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

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

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

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

polynomial approximations of functions with endpoint singularities and product integration formulas several problems of mathematical physics lead to fredholm integral equations of the second kind where the kernels are either weakly or strongly sin gular and the known terms are smooth these equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities in this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function these esti mates are then used to obtain bounds for the remainder terms of interpolatory product rules based on the zeros of classical jacobi orthogonal polynomials that have been proposed for the discretization of integrals of the form k x y f x dx appearing in the integral equations mentioned above

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

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

bending vibration of axially loaded timoshenko beams with locally distributed kelvin voigt damping abstract utilizing the timoshenko beam theory and applying hamilton s principle the bending vibration equations of an axially loaded beam with locally distributed internal damping of the kelvin voigt type are established the partial differential equations of motion are then discretized into linear second order ordinary differential equations based on a finite element method a quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams the effects of the internal damping sizes and locations of damped segment axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated it is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics

associated generalized jacobi functions and polynomials abstract pointwise estimates are obtained for associated functions and polynomials corresponding to generalized jacobi weights these estimates hold uniformly in the interval of orthogonality

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

meshless rbfs method for numerical solutions of two dimensional high order fractional sobolev equations abstract in this paper meshless rbfs method is proposed to solve two dimensional time fractional sobolev equations the proposed method uses rbfs for approximation of spatial operator finite difference formula of o t 2 0 1 is used for time fractional derivative approximation while rule 0 1 as time stepping scheme for the advancement of solution in time validation of the proposed method is made by considering various test examples from literature simulated results are found in very good agreement with available exact solutions a rigorous comparative analysis made with other methods testifies proposed method s superiority in higher dimensions efficiency and accuracy of rbfs method are examined by varying number of nodes n in the domain of influence time step size t as well as l 2 l and l rms error norms linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis

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

non local finite element analysis of damped beams non local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method unlike local damping models the internal force of the non local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure in the finite element analysis the interpolating shape functions of the element displacement field are identical to those of standard two node beam elements however for non local damping nodes remote from the element do have an effect on the energy expressions and hence on the damping matrix the expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and euler bernoulli beam theory alternatively numerical integration may be applied to obtain solutions examples are given where the eigenvalues are compared to the exact solution for a pinned pinned beam to demonstrate the convergence of the finite element method the results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique 2007 elsevier ltd all rights reserved

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

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

asymptotic error bounds of product formulas for functions with interior singularity this paper is concerned with the convergence of product quadrature formulas of interpolatory type based on the zeros of jacobi polynomials for the approximation of integrals of the type f x is supposed to be of the form vert x x 0 vert sigma sigma 0 not an integer x 0 in 1 1 the kernel k x y can be a smooth one or it can contain an algebraic or a logarithmic singularity

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

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

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

improved approximate methods for calculating frequency response function matrix and response of mdof systems with viscoelastic hereditary terms as we know it is difficult and unnecessary to obtain all the eigenpairs of a large scaled viscoelastic nonviscous or hysteretic damping systems which means that the mode truncation scheme is generally used and the mode truncated error is therefore introduced this study is aimed at eliminating the influence of the unavailable modes on the dynamic response of mdof systems with viscoelastic hereditary terms the energy dissipation terms of the system depend on the past history of motion via convolution integrals over some kernel functions therefore the system is a nonviscously damped system which has been considered as the most generalized damping model within the scope of a linear mechanical analysis to approximate frequency response function frf matrix and response without using the unavailable modes we suggest two methods which attempt to approximate the influence of the unavailable modes in terms of the lower modes and system matrices by using the first one or two terms of neumann expansion of the contribution of the unavailable modes in contrast with the frf matrix approximated in terms of the first two terms of neumann expansion these procedures cannot be extended to further high order terms since all of them will be affected by the frequency dependent variation of damping matrix from previous terms finally an example is shown that the two presented methods can make the mode truncated error reduce and may be used to approximate the influence of nonviscous modes contributed to frf matrix due to the fact that the nonviscous modes are difficult to be obtained accurately even if a small scaled model is used for some eigensolution methods

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

a galerkin method for distributed systems with non local damping in this paper a non local damping model including time and spatial hysteresis effects is used for the dynamic analysis of structures consisting of euler bernoulli beams and kirchoff plates unlike ordinary local damping models the damping force in a non local model is obtained as a weighted average of the velocity field over the spatial domain determined by a kernel function based on distance measures the resulting equation of motion for the beam or plate structures is an integro partial differential equation rather than the partial differential equation obtained for a local damping model approximate solutions for the complex eigenvalues and modes with non local damping are obtained using the galerkin method numerical examples demonstrate the efficiency of the proposed method for beam and plate structures with simple boundary conditions for non local and non viscous damping models and different kernel functions

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

numerical solutions for fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method abstract we investigate the spectral collocation for fredholm integral equations of the second kind with weakly singular kernel the jacobi gauss quadrature formula is used to approximate the integral operator in the numerical implementation we obtain the convergence rates for the approximated solution of weakly singular fredholm integral equations which show that the errors of the approximate solution decay exponentially in l norm and weighted l2 norm some numerical examples are given to illustrate the theoretical results

a note on the power spectrum of the iterates of feigenbaum s function we give rigorous bounds on the scaling behaviour of the power spectrum for feigenbaum s map

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

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

convergence analysis of space time jacobi spectral collocation method for solving time fractional schrodinger equations abstract in this paper the space time jacobi spectral collocation method jsc method is used to solve the time fractional nonlinear schr o dinger equations subject to the appropriate initial and boundary conditions at first the considered problem is transformed into the associated system of nonlinear volterra integro partial differential equations pdes with weakly singular kernels by the definition and related properties of fractional derivative and integral operators therefore by collocating the associated system of integro pdes in both of the space and time variables together with approximating the existing integral in the equation using the jacobi gauss type quadrature formula then the problem is reduced to a set of nonlinear algebraic equations we can consider solving the system by some robust iterative solvers in order to support the convergence of the proposed method we provided some numerical examples and calculated their l norm and weighted l2 norm at the end of the article

vibration analysis of beams with non local foundations using the finite element method in this paper a non local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method unlike local foundation models the reaction of the non local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure in the finite element analysis the interpolating shape functions of the element displacement field are identical to those of standard two node beam elements however for non local elasticity or damping nodes remote from the element do have an effect on the energy expressions and hence the damping and stiffness matrices the expressions of these direct and cross matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions alternatively numerical integration may be applied to obtain solutions numerical results for eigenvalues and associated eigenmodes of euler bernoulli beams are presented and compared where possible with results in literature using exact solutions and galerkin approximations the examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non local viscoelastic foundations copyright 2007 john wiley sons ltd

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

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

identification of damping part 1 viscous damping abstract characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics the most common approach is to use viscous damping where the instantaneous generalized velocities are the only relevant state variables that affect damping forces however viscous damping is by no means the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model this paper and its companion see pp 63 88 of this issue are devoted to developing methodologies for identification of such general damping models responsible for energy dissipation in a vibrating structure this paper considers identification of viscous damping under circumstances when the actual damping model in the structure is non viscous a method is presented to obtain a full non proportional viscous damping matrix from complex modes and complex natural frequencies it is assumed that the damping is small so that a first order perturbation method is applicable the proposed method and several related issues are discussed by considering numerical examples based on a linear array of damped spring mass oscillators it is shown that the method can predict the spatial location of damping with good accuracy and also give some indication of the correct mechanism of damping

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

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

numerical treatment of second kind fredholm integral equations systems on bounded intervals in this paper the authors propose numerical methods to approximate the solutions of systems of second kind fredholm integral equations they prove that such methods are stable and convergent error estimates in weighted l p norm 1

remez type inequalities and their applications abstract the remez inequality gives a sharp uniform bound on 1 1 for real algebraic polynomials p of degree at most n if the lebesgue measure of the subset of 1 1 where p is at most 1 is known remez type inequalities give bounds for classes of functions on a line segment on a curve or on a region of the complex plane given that the modulus of the functions is bounded by 1 on some subset of prescribed measure this paper offers a survey of the extensive recent research on remez type inequalities for polynomials generalized nonnegative polynomials exponentials of logarithmic potentials and muntz polynomials remez type inequalities play a central role in proving other important inequalities for the above classes the paper illustrates the power of remez type inequalities by giving a number of applications

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

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

rigorous bounds on the power spectrum of arbitrary prime eta order renormalisation group equations the authors give rigorous bound inequalities for the universal scaling factors with their numerical values n 3 4 5 of the power spectrum for a renormalisation group equation of arbitrary prime order eta by using a generalisation of the method of collet eckmann thomas

eigensensitivity based optimal distribution of a viscoelastic damping layer for a flexible beam abstract in this paper optimal distribution of a viscoelastic damping layer is sought for suppression of the transient vibration of a flexible beam for the damping design eigenvalues in the range of interest are taken as design criteria rather than the responses at a specific frequency two eigensensitivity based optimizing procedures are proposed which are analogous to the pole placement technique and optimal control theory for dynamic system design for the eigenanalysis of the structure with frequency dependent material golla hughes mctavish ghm model is used in expressing the viscoelastic material property and an approximate eigensolution is employed to avoid the intensity of iterative computation in the optimization process which is caused by additional degrees of freedom due to ghm modelling optimized partial coverage configurations are illustrated and compared to the full coverage configuration demonstrating the improved vibration characteristic of the optimally layered structure

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

symmetric state space method for a class of nonviscously damped systems multiple degree of freedom linear nonviscously damped systems are considered it is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions the traditional state space approach well known for viscously damped systems is extended to such nonviscously damped systems using a set of internal variables suitable numerical examples are provided to illustrate the proposed approach i introduction v iscousdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems this model e rst introduced by rayleigh 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping viscous damping models are used widely for their simplicity and mathematical convenience even though the behavior of real structural materials is at best poorly mimicked by simple viscous models forthisreason itiswellrecognizedthat ingeneral aphysically realistic model of damping will not be viscous damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models mathematically any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model clearly a wide range of choice is possible either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments here we will use a particular type of damping model that is not viscous and throughout the paper the terminology nonviscous damping will refer to this specie c model only possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions 2 the equations of motion of an n degree of freedom linear system with such damping can be expressed by

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

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

legendre polynomials of the second kind fourier series and lagrange interpolation abstract the distribution of the zeros of jacobi polynomials of the second kind p m w is studied and lower and upper bounds for the corresponding christoffel functions are given fourier expansions in the system p m w are also considered moreover some theorems on the convergence of lagrange interpolating polynomials based on the zeros of p m w are proved

spectral collocation methods for nonlinear coupled time fractional nernst planck equations in two dimensions and its convergence analysis abstract in this paper a nonlinear coupled time fractional nernst planck equation in two dimensions space is considered using properties of the caputo fractional derivative and riemann liouville fractional integral the time fractional nernst planck problem is transformed into a volterra integral model with singular kernel equivalently we propose a novelty spectral collocation method in time and space with a spectral expansion of jacobi interpolation polynomial to discretize the model we also derive the optimal rate of convergence of this method numerical results are presented to verify efficiency of collocation methods and spectral accuracy theory

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

a new analytical technique for vibration analysis of non proportionally damped beams abstract vibrating linear mechanical systems in particular continuous systems are often modelled considering proportional damping distributions only although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy in this paper an analytical method is given to take into account the effects of a more general viscous damping model referred to as non proportional damping on a class of vibrating continuous systems a state form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form i e complex modes corresponding to non synchronous motions numerical examples are included in order to show the efficiency of the proposed method non proportional damping distributions of different type such as internal and external lumped or distributed viscous damping are tested on non homogeneous euler bernoulli beams in bending vibration with different boundary conditions finally a discussion on root locus diagrams behaviour and on modal damping ratio significance for non proportionally damped systems is presented

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

mean convergence of lagrange interpolation for freud s weights with application to product integration rules the connection between convergence of product integration rules and mean convergence of lagrange interpolation in l 1 p 00 has been thoroughly analysed by sloan and smith 37 motivated by this connection we investigate mean convergence of lagrange interpolation at the zeros of orthogonal polynomials associated with freud weights on r our results apply to the weights exp x 2 m 2 4 6 and for the hermite weight m 2 extend results of nevai 28 and bonan 2 in at least one direction the results are sharp in l 1 p g 2 as a consequence we can improve results of smith sloan and opie 38 on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the hermite weight in the process we prove a new markov stieltjes inequality for gauss quadrature sums and solve a problem of nevai on how to estimate certain quadrature sums