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

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

an optimal order collocation method for first kind boundary integral equations on polygons this paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in bbb r 2 before discretization the equation is transformed to an equivalent equation over pi pi using a nonlinear parametrization of the polygon which varies more slowly than arc length near each corner this has the effect of producing a transformed equation with a solution which is smooth on pi pi this latter integral equation is shown to be well posed in appropriate sobolev spaces the structure of the integral operator is described in detail and can be written in terms of certain non standard mellin convolution operators using this information we are able to show that the collocation method using splines of order k degree k 1 converges with optimal order o h k the collocation points are the mid points of subintervals when k is odd and the break points when k is even and stability is shown under the assumption that the method may be modified slightly using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method the convergence results are illustrated with some numerical examples

partial stability of catastrophe sections and its application to the cuspoids and conic umbilics the control space of a system described by an elementary catastrophe frequently has locally at least a cartesian product structure caused for example by physical distinctions among control variables or by observational necessity the effect of this structure like that of symmetries can be to stabilise singular catastrophe sections partial unfoldings the authors define partial stability to be the structural stability of a stratum of a catastrophe section under perturbations respecting the product structure these ideas are illustrated by reference to a recent observation of double cusp diffraction and then applied to previous studies of singular catastrophe sections

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

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

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

numerical methods for integral equations of mellin type we present a survey of numerical methods based on piecewise polynomial approximation for integral equations of mellin type including examples arising in boundary integral methods for partial differential equations on polygonal domains

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

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

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

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

minimum stability control problem and time optimal control problem for fuzzy linear control systems abstract in this paper we give a new concept that is the minimum stability control problems for fuzzy linear control systems and our observations revolve around this object at the same time study the existence of solutions for some kinds of this problem by generalized hukuhara derivative in addition we present the time optimal fuzzy control problem for this problems an algorithm to find the time optimal is described and illustrated by a basic example

a numerical method and efficient preconditioner for generalized airfoil equations in this paper the generalized airfoil equations with cauchy algebraic singularity are researched by introducing the collocation method to conquer the singularity of the integral equation theoretical analysis and examples showed this method has accuracy o h m 1 with m the order of the interpolate polynomial in addition by separating the singular part into two so that they can offset against each other in the collocation points a new equivalent integrate function is also constructed moreover to accelerate the convergence of solving the obtained linear system a tridiagonal preconditioner is also presented for the singular integral operator by the operator splitting idea compared to the traditional tridiagonal preconditioner our method is more easily computed by an explicit inverse finally some numerical examples are presented to illustrate the behavior of the collocation method and this preconditioner

global polynomial approximation for symm s equation on polygons summary to solve 1d linear integral equations on bounded intervals with nonsmooth input functions and solutions we have recently proposed a quite general procedure that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials in particular the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap in these cases we have obtained arbitrarily high orders of convergence through the solution of very well conditioned linear systems in this paper to enlarge the domain of applicability of our technique we show how the above procedure can be successfully used also to solve the classical symm s equation on a piecewise smooth curve the collocation method we propose applied to the transformed equation and based on chebyshev polynomials of the first kind has shown to be stable and convergent a comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive

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

transformations for evaluating singular boundary element integrals accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights this paper presents a straightforward transformation to improve the accuracy of evaluating singular integrals the transformation is in a sense a generalisation of the popular method of telles with the underlying idea being to utilise the same gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation the new transformation is also shown to be equivalent to other existing transformations in certain situations comparison of the new method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained the technique can also be extended to evaluate certain hadamard finite part integrals based on the observation of several integrals considered guidelines are suggested for the best transformation order to use i e the degree to which nodes should be clustered near the singular point

a modified non linear transformation method for evaluating weakly singular boundary integrals accurate numerical evaluation of boundary integrals is fundamental to producing useful results with the boundary element method this paper introduces a generalisation of a recently introduced combined method subtraction of singularity followed by a non linear transformation which takes into account the effect of the basis functions the new method is applied to solve weakly singular integrals which arise in the solution of the two dimensional laplace equation the new method was found in the cases considered to be numerically superior to both the combined method and any of the non linear transformation methods

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

high order methods for linear functionals of solutions of second kind integral equations the authors show that linear functionals e g inner products or point functionals of solutions of second kind fredholm integral equations may be found within o 1 n 2r by very simple computational schemes using the nystrom or quadrature method based on piecewise polynomials of order r on an n 1 point mesh our theory covers a class of nonsmooth wiener hopf type equations and applications include the calculation of crack energy and stress intensity factors in certain linear elastic fracture problems and the use of the boundary integral method to compute various superconvergent solutions to boundary value problems in polygonal domains a proof of the uniform convergence of nystrom s method for certain boundary integral equations on polygonal domains is also included

a comparison of transformation methods for evaluating two dimensional weakly singular integrals accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique in this paper a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of laplace s equation in three dimensions and poisson s equation in two dimensions both are two dimensional weakly singular integrals and are evaluated using in a product fashion methods which have recently been used for evaluating one dimensional weakly singular integrals arising in the boundary element method the methods used are based on various polynomial transformations of conventional gaussian quadrature points where the transformation polynomial has zero jacobian at the singular point methods which split the region of integration into sub regions are considered as well as non splitting methods in particular the newly introduced and highly accurate generalized composite subtraction of singularity and non linear transformation approach gssnt is applied to various two dimensional weakly singular integrals a study of the different methods reveals complex relationships between transformation orders position of the singular point integration kernel and basis function it is concluded that the gssnt method gives the best overall results for the two dimensional weakly singular integrals studied

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

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

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 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

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

product integration collocation methods for noncompact integral operator equations we discuss the numerical solution of a class of second kind integral equations in which the integral operator is not compact such equations arise for example when boundary integral methods are applied to potential problems in a two dimensional domain with corners in the boundary we are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes provided some simple modifications are made to the underlying basis functions these are sufficient to ensure stability but do not damage the rate of convergence numerical experiments show that such modifications are necessary in certain circumstances

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

a polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non smooth boundaries the aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1d and 2d weakly singular and singular integral equations the introduction of a smoothing transformation besides smoothing the solution allows also the use of a very simple and efficient collocation method based on chebyshev polynomials of the first kind and their zeros further we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements which are given by weakly singular singular or nearly singular integrals several numerical tests are given to point out the efficiency of the numerical approach we propose copyright 2003 john wiley sons ltd

singularities of plane curves which occur as singular sections of the bifurcation sets of the cuspoid catastrophes plane curves specified parametrically by y sp x sq p and q integral p or q 0 are either graphs of monomials or have a singularity at the origin which can be one of four types cusp bend kink or end these types are classified by p q even odd or by a winding number index singular sections of the cuspoid bifurcation sets in which all but two of the control variables are set to zero are either cusps bends or one of four more singular curves derived from the above by the addition of the tangent at the origin an explicit classification is given of the curve types arising in codimension up to 7 only in such sections can the equations of the bifurcation sets generally be written explicitly

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

a collocation method for the generalized airfoil equation for an airfoil with a flap we are concerned with the numerical solution of the generalized airfoil equation for an airfoil with a flap by means of global algebraic polynomial approximants this problem has been considered in a recent paper by monegato and sloan siam j numer anal 34 1997 pp 2288 2305 where the authors proved the stability and convergence of a galerkin method based on high order polynomials they also presented very promising numerical results for the corresponding collocation method but left the theoretical results of stability and convergence as an open problem we have succeeded in proving the stability and convergence properties of this collocation method by slightly modifying it in the neighborhood of the flap the rate of convergence we have derived for our collocation method is very similar to that of the above galerkin method

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

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

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

a discrete collocation method for symm s integral equation on curves with corners this paper is devoted to the approximate solution of the classical first kind boundary integral equation with logarithmic kernel symm s equation on a closed polygonal boundary in 2 we propose a fully discrete method with a trial space of trigonometric polynomials combined with a trapezoidal rule approximation of the integrals before discretization the equation is transformed using a nonlinear mesh grading parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions the convergence results are illustrated with some numerical examples

a generalisation of telles method for evaluating weakly singular boundary element integrals abstract the accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights this paper presents a simple transformation to improve the accuracy of evaluating weakly singular integrals the transformation is in a sense a generalisation of the popular method of telles internat j numer methods eng 24 1987 959 973 with the underlying idea being to utilise the same gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation comparison of the generalised method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained based on observation of several integrals considered guidelines are suggested for the best transformation order to use i e the degree to which nodes should be clustered at the singular points an analysis of the generalised approach is also presented where asymptotic estimates for the truncation error after evaluating the transformed integrals using gauss legendre quadrature are determined further numerical experimentation shows that these are very good estimates

on polynomial collocation for second kind integral equations with fixed singularities of mellin type we consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of mellin convolution type using a stability result by junghanns and one of the authors we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and using the asymptotics of the solution we derive the rates of convergence finally we describe an algorithm to compute the stiffness matrix based on simple gaus quadratures and an alternative algorithm based on a recursion in the spirit of monegato and palamara orsi all together an almost best approximation to the solution of the integral equation can be computed with n2 log n 2 resp n2 operations where n is the dimension of the polynomial trial space

singular coordinate sections of the conic umbilic catastrophes studies the intersection of a normal form for the conic umbilic catastrophe of general codimension k with planes in control space having all but two control variables set to zero and display in a set of figures and a table the geometry in which strata of the bifurcation set intersect these planes and the singularity types occurring on them five distinct curve forms arise the authors compare these sections with those of the cuspoids and discuss their occurrence as optical caustics

numerical solution of the generalized airfoil equation for an airfoil with a flap in this paper we are concerned with the generalized airfoil equation which governs the pressure across an airfoil oscillating in a wind tunnel in particular we examine the case of an airfoil which contains a flap so that the input function has a simple jump at a point x0 1 x0 1 this generates a logarithmic singularity in the solution u at the same point x0 and considerably complicates the problem of the numerical approximation of u by introducing a nonlinear transformation of the independent variable in the integral equation we improve the behavior of the unknown function u through incorporation of the jacobian of the transformation the price is that the kernel of the new integral equation becomes more complicated nevertheless we are able to show that a galerkin method based on high order polynomials is both stable and convergent we also present some promising numerical results for the corresponding collocation method for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the galerkin method

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

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

theory and applications of fuzzy volterra integral equations formulations of fuzzy integral equations in terms of the aumann integral do not reflect the behavior of corresponding crisp models consequently they are ill adapted to describe physical phenomena even when vagueness and uncertainty are present a similar situation for fuzzy odes has been obviated by interpretation in terms of families of differential inclusions the paper extends this formalism to fuzzy integral equations and shows that the resulting solution sets and attainability sets are fuzzy and far better descriptions of uncertain models involving integral equations the investigation is restricted to volterra type equations with mildly restrictive conditions but the methods are capable of extensive generalization to other types and more general assumptions the results are illustrated by integral equations relating to control models with fuzzy uncertainties

convergence and stability of a collocation method for the generalized airfoil equation we establish the mean square convergence and numerical stability of a collocation method for solving a class of cauchy singular integral equations the method uses the jacobi polynomials p 1 2 1 2 n as basis elements and the zeros of the polynomial p 1 2 1 2 n 1 as collocation points uniform convergence is shown to hold for the first iterate convergence is proved via vainikko s embedding method and stability is demonstrated by generalizing the approach of atkinson for integral equations of the second kind

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

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

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

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 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