barycentric coordinates for polytopes in wachspress 1975 1 theory was developed for constructing rational basis functions for convex polygons and polyhedra these barycentric coordinates were positive within the elements generalization to higher space dimensions is described here the gadj algorithm developed by dasgupta 2003 5 and in dasgupta and wachspress 2008 6 is crucial for simple construction of rational barycentric basis functions

two stage approximation methods with extended b splines we develop and analyze a framework for two stage methods with eb splines applicable to continuous and discrete approximation problems in particular we propose a weighted discrete least squares fit that yields optimal convergence rates for sufficiently dense data on lipschitz domains in rd

statistical fuzzy approximation by fuzzy positive linear operators in this paper we prove a korovkin type approximation theorem for fuzzy positive linear operators by using the notion of a statistical convergence where a is a non negative regular summability matrix this type of approximation enables us to obtain more powerful results than in the classical aspects of approximation theory settings an application of this result is also given furthermore we compute the rates of this statistical fuzzy convergence of the operators via the fuzzy modulus of continuity

etching a two dimensional mathematical approach a mathematical model for the etching of a semi infinite active surface is presented the model assumes that the transport of the active species occurs solely by diffusion it is shown that when the diffusion field propagates much faster than the etched surface the problem can be solved by a singular perturbation technique which distinguishes a near field in the area where the moving surface and the non etchable mask meet and a far field where edge effects may be disregarded to first order the leading terms of a composite expansion are given from which the shape of the moving boundary can be determined at all times

quadratic serendipity finite elements over convex polyhedra summary the conventional approach to construct quadratic elements for an n sided polygon will yield n n 1 2 shape functions which increases the computational effort it is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion floater and lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates this approach ensures 2n shape functions with nodes only on the boundary of the polygon in this paper we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra this is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates the quadratic shape functions possess kronecker delta property and satisfy constant linear and quadratic precision the accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests the numerical results show that the quadratic serendipity elements pass the patch test yield optimal convergence rate and can tolerate extreme mesh distortion this article is protected by copyright all rights reserved

numerical solution of an etching problem abstract after a short description of the chemical background of etching processes a simplified mathematical model is formulated this contains two parameters that determine the etching profile two approaches for obtaining a numerical solution are discussed first the problem in terms of a variational inequality is reformulated and some new results on its numerical solution are presented then how the problem can be solved by means of a moving grid method as well is explored results obtained by both methods are presented and compared the latter method is applicable to a much wider class of boundary conditions but the variational inequality approach seems attractive for other reasons finally this work is compared with some experimental results

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

lacunary ideal convergence in intuitionistic fuzzy normed linear spaces the aim of this article is to introduce and study the notion of lacunary ideal convergence in intuitionistic fuzzy normed linear spaces as a variant of the notion of ideal convergence also a new concept called lacunary convergence is introduced using these two notions lacunary i limit points and lacunary i cluster points have been defined and the relation between them has been established characterization for lacunary ideal convergence preserving linear operators has been given furthermore the notions of lacunary cauchy and lacunary i cauchy sequences are introduced and studied

univariate fuzzy random neural network approximation operators in this article we study the rate of pointwise convergence in the q mean to thefuzzy random unit operator of very precise univariate fuzzy random neural network operators of cardaliaguet euvrard and squashing types these fuzzy random operators arise in a natural and common way among fuzzy random neural networks these rates are given through probabilistic jackson type inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function or its fuzzy derivatives also several interesting new results in fuzzy random analysis are given of independent merit which are used then in the proofs of the main results of the paper

rate of convergence of basic neural network operators to the unit univariate case this chapter deals with the determination of the rate of convergence to the unit of some neural network operators namely the normalized bell and squashing type operators

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

on the ideal convergence of sequences of fuzzy numbers in this paper we introduce and study the concepts of i convergence i convergence and i cauchy sequence for sequences of fuzzy numbers where i denotes the ideal of subsets of n the set of positive integers

meshless galerkin method using radial basis functions based on domain decomposition in this present paper we will give an overview about the combination of rbf meshless method with domain decomposition especially galerkin method based on the analysis for this method before we will consider the combination of projection domain decomposition method with radial basis function both of galerkin approximation and collocation method are discussed some examples are given to show the efficiency of this method

a comparison of numerical models for one dimensional stefan problems in this paper we present a critical comparison of the suitability of several numerical methods level set moving grid and phase field model to address two well known stefan problems in phase transformation studies melting of a pure phase and diffusional solid state phase transformations in a binary system similarity solutions are applied to verify the numerical results the comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques finally it is shown both numerically and analytically that the solid solid phase transformation is a limiting case of the solid liquid transformation

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

fitting monotone surfaces to scattered data using c 1 piecewise cubics we derive sufficient conditions on the bezier net of a bernstein bezier polynomial defined on a triangle in the plane to insure that the corresponding surface is monotone we then apply these conditions to construct a new algorithm for fitting a monotone surface to gridded data the method uses c1 cubic splines defined on the triangulation obtained by drawing both diagonals of each subrectangle in addition we present an algorithm for the monotone scattered data interpolation problem which is based on a method for creating gridded data from the scattered data numerical results for several test examples are presented

multilevel schwarz methods for the biharmonic dirichlet problem the author considers the solution of the algebraic system of equations that result from the finite element discretization of the biharmonic equation some multilevel algorithms are designed and analyzed using a schwarz framework both additive and multiplicative variants of the algorithms are considered and condition number estimates for the additive algorithms and the energy norm estimates for the error propagation operator of the multiplicative algorithms are given it is noted that for a proper ordering the iterative operators of the multiplicative algorithms correspond to the error propagation operators of certain v cycle multigrid methods

asymptotic and numerical solutions for the two dimensional solidification of a liquid half space this paper concerns the twodimensional inward solidification of a halfspace filled with molten material at its fusion temperature the temperature along half of the boundary surface is instantaneou

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

on schwarz s domain decomposition methods for elliptic boundary value problems summary we study the additive and multiplicative schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness r 1 the finite element method reduces an elliptic boundary value problem to a linear system of equations it is well known that as the number of triangles in the underlying triangulation is increased which is indispensable for increasing the accuracy of the approximate solution the size and condition number of the linear system increases the schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size we shall show in this paper that the approximate solutions from the multiplicative schwarz domain decomposition method converge to the exact solution of the linear system geometrically we also show that the additive schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method we tested these methods for the biharmonic equation with dirichlet boundary condition over an arbitrary polygonal domain using c 1 cubic spline functions over a quadrangulation of the given domain the computer experiments agree with our theoretical results

meshless collocation method based on dirichlet neumann substructure iteration in this paper we will combine the dirichlet neumann substructure iteration with so called meshless collocation method for pde using rbf the convergence of this method and numerical examples are given to show the efficiency

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

etching through a slit the problem of etching through a slit shaped orifice is formulated in mathematical terms although a full solution to this problem can be obtained only by the application of numerical methods it is shown that certain aspects of it can be dealt with by analytical means asymptotic solutions that are valid in certain parts of the time domain are derived early intermediate and late asymptotes are considered these analytical results are compared with numerical ones and a satisfactory agreement is found emphasizing the validity of the two approaches

a domain decomposition method for computing bivariate spline fits of scattered data a domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy discrete least squares and penalized least squares splines is described the method is based on splitting the domain into smaller domains solving the associated smaller fitting problems and combining the coefficients to get a global fit explicit error bounds are established for how well our locally constructed spline fits approximate the global fits some numerical examples are given to illustrate the effectiveness of the method

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

meshless galerkin method based on regions partitioned into subdomains in this paper we will combine the domain decomposition method with so called meshless galerkin method for pde using rbf the convergence of this new method and numerical examples are given

scaled boundary polygons for linear elastodynamics abstract a polygonal scaled boundary finite element method sbfem is proposed for linear elastodynamics in two dimensions the domain is divided into non overlapping polygonal elements and the scaled boundary finite element approach is employed over each polygon the advantages are that an arbitrary interpolation order can be used on each polygon and we can choose the optimal element order on each edge individually moreover the sbfem simplifies the numerical integration over the polygons when compared to conventional approaches the dynamic stiffness matrix is computed by employing the continued fraction expansion the influence of the shape of the polygon and the element order on the boundary of each polygon are studied the robustness and the accuracy of the approach are demonstrated with numerical examples

scattered data interpolation by bivariate splines with higher approximation order given a set of scattered data we usually use a minimal energy method to find a lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations it is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation to improve this order of approximation we propose several new schemes in this paper mainly we follow the ideas of clamped cubic interpolatory splines and not a knot interpolatory splines in the univariate setting and extend them to the bivariate setting in addition instead of the energy functional of the second order we propose to use higher order versions we shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

range restricted scattered data interpolation using convex combination of cubic bezier triangles abstract the construction of range restricted bivariate c 1 interpolants to scattered data is considered sufficient nonnegativity conditions on the bezier ordinates are derived to ensure that the nonnegativity of a cubic bezier triangular patch the gradients at the data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled the interpolating surface is piecewise a convex combination of three cubic bezier triangular patches with the same set of boundary bezier ordinates its construction is local and is easily extended to include as upper and lower constraints to the interpolant surfaces of the form z c x y where c x y is a constant linear quadratic or cubic polynomial moreover c 1 piecewise polynomial surfaces consisting of polynomial pieces of the form z c x y on the triangulation of the data sites are also admissible constraints a number of numerical examples are presented graphically

asymptotic solutions to the stefan problem with a constant heat source at the moving boundary in 1989 huppert formulated mathematical models of the melting and or freezing that occurs when a hot fluid begins to flow turbulently over a cold substrate he solved the resulting stefan problems numerically and presented certain asymptotic results this note revisits one of his problems and adds to the mathematical analysis of its asymptotic structure generalizations and relationships with other results are also noted

shape preserving bivariate interpolation a method for constructing shape preserving surfaces interpolating arbitrary sets of data on rectangular grids is presented the surfaces are tensor product splines of arbitrary continuity class which extend some properties of univariate shape preserving splines given in earlier papers

nonnegative data interpolation by spherical splines abstract we present a spherical spline method for scattered data interpolation over the unit sphere which preserves nonnegativity of the data values the method is based on a classic constrained minimization approach the usual side conditions of smoothness and data interpolation are supplemented by non negativity constraints we establish existence and uniqueness of non negative minimizers in three cases c 1 spline spaces of odd degree greater than or equal to five over generic triangulations c 1 cubic spline spaces over clough tocher triangulations c 1 cubic spline spaces over triangulations of convex quadrangulations we present the results on approximation order of nonnegative minimizers as well the method extends to range restricted interpolation we establish sufficient conditions on the spline coefficients that guarantee range restrictions on the spherical splines numerical solutions are computed by means of a projected gradient method numerical examples illustrate performance of non negative and range restricted data fitting

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

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

positivity preserving scattered data interpolation scheme using the side vertex method abstract a local surface interpolation scheme has been developed which interpolates and preserves the shape of the positive scattered data arranged over the triangular domain the proposed interpolating surface has been constructed by using the side vertex method a c 1 piecewise rational cubic function with four parameters has been used for interpolation along each edge of a triangle and along line segments joining a vertex to the opposite edge simple sufficient data dependent constraints have been derived on the free parameters in the description of rational cubic function to ensure the shape of positive scattered data

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

a new semi analytical method for phase transformations in binary alloys in the present paper a new semi analytical method is developed to cover a wide range of phase transformation problems and their practical applications the solution procedure consists of two parts first determination of the position of the moving boundary named the homogenous part and second determination of the concentration named the non homogenous part the homogenous part leads to a system of homogenous linear equations based on the mathematical fact that a homogenous system has a non trivial solution if the determinant of the coefficient matrix equals zero this determinant leads to an ordinary differential equation for the moving boundary and its solution leads to a closed form formula for the position of the moving boundary the non homogenous part transforms the governing equations to a non homogenous linear system of equations having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method solution of the non homogenous system leads to a value of these unknowns once these unknowns are computed the concentration at any time and at any point can be found easily

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

radial basis functions and level set method for image segmentation using partial differential equation combining nonlinear evolution equations which arise from image segmentation using partial differential equation based level set method using radial basis functions a meshless numerical algorithm is presented for image segmentation in this paper both globally supported and compactly supported radial basis functions are used to interpolate the level set function of the evolution equation with a high level of accuracy and smoothness the nonlinear evolution equation is finally cast into ordinary differential equations and euler s scheme is employed compared with traditional level set approaches the presented algorithm is robust to initialization or even free of manual initialization and avoids the complex and costly re initialization of the level set function the capability of the presented algorithm is demonstrated through some numerical experiments

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

compactly supported radial basis function based meshless method for photon propagation model of fluorescence molecular tomography fluorescence molecular tomography fmt is a powerful imaging modality for the research of cancer diagnosis disease treatment and drug discovery via three dimensional 3 d imaging reconstruction it can quantitatively and noninvasively obtain the distribution of fluorescent probes in biological tissues currently photon propagation of fmt is conventionally described by the finite element method fem and it can obtain acceptable image quality however there are still some inherent inadequacies in fem such as time consuming discretization error and inflexibility in mesh generation which partly limit its imaging accuracy to further improve the solving accuracy of photon propagation model ppm we propose a novel compactly supported radial basis functions csrbfs based meshless method mm to implement the ppm of fmt we introduced a series of independent nodes and continuous csrbfs to interpolate the ppm which can avoid complicated mesh generation to analyze the performance of the proposed mm we carried out numerical heterogeneous mouse simulation to validate the simulated surface fluorescent measurement then we performed an in vivo experiment to observe the tomographic reconstruction the experimental results confirmed that our proposed mm could obtain more similar surface fluorescence measurement with the golden standard monte carlo method and more accurate reconstruction result was achieved via mm in in vivo application

boundary concentrated finite element methods a method with optimal up to logarithmic terms complexity for solving elliptic problems is proposed the method relies on interior regularity but the solution may have globally low regularity due to rough boundary data or geometries elliptic regularity results high order approximation results and an efficient preconditioner are presented the method is utilized to realize with linear logarithmic complexity an accurate and data sparse approximation to the associated elliptic poincare steklov operators further applications include the treatment of exterior boundary value problems and the solution of problems in the framework of domain decomposition methods

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

analytical and numerical solutions to two dimensional moving interface problems with applications to the solidification of killed steel ingots this paper is concerned with the development of methods for investigating moving interface problems that involve more than one space variable the work was motivated by problems arising from the british steel corporation scunthorpe concerning the solidification of killed steel ingots first we investigate certain two dimensional analytical solutions which simulate particular aspects of ingot solidification these relate to the inward solidification of a liquid half space either when the wall temperature varies with position along the wall or when newton cooling takes place at the wall with the local heat transfer coefficient a function of wall position other approximate analytical approaches with use of the heat balance integral method and formulation in terms of an integro differential equation are discussed briefly and their limited application to two dimensional problems noted a computer procedure is then developed for solving these half space problems and this yields results in exact agreement with the analytical solutions in the work special attention is given to obtaining reliable information for small stefan number beta l c p t f t 0 since this relates to values found in the steel making process in the major part of the paper a two dimensional mathematical model of ingot solidification is developed numerical solutions are given for an ingot when the mould surface varies in geometry and in composition good agreement is obtained with radio isotope tracer tests performed at the british steel corporation scunthorpe

the spectral projection decomposition method for elliptic equations in two dimensions the projection decomposition method pdm is invoked to extend the application area of the spectral collocation method to elliptic problems in domains compounded of rectangles theoretical and numerical results are presented demonstrating the high accuracy of the resulting method as well as its computational efficiency

quadratic serendipity finite elements on polygons using generalized barycentric coordinates we introduce a nite element construction for use on the class of convex planar polygons and show it obtains a quadratic error convergence estimate on a convex n gon satisfying simple geometric criteria our construction produces 2n basis functions associated in a lagrange like fashion to each vertex and each edge midpoint by transforming and combining a set of n n 1 2 basis functions known to obtain quadratic convergence the technique broadens the scope of the so called serendipity elements previously studied only for quadrilateral and regular hexahedral meshes by employing the theory of generalized barycentric coordinates uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges numerical evidence is provided on a trapezoidal quadrilateral mesh previously not amenable to serendipity constructions and applications to adaptive meshing are discussed

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

a conserving discretization for the free boundary in a two dimensional stefan problem the dissolution of a disk likeal2cuparticle is considered a characteristic property is that initially the particle has a nonsmooth boundary the mathematical model of this dissolution process contains a description of the particle interface of which the position varies in time such a model is called a stefan problem it is impossible to obtain an analytical solution for a general two dimensional stefan problem so we use the finite element method to solve this problem numerically first we apply a classical moving mesh method computations show that after some time steps the predicted particle interface becomes very unrealistic therefore we derive a new method for the displacement of the free boundary based on the balance of atoms this method leads to good results also for nonsmooth boundaries some numerical experiments are given for the dissolution of anal2cuparticle in anal cualloy

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

polygonal spline spaces and the numerical solution of the poisson equation it is known that generalized barycentric coordinates can be used to form bernstein polynomial like functions over a polygon with any number of sides we propose to use these functions to form a space of continuous polygonal splines piecewise defined functions of order d over a partition consisting of polygons which is able to reproduce all polynomials of degree d locally supported basis functions for the space are constructed for order d ge 2 the construction for d 2 is simpler than the serendipity quadratic finite elements that have appeared in the recent literature the number of basis functions is similar to but fewer than those of the virtual element method we use them for the numerical solution of the poisson equation on two special types of nontriangular partitions to present a proof of concept for solving pdes over polygonal partitions numerical solutions based on quadrangulations and pentagonal partitions are demonstrated to show the efficiency of these polygonal spline functions

spherical splines for data interpolation and fitting we study minimal energy interpolation and discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines

fuzzy approximation by fuzzy convolution type operators here we introduce and study four sequences of naturally arising fuzzy integral operatorsof convolution type that are integral analogs of known fuzzy wavelet type operators defined via a scaling function their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity also their high order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the n t h order n 1 h fuzzy derivative of the engaged fuzzy number valued function the fuzzy global smoothness preservation property of these operators is demonstrated also

arbitrary order nodal mimetic discretizations of elliptic problems on polygonal meshes we develop and analyze a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form these methods are derived from the local consistency condition that is exact for polynomials of any degree m geq1 the degrees of freedom are a solution values at the quadrature nodes of the gauss lobatto formulas on each mesh edge and b solution moments inside polygons the convergence of the method is proven theoretically and an optimal error estimate is derived in a mesh dependent norm that mimics the energy norm numerical experiments confirm the convergence rate that is expected from the theory

a mixed h 1 conforming finite element method for solving maxwell s equations with non h 1 solution in this paper we propose and analyze a mixed h 1 conforming finite element method for solving maxwell s equations in terms of electric field and lagrange multiplier where the multiplier is intr

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

a direct method for solving two dimensional one phase stefan problems abstract a new algorithm is proposed for solving the stefan problem for the case when the initial region occupied by the medium in a given phase is a starshaped two dimensional domain the evolution of this region as time increases is determined by plotting the free boundary directly at each time step this is approximated by a polygon in the interior of which the heat equation is solved with piecewise linear finite elements numerical experiments indicate that the method is stable and that convergence is to be expected under suitable assumptions on the regularity of both the boundary and initial data of the problem

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

high order fuzzy approximation by fuzzywavelet type and neural network operators here are studied in terms of fuzzy high approximation to the unit several basic sequences of fuzzy wavelet type operators and fuzzy neural network operators these operators are fuzzy analogs of earlier studied real ones the produced results generalize earlier real ones into the fuzzy setting here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the nth order n 1 h fuzzy derivative of the engaged fuzzy number valued function at the end we present a related lp result for fuzzy neural network operators

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

error estimates in w 2 semi norms for discrete interpolating d 2 splines we derive error estimates in w 2 semi norms for multivariate discrete d 2 splines that interpolate an unknown function at the vertices of given triangulations these results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

quasi interpolation projectors for box splines we consider box spline quasi interpolants based on local linear functionals of point evaluator and integral type the approximations are easy to compute and reproduce the whole spline space in question

on henstock integral of fuzzy number valued functions i abstract in this paper we define and discuss the fh integral for fuzzy number valued functions using a concrete structure into which we embed the fuzzy number space e 1 several necessary and sufficient conditions of integrability for fuzzy number valued functions are given by means of abstract function theory

norms of projectors onto spaces with riesz bases abstract many important linear operators p x s of a linear space x onto a subspace s of x are defined by the minimum problem f p f min f u u s f x where the norm on x is induced by an inner product we study the l norm of such operators p when s has a riesz basis b then we apply the results in detail to the best l2 and l2 approximations for univariate polynomial splines spaces and to box spline series

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables