spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

some quadrature formulae with nonstandard weights in this paper the authors study truncated quadrature rules based on the zeros of generalized laguerre polynomials then they prove the stability and the convergence of the introduced integration rules some numerical tests confirm the theoretical results

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

numerical treatment of a class of systems of fredholm integral equations on the real line in this paper the authors propose a nystrom method based on a truncated gaussian rule to solve systems of fredholm integral equations on the real line they prove that it is stable and convergent and that the matrices of the solved linear systems are well conditioned moreover they give error estimates in weighted uniform norm and show some numerical tests fredholm integral equations nystrom method truncated gaussian rule

error analysis of nitsche s mortar method optimal a priori and a posteriori error estimates are derived for three variants of nitsche s mortar finite elements the analysis is based on the equivalence of nitsche s method and the stabilised mixed method nitsche s method is defined so that it is robust with respect to large jumps in the material and mesh parameters over the interface numerical results demonstrate the robustness of the a posteriori error estimators

error estimates in the numerical evaluation of some bem singular integrals in some applications of galerkin boundary element methods one has to compute integrals which after proper normalization are of the form b 2 x y dxdy a 1 x y y where a b 1 1 or a b a 1 or a b 1 b and f x y is a smooth function in this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p or h p formulation of a galerkin method is used this approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the cauchy kernel and the outer integral by a rule which takes into account the log endpoint singularities of its integrand some numerical examples are also given

a lagrange type projector on the real line we introduce an interpolation process based on some of the zeros of the mth generalized freud polynomial convergence results and error estimates are given in particular we show that in some important function spaces the interpolating polynomial behaves like the best approximation moreover the stability and the convergence of some quadrature rules are proved

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

gaussian quadrature rules with exponential weights on 1 1 we study the behavior of some truncated gaussian rules based on the zeros of pollaczek type polynomials these formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces moreover we apply these results to the related lagrange interpolation process and to prove the stability and the convergence of a nystrom method for fredholm integral equations of the second kind finally some numerical examples are shown

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

approximation of the hilbert transform on the real line using hermite zeros the authors study the hilbert transform on the real line they introduce some polynomial approximations and some algorithms for its numerical evaluation error estimates in uniform norm are given

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

convergence of product integration rules over 0 for functions with weak singularities at the origin in this paper we consider integrals of the form 0 e x k x y f x dx with f c p 0 c q 0 q p 0 and x i f p 1 x c 0 i 1 q p when q p they appear for instance in certain wiener hopf integral equations and are of interest if one wants to solve these by a nystrom method to discretize the integral above we propose to use a product rule of interpolatory type based on the zeros of laguerre polynomials for this rule we derive weighted uniform convergence estimates and present some numerical examples

on mean convergence of hermite fejer and hermite interpolation for erdos weights abstract we investigate convergence of hermite fejer and hermite interpolation polynomials in l p 0 for erdos weights

on the convergence of t sum series of fuzzy numbers abstract in this paper we study the existence of the limit of the series of fuzzy numbers with different spreads and different shape functions where addition is defined by the sup t norm and we show the uniform continuity of the limit this generalizes the earlier results of hong fuzzy sets and systems 77 1996 253 254

stieltjes polynomials and lagrange interpolation bounds are proved for the stieltjes polynomial e n 1 and lower bounds are proved for the distances of consecutive zeros of the stieltjes polynomials and the legendre polynomials p n this sharpens a known interlacing result of szego as a byproduct bounds are obtained for the geronimus polynomials g n applying these results convergence theorems are proved for the lagrange interpolation process with respect to the zeros of e n 1 and for the extended lagrange interpolation process with respect to the zeros of p n e n 1 in the uniform and weighted l p norms the corresponding lebesgue constants are of optimal order

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

projection methods and condition numbers in uniform norm for fredholm and cauchy singular integral equations in this paper the authors propose a numerical method for the approximate solution of some classes of fredholm and cauchy integral equations including the discrete collocation and collocation methods

symmetric and non symmetric variants of nitsche s method for contact problems in elasticity theory and numerical experiments a general nitsche method which encompasses symmetric and non symmetric variants is proposed for frictionless unilateral contact problems in elasticity the optimal convergence of the method is established both for two and three dimensional problems and lagrange affine and quadratic finite element methods two and three dimensional numerical experiments illustrate the theory

noninterpolatory integration rules for cauchy principal value integrals etude de la convergence des regles d integration interpolat pour l approximation des integrales de cauchy

the numerical computation of some integrals on the real line abstract we discuss the approximation of integrals of type i f t r f x k x t e x 2 x d x 1 k is the weakly singular algebraic kernel x t 1 for large value of the parameter t moreover we consider strongly oscillatory kernels of type k 1 x t sin tx 2 k 2 x t cos tx 2 weighted error estimates in uniform and l1 norm are stated and numerical examples to confirm the efficiency of the proposed procedures are given

some weighted polynomial inequalities abstract the author considers a weight function of sub exponential type with zeros and singularities in 1 1 some inequalities of remez bernstein markov and nikolsky type are proved the results hold for polynomials and generalized polynomials moreover an upper and a lower bound for the christoffel functions are derived

on the convergence of product sum series of l r fuzzy numbers we study series a1 a2 an of l r fuzzy numbers where addition is defined via the sup product norm convulation and obtain some convergence theorems which generalize earlier results by fuller cf 2

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

the penalty free nitsche method and nonconforming finite elements for the signorini problem we design and analyse a nitsche method for contact problems compared to the seminal work of chouly and hild siam j numer anal 51 2013 pp 1295 1307 our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces the contact condition is then imposed using the nonsymmetric variant of nitsche s method that does not require a penalty term for stability nonconforming piecewise affine elements are considered for the bulk discretization we prove optimal error estimates in the energy norm

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

numerical methods for fredholm integral equations on the square abstract in this paper we shall investigate the numerical solution of two dimensional fredholm integral equations by nystrom and collocation methods based on the zeros of jacobi orthogonal polynomials the convergence stability and well conditioning of the method are proved in suitable weighted spaces of functions some numerical examples illustrate the efficiency of the methods

a note on the convergence of t sum series of l r fuzzy numbers abstract we prove the existence of the limit of series a 1 a 2 a n of l r fuzzy numbers where addition is defined via the sup t norm convolution and show uniform continuity of the limit this generalizes the earlier result by triesch 1993

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

nystrom methods for bivariate fredholm integral equations on unbounded domains abstract in this paper we propose a numerical procedure in order to approximate the solution of two dimensional fredholm integral equations on unbounded domains like strips half planes or the whole real plane we consider global methods of nystrom types which are based on the zeros of suitable orthogonal polynomials one of the main interesting aspects of our procedures regards the quality of the involved functions since we can successfully manage functions which are singular on the finite boundaries and can have an exponential growth on the infinite boundaries of the domains moreover the errors of the methods are comparable with the error of best polynomial approximation in the weighted spaces of functions that we go to treat the convergence and the stability of the methods and the well conditioning of the final linear systems are proved and some numerical tests which confirm the theoretical estimates are given

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

uniform convergence of derivatives of lagrange interpolation abstract uniform convergence of derivatives of lagrange interpolation at the union of zeros of jacobi polynomials and some additional points is investigated

extended lagrange interpolation on the real line let p m w a m be the sequence of the polynomials orthonormal w r t the sonin markov weight w a x e x 2 x a the authors study extended lagrange interpolation processes essentially based on the zeros of p m w a p m 1 w a determining the conditions under which the lebesgue constants in some weighted uniform spaces are optimal

mean convergence of hermite fejer interpolation abstract weighted l p convergence of hermite fejer interpolation based on the zeros of generalized jacobi polynomials is investigated the main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

convergence of product integration rules for functions with interior and end point singularities over bounded and unbounded intervals the convergence of product integration rules based on gaussian quadrature points is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals the investigation is based on a new convergence result for lagrangian interpolation and gaussian quadrature of singular integrands

inverse inequalities on non quasi uniform meshes and application to the mortar element method we present a range of mesh dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape regular but possibly non quasi uniform meshes these inequalities involve norms of the form h u w s p for positive and negative s and where h is a function which reflects the local mesh diameter in an appropriate way the only global parameter involved is n the total number of degrees of freedom in the finite element space and we avoid estimates involving either the global maximum or minimum mesh diameter our inequalities include new variants of inverse inequalities as well as trace and extension theorems they can be used in several areas of finite element analysis to extend results previously known only for quasi uniform meshes to the locally refined case here we describe applications to i the theory of nonlinear approximation and ii the stability of the mortar element method for locally refined meshes

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

mixed and stabilized finite element methods for the obstacle problem we discretize the lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods a priori and a posteriori error estimates are derived and numerically verified

a survey on mean convergence of interpolatory processes abstract a detailed account of what happened in the theory of mean convergence of lagrange and hermite fejer type interpolation in the last fifty five years is given starting from the classical result of erd s and turan hopefully all important developments are considered in some cases with an indication of the method of proof even some yet unpublished results are included a list of references helps to orientate those interested in the details

nystrom method for cauchy singular integral equations with negative index in this paper the authors propose a nystrom method to approximate the solutions of cauchy singular integral equations with constant coefficients having a negative index they consider the equations in spaces of continuous functions with weighted uniform norm they prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates

truncated quadrature rules over 0 infty and nystrom type methods we propose replacing the classical gauss laguerre quadrature formula by a truncated version of it obtained by ignoring the last part of its nodes this has the effect of obtaining optimal orders of convergence corresponding quadrature rules with kernels are then considered and optimal error estimates are derived also for them these rules are finally used to define stable nystrom type interpolants for a second kind of integral equation on the real semiaxis whose solutions decay exponentially at infty

markov sonin gaussian rule for singular functions the paper deals with the approximation of integrals r fw where w is a markov sonin weight and f can be singular at the origin gaussian type quadrature roles having a better behaviour w r t the ordinary gaussian rule are introduced error estimates in weighted l1 norm and some numerical tests are given

product integration over infinite intervals i rules based on the zeros of hermite polynomials the paper discusses both theoretical properties and practical implementation of product integration rules of the form n j k x f x dx wnlj xnj o i l where f is continuous k is absolutely integrable the nodes xn are roots of the hermite polynomials hn x and the weights wn are chosen so that the rule is exact if f is any polynomial of degree n convergence of the rule to the exact integral as n oo is proved for a wide class of functions f and k including singular or oscillatory functions k and rates of convergence are estimated the rules are shown to have the property of asymptotic positivity and as a consequence exhibit good numerical stability numerical calculations for some practical cases are presented which show the method to be computationally effective for integrands including highly oscillatory ones that decay suitably at infinity applications of the method to integration over 0 oo are also discussed

on the connection between the stabilized lagrange multiplier and nitsche s methods we derive nitsche s method for the domain decomposition through the stabilized lagrange multiplier method taking material parameters carefully into account this derivation naturally introduces parameter weighted average flux and stabilizing terms to nitsche s method we show stability and a priori analyses in the mesh dependent norms for both the stabilized method and nitsche s method and discuss connections between the proposed methods

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

approximation of the hilbert transform on the real semiaxis using laguerre zeros the authors propose two new algorithms for the computation of cauchy principal value integrals on the real semiaxis the proposed quadrature rules use zeros of laguerre polynomials theoretical error estimates are proved and some numerical examples are showed