precision using abstract interpretation techniques to model abstraction allows us to state very simple conditions on the observables which guarantee the validity of several general theorems

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

uncertain systems and fuzzy differential equations the study of fuzzy differential equations is initiated and some typical results are given in order to understand the intricacies involved in incorporating fuzziness in the terms of differential equations

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

time stepping in petrov galerkin methods based on cubic b splines for compactons abstract four numerical methods with first to fourth order of accuracy have been developed for the time integration of the rosenau hyman k 2 2 equation the error in the solution and the invariants for the propagation of one compacton and the stability in collisions among compactons have been studied using these methods numerically induced radiation has also been characterized by means of wavefront velocity and wavefront amplitude showing that the self similarity of the radiation wavepackets observed in the numerical results is a consequence of the time stepping method among the four methods studied in this paper the best results in terms of accuracy computational cost and stability have been obtained by means of using the second order time integration method

focusing and polarization in linear intuitionistic and classical logics a focused proof system provides a normal form to cut free proofs in which the application of invertible and non invertible inference rules is structured within linear logic the focused proof system of andreoli provides an elegant and comprehensive normal form for cut free proofs within intuitionistic and classical logics there are various different proof systems in the literature that exhibit focusing behavior these focused proof systems have been applied to both the proof search and the proof normalization approaches to computation we present a new focused proof system for intuitionistic logic called ljf and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing we also use ljf to design a focused proof system lkf for classical logic our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in girard s lc and lu proof 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

existence and interrelation between set and fuzzy differential equations abstract an existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general then the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it is studied

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

quartic parameters for acoustic applications of lattice boltzmann scheme using the taylor expansion method we show that it is possible to improve the lattice boltzmann method for acoustic applications we derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy the corresponding discrete equations are solved with the help of symbolic manipulation the solutions are obtained in the case of d3q27 lattice boltzmann scheme various numerical tests support the coherence of this approach

the initial value problem for interval valued second order differential equations under generalized h differentiability in this paper the interval valued second order differential equations under generalized hukuhara differentiability isdes are introduced under suitable conditions we obtain the existence and uniqueness results of solutions to isdes to prove this assertion we use idea of contraction principle and successive approximations furthermore we use the method based on properties of linear transformations ltm to find the explicit solution of initial value problem for linear second order differential equation with interval valued forcing function and with interval initial values iivp we apply the linear transformations method to two example problems including a vibrating mass and an electrical circuit finally the method based on the analysis of a solution to real valued second order differential equation is investigated to solve interval valued second order differential equations with interval valued coefficients interval valued forcing function and interval initial values some examples are presented to illustrate applicability of the proposed method

numerical methods based on modified equations for nonlinear evolution equations with compactons abstract compactons are traveling wave solutions with compact support resulting from the balance of both nonlinearity and nonlinear dispersion numerical methods with second fourth sixth and eighth order approximations to the spatial derivatives obtained by means of the method of modified equations applied to the ismail taha finite difference scheme for the rosenau hyman equation are developed the whole set of methods is compared among them in accuracy invariant conservation and in compacton collisions the best method among those studied in terms of the tradeoff between accuracy and computational cost is determined

high order accurate finite difference schemes based on symmetry preservation abstract in this paper we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve lie symmetry groups of underlying partial differential equations pdes in the proposed approach invariant or symmetry preserving numerical schemes with a desired or fixed order of accuracy are constructed from some non invariant base numerical schemes modified forms of pdes are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of pdes these defect correction terms of modified pdes that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames this feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations the proposed method is demonstrated via construction of invariant numerical schemes with fixed and higher order of accuracy for some common linear and nonlinear problems including the linear advection diffusion equation in 1d and 2d inviscid burgers equation and viscous burgers equation and the performance of these invariant numerical schemes is further evaluated our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying pdes

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

the von neumann analysis and modified equation approach for finite difference schemes the von neumann discrete fourier analysis and modified equation technique have been proven to be two effective tools in the design and analysis of finite difference schemes for linear and nonlinear problems the former has merits of simplicity and intuition in practical applications but only restricted to problems of linear equations with constant coefficients and periodic boundary conditions the later pde approach has more extensive potential to nonlinear problems and error analysis despite its kind of relative complexity the dissipation and dispersion properties can be observed directly from the pde point of view even order terms supply dissipation and odd order terms reflect dispersion in this paper we will show rigorously their full equivalence via the construction of modified equation of two level finite difference schemes around any wave number only in terms of the amplification factor used in the von neumann analysis such a conclusion fills in the gap between these two approaches in literatures

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

asymptotic analysis of finite difference methods with this article we want to advocate the use of asymptotic methods for the analysis of finite difference schemes we present several examples to demonstrate the applicability of the approach advantages over the modified equation and truncation error analysis are pointed out