differential equations for multiple orthogonal polynomials with respect to classical weights raising and lowering operators we obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to r n classical weights these multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials combining the lowering operator with the raising operators which have been obtained earlier in the literature we then also obtain a linear differential equation of order r 1

a weak star convergence result for orthogonal rational functions we prove a weak star convergence result for orthogonal rational functions on the interval 1 1 which generalizes a well known result by rakhmanov math ussr sb 32 1977 199 as an application we use this result to approximate a certain integral which occurs in the computation of orthogonal rational functions

a quadrature formula for integrals with nearby singularities the purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities in the boundary element method the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point especially field variables and their derivatives at a field point near a boundary cannot be computed accurately in the present paper a quadrature formula for l isolated singularities near the integration interval based on lagrange interpolatory polynomials is obtained the error estimation of the proposed formula is also given quadrature formulas for regular and singular integrals with conjugate poles are derived numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy

efficiency and accuracy of numerical solutions to the time dependent schrodinger equation recent significant improvements of the numerical solutions of the time dependent schrodinger equation beg the question as to whether these recent methods are comparable in efficacy in terms of accuracy and computational time to the current method of choice i e the chebyshev expansion of the time evolution operator and the fast fourier transform method of determining the kinetic energy in this paper we review the methods in question and by studying the time development of a coherent wave packet in an oscillator well we are able to assess the effectiveness of the various methods it turns out that the new generalizations come close to within an order of magnitude to being able to generate solutions as precisely and efficiently as the chebyshev fast fourier transform method the strict unitarity of the generalized methods may be an advantage we also show that the fast fourier transform approach to calculating the kinetic energy can be replaced by straightforward numerical differentiation to obtain the same precision

construction of a probabilistic model for impedance matrices impedance matrices allow for the coupling of domains with very different properties and possibly modeled with different methods this paper presents the construction of a probabilistic model for such matrices using a nonparametric method that allows for the consideration of data errors as well as model errors to enable the application of this method in the case of a domain for which no finite element model is available the identification of a hidden state variables model from the knowledge of the impedance matrix at a discrete set of frequencies is also described finally the construction of the probabilistic model of the impedance of a pile foundation on a layered unbounded soil illustrates the capabilities of the method

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

numerical solutions of the time dependent schrodinger equation reduction of the error due to space discretization we present an improved space discretization scheme for the numerical solutions of the time dependent schr odinger equation compared to the scheme of w van dijk and f m toyama phys rev e 75 036707 2007 the present one which contains more terms of second order partial derivatives greatly reduces the error resulting from the spatial integration for a 2l 1 point formula with 2l 1 terms of second order partial derivatives the local truncation error can decrease from the order of ensuremath delta x 2l to ensuremath delta x 4l while the previous one contains only one term of second order partial derivative two well known numerical examples and the corresponding error analysis demonstrate that the present scheme has an advantage in the precision and efficiency over the previous one

on non linear transformations for accurate numerical evaluation of weakly singular boundary integrals this paper presents a study of the performance of the non linear co ordinate transformations in the numerical integration of weakly singular boundary integrals a comparison of the smoothing property numerical convergence and accuracy of the available non linear polynomial transformations is presented for two dimensional problems effectiveness of generalized transformations valid for any type and location of singularity has been investigated it is found that weakly singular integrals are more efficiently handled with transformations valid for end point singularities by partitioning the element at the singular point further transformations which are excellent for cpv integrals are not as accurate for weakly singular integrals connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated cubic transformation is seen to be the optimum choice for single precision and quartic or quintic one for double precision computations a new approach which combines the method of singularity subtraction with non linear transformation has been proposed this composite approach is found to be more accurate efficient and robust than the singularity subtraction method and the non linear transformation methods copyright 2001 john wiley sons ltd

non singular somigliana stress identities in elasticity the paper presents two fully equivalent and regular forms of the hypersingular somigliana stress identity in elasticity that are appropriate for problems in which the displacement field and resulting stresses is c1 continuous each form is found as the result of a single decomposition process on the kernels of the somigliana stress identity in three dimensions the results show that the use of a simple stress state for regularization arises in a direct manner from the somigliana stress identity just as the use of a constant displacement state regularization arose naturally for the somigliana displacement identity the results also show that the same construction leads naturally to a finite part form of the same identity while various indirect constructions of the equivalents to these findings are published none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular somigliana stress identity that arise at corners and edges these new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular somigliana stress identity are replaced by a sole requirement on displacement field continuity the resulting regularized and finite part forms of the somigliana stress identity leads to a regularized form of the stress boundary integral equation stress bie the regularized stress bie is shown to properly allow piecewise discontinuity of the boundary data subject only to c1 continuity of the underlying displacement field the importance of the findings is in their application to boundary element modeling of the hypersingular problem the piecewise discontinuity derivation for corners is found to provide a rigorous and non singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress bie the boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non conforming elements for the hypersingular equations the analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress bie that have been the basis of boundary element analysis previously published by the first author of the current paper

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

non local boundary integral formulation for softening damage a strongly non local boundary element method bem for structures with strain softening damage treated by an integral type operator is developed a plasticity model with yield limit degradation is implemented in a boundary element program using the initial stress boundary element method with iterations in each load increment regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals a numerical example is solved to verify the physical correctness and efficiency of the proposed formulation the example consists of a softening strip perforated by a circular hole subjected to tension the strain softening damage is described by a plasticity model with a negative hardening parameter the local formulation is shown to exhibit spurious sensitivity to cell mesh refinements localization of softening damage into a band of single cell width and excessive dependence of energy dissipation on the cell size by contrast the results for the non local theory are shown to be free of these physically incorrect features compared to the classical non local finite element approach an additional advantage is that the internal cells need to be introduced only within the small zone or band in which the strain softening damage tends to localize within the structure copyright 2003 john wiley sons ltd

coincident collocation of displacement and tangent derivative boundary integral equations in elasticity the regular boundary integral equations of elastostatics are combined with regularized versions of the tangent derivative equations and collocated at the same points to formulate the elasticity problem in terms of displacements tractions and the tangential displacement gradients hermitian cubic polynomials are used for functional interpolation on certain elements to formulate the boundary element method in terms of displacements tractions and their tangent derivatives commensurate accuracy of nodal values of these functions and the tangent derivatives is obtained and makes possible the accurate and immediate recovery of all stress components an example problem demonstrates the accuracy and atility of the approach

use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators in this work a new global reanalysis technique for the efficient computation of stresses and error indicators in two dimensional elastostatic problems is presented in the context of the boundary element method the global reanalysis technique can be viewed as a post processing activity that is carried out once an analysis using lagrangian elements has been performed to do the reanalysis the functional representation for the displacements is changed from lagrangian to hermite introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom next assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using lagrangian elements the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced under this scheme a second system of equations is generated and once it is solved the nodal values of the tangential derivatives of the displacements are obtained this approach gives more accurate results for the stresses at the nodes since it avoids the need to differentiate the shape functions in order to obtain the normal strain in the tangential direction when compared with the use of hermite elements the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements another important feature of the proposed approach is that an efficient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of lagrangian elements and the global reanalysis technique

boundary integral evaluation of surface derivatives in boundary element analysis first order function derivatives e g boundary potential gradient or stress tensor can be accurately computed by evaluating the hypersingular integral equation for these quantities however this approach requires a complete integration over the boundary and is therefore computationally quite expensive herein it is shown that this method can be significantly simplified only local singular integrals need to be evaluated the procedure is based upon defining the singular integrals as a limit to the boundary and exploiting the ability to use both interior and exterior boundary limits test calculations for two and three dimensional problems demonstrate the accuracy of the method

orthogonal basis functions in discrete least squares rational approximation we consider a problem that arises in the field of frequency domain system identification if a discrete time system has an input output relation y z g z u z with transter function g then the problem is to find a rational approximation n for g the data given are measurements of input and output spectra in the frequency points zk u zk y zk nk 1 together with some weight the approximation criterion is to minimize the weighted discrete least squares norm of the vector obtained by evaluating g n in the measurement points if the poles of the system are fixed then the problem reduces to a linear least squares problem in two possible ways by multiplying out the denominators and hide these in the weight which leads to the construction of orthogonal vector polynomials or the problem can be solved directly using an orthogonal basis of rational functions the orthogonality of the basis is important because if the transfer function n is represented with respect to a nonorthogonal basis then this least squares problem can be very ill conditioned even if an orthogonal basis is used but with respect to the wrong inner product e g the lebesgue measure on the unit circle numerical instability can be fatal in practice we show that both approaches lead to an inverse eigenvalue problem which forms the common framework in which fast and numerically stable algorithms can be designed for the computation of the orthonormal basis

asymptotics of multiple orthogonal polynomials associated with the modified bessel functions of the first kind we consider polynomials which are orthogonal with respect to weight functions which are defined in terms of the modified bessel functions i and i 1 these multiple orthogonal polynomials pn c of type ii were introduced earlier by the authors the aim of this paper is to study some asymptotics of these polynomials such as plancherel rotach asymptotics as a consequence of these results we obtain the accumulation set of the scaled zeros of pn c we also give a mehler heine type formula for these polynomials which is valid on compact subsets of c hence we obtain a more precise result about the asymptotic behavior of the small zeros of pn c

multiple little q jacobi polynomials we introduce two kinds of multiple little q jacobi polynomials pn with multi index nrarr n1 n2 nr and degree n n1 n2 nr by imposing orthogonality conditions with respect to r discrete little q jacobi measures on the exponential lattice qk k 0 1 2 3 where 0 q 1 we show that these multiple little q jacobi polynomials have useful q difference properties such as a rodrigues formula consisting of a product of r difference operators some properties of the zeros of these polynomials and some asymptotic properties will be given as well

non singular boundary integral formulations for plane interior potential problems in this article a non singular formulation of the boundary integral equation is developed to solve smooth and non smooth interior potential problems in two dimensions the subtracting and adding back technique is used toregularize the singularity of green s function and to simplify the calculation of the normal derivative of green s function after that a global numerical integration is directly applied at the boundary and those integration points are also taken as collocation points to simplify the algorithm of computation the result indicates that this simple method gives the convergence speed of order n 3 in the smooth boundary cases for both dirichlet and mix type problems for the non smooth cases the convergence speed drops at o n 1 2 for the dirichlet problems

iterative solution of hermite boundary integral equations an efficient iterative method for the solution of the linear equations arising from a hermite boundary integral approximation has been developed along with equations for the boundary unknowns the hermite system incorporates equations for the first order surface derivatives gradient of the potential and is therefore substantially larger than the matrix for a corresponding linear approximation however by exploiting the structure of the hermite matrix a two level iterative algorithm has been shown to provide a very efficient solution algorithm in this approach the boundary function unknowns are treated separately from the gradient taking advantage of the sparsity and near positive definiteness of the gradient equations in test problems the new algorithm significantly reduced computation time compared to iterative solution applied to the full matrix this approach should prove to be even more effective for the larger systems encountered in three dimensional analysis and increased efficiency should come from pre conditioning of the non sparse matrix component

on the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations complex boundary integral equations fredholm type regular or cauchy type singular or even hadamard mangler type hypersingular have been used for the numerical solution of general plane isotropic elasticity problems the related muskhelishvili and particularly lauricella sherman equations are famous in the literature but several more extensions of the lauricella sherman equations have also been proposed in this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries for anyone of the above equations regular or singular or hypersingular but with a prerequisite their actual numerical solution through the appropriate use of the even more classical elementary cauchy theorem in complex analysis this approach has been already used for the accurate numerical computation of analytic functions and their derivatives by ioakimidis papadakis and perdios bit 1991 31 276 285 without applications to elasticity problems but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non analytic complex density function such as an edge dislocation loading distribution density is numerically available on the boundary the present results are also directly applicable to inclusion problems anisotropic elasticity antiplane elasticity and classical two dimensional fluid dynamics but unfortunately not to crack problems in fracture mechanics brief numerical results for the complex potentials showing the dramatic increase of the computational accuracy are also displayed and few generalizations proposed copyright 2000 john wiley sons ltd

an interpolation algorithm for orthogonal rational functions let a 1 2 be a sequence of numbers on the extended real line r r and a positive bounded borel measure with support in a subset of r we introduce rational functions n with poles 1 n that are orthogonal with respect to if all poles are at infinity we recover the polynomial situation it is well known that under certain conditions on the location of the poles the system n is regular such that the orthogonal functions satisfy a three term recurrence relation similar to the one for orthogonal polynomials to compute the recurrence coefficients one can use explicit formulas involving inner products we present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the riesz herglotz nevanlinna transform of the measure error bounds are derived and some examples serve as illustration

weakly singular stress bem for 2d elastostatics a weakly singular stress bem is presented in which the linear state regularizing field is extended over the entire surface the algorithm employs standard conforming c0 elements with lagrangian interpolations and exclusively uses gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space the linear state stress bie on which the algorithm is based has no free term so that the bem treatment of external corners requires no special consideration other than to admit traction discontinuities the self regularizing nature of the somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary a key observation is the relation between bie density functions and successful interpolation orders numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations additionally the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions copyright 1999 john wiley sons ltd

numerical p version refinement studies for the regularized stress bem in the development of the boundary element method bem and the finite element method fem researchers have typically selected similar basis functions that is both methods typically employ low order interpolations such as piece wise linear or piece wise quadratic and rely on h version refinement to increase accuracy as required in the case of the fem the decision to use low order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system however in many bem formulations low order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations unlike their efficient use in the fem fine meshes of low order elements in the bem are highly inefficient from a computational standpoint given the dense nature of bem systems moreover owing to singularities in the kernel functions the bem should be expected to benefit more so than the fem from very high levels of local accuracy through the use of regularized algorithms which only require numerical integration p version refinement in the bem is easily extended to include any set of basis functions with no significant increase in programming complexity numerical results show that by using interpolations as high as 12th and 16th order one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size for two dimensional problems it is also shown that for a given level of error one can expect reductions in system size by an order of magnitude thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude copyright 2003 john wiley sons ltd

galerkin boundary integral analysis for the axisymmetric laplace equation the boundary integral equation for the axisymmetric laplace equation is solved by employing modified galerkin weight functions the alternative weights smooth out the singularity of the green s function at the symmetry axis and restore symmetry to the formulation as a consequence special treatment of the axis equations is avoided and a symmetric galerkin formulation would be possible for the singular integration the integrals containing a logarithmic singularity are converted to a non singular form and evaluated partially analytically and partially numerically the modified weight functions together with a boundary limit definition also result in a simple algorithm for the post processing of the surface gradient published in 2005 by john wiley sons ltd

the computation of orthogonal rational functions on an interval a method is given to compute the recurrence coefficients for high degree orthogonal rational functions on an interval of the real line and some asymptotic results are used to derive computable error bounds we also prove a density result for these functions several examples are given as illustrations

symmetric weak form integral equation method for three dimensional fracture analysis a symmetric galerkin boundary element method is developed for the analysis of linearly elastic isotropic three dimensional solids containing fractures the formulation is based upon a weak form displacement integral equation and a weak form traction integral equation recently developed by li and mear 1997 these integral equations are only weakly singular and their validity requires only that the boundary displacement data be continuous hence allowing standard co elements to be employed as part of the numerical implementation a special crack tip element is developed which has a novel feature in that there exist degrees of freedom associated with the nodes at the crack front as a result a higher degree of approximation is achieved for the relevant displacement data on the crack and further the stress intensity factors are obtained directly in terms of the crack front nodal data various examples are treated for cracks in unbounded domains and for cracks in finite domains including both embedded and surface breaking cracks and it is demonstrated that highly accurate results can be achieved using relatively coarse meshes

evaluation of 2 d green s boundary formula and its normal derivative using legendre polynomials with an application to acoustic scattering problems this paper presents the non singular forms in a global sense of two dimensional green s boundary formula and its normal derivative the main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain that is the proposed element free method requires only nodal data the approach includes expressing the unknown function as a truncated fourier legendre series together with transforming the integration interval a b to 1 1 the series coefficients are thus to be determined the hypersingular integral interpreted in the hadamard finite part sense and some weakly singular integrals can be evaluated analytically the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point the effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions the regularization is further applied to acoustic scattering problems the well known burton miller method using a linear combination of the surface helmholtz integral equation and its normal derivative is adopted to overcome the non uniqueness problem a general non singular form of the composite equation is derived comparisons with analytical solutions for acoustically soft and hard circular cylinders are made copyright 2001 john wiley sons ltd

a direct proof of the christoffel darboux identity and its equivalence to the recurrence relationship abstract first a direct proof of the christoffel darboux identity for orthogonal polynomials is given this proof does not make use of the three terms recurrence relationship then it is proved that if a family of polynomials satisfies the christoffel darboux identity then it also satisfies a three terms recurrence relationship and thus it forms a family of orthogonal polynomials which shows the equivalence between both relations

a generalized non linear transformation for evaluating singular integrals in this paper we propose a function which according to a parameter included therein generates a new sigmoidal transformation and converges to the well known sato polynomial transformation following well established procedures in the literature we employ the present transformation in the numerical evaluation of weakly singular integrals cauchy principal value integrals and hadamard finite part integrals by several numerical examples it is shown that the present transformation is available for all kinds of singular integrals mentioned above and in some cases gives better results compared with those of traditional non linear transformations copyright 2005 john wiley sons ltd

multiple orthogonal polynomials for classical weights a new set of special functions which has a wide range of applications from number theory to integrability of nonlinear dynamical systems is described we study multiple orthogonal polynomials with respect to p 1 weights satisfying pearson s equation in particular we give a classification of multiple orthogonal polynomials with respect to classical weights which is based on properties of the corresponding rodrigues operators we show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p 1 we also obtain explicit formulas and recurrence relations for these polynomials

orthogonal rational functions and quadrature on an interval rational functions with real poles and poles in the complex lower half plane orthogonal on the real line are well known quadrature formulas similar to the gauss formulas for orthogonal polynomials have been studied we generalize to the case of arbitrary complex poles and study orthogonality on a finite interval the zeros of the orthogonal rational functions are shown to satisfy a quadratic eigenvalue problem in the case of real poles these zeros are used as nodes in the quadrature formulas

remarks on the gauss quadrature rule for a particular class of finite part integrals the method of finite part hadamard type or hypersingular integrals is already a standard approach in applied numerical methods in engineering and mainly in acoustics fluid dynamics elasticity and fracture mechanics here we study the convergence of the non classical possessing a negative node outside the integration interval as well as a negative weight corresponding to this node gauss legendre quadrature rule for finite part integrals on the interval 0 1 with the weight function 1 y and we prove its validity for integrands possessing a continuous first derivative the case of integrands possessing a higher order derivative is also considered and the rate of convergence is established these results are based on the derivation of a bound for the negative node in the above quadrature rule and further on the theory of convergence of the classical gauss quadrature rule numerical results verifying the results of this paper are also displayed the present results can be generalized to the gauss laguerre similarly non classical quadrature rule for finite part integrals on the interval 0 as well as to the two special but important in engineering applications non classical gaussian quadrature rules for cauchy type principal value integrals and mangler type finite part integrals have been proposed by tsamasphyros and dimou

some discrete multiple orthogonal polynomials in this paper we extend the theory of discrete orthogonal polynomials on a linear lattice to polynomials satisfying orthogonality conditions with respect to r positive discrete measures first we recall the known results of the classical orthogonal polynomials of charlier meixner kravchuk and hahn t s chihara an introduction to orthogonal polynomials gordon and breach new york 1978 r koekoek and r f swarttouw reports of the faculty of technical mathematics and informatics no 98 17 delft 1998 a f nikiforov et al classical orthogonal polynomials of a discrete variable springer berlin 1991 these polynomials have a lowering and raising operator which give rise to a rodrigues formula a second order difference equation and an explicit expression from which the coefficients of the three term recurrence relation can be obtained then we consider r positive discrete measures and define two types of multiple orthogonal polynomials the continuous case jacobi laguerre hermite etc was studied by van assche and coussement j comput appl math 127 2001 317 347 and aptekarev et al multiple orthogonal polynomials for classical weights manuscript the families of multiple orthogonal polynomials of type ii that we will study have a raising operator and hence a rodrigues formula this will give us an explicit formula for the polynomials finally there also exists a recurrence relation of order r 1 for these multiple orthogonal polynomials of type ii we compute the coefficients of the recurrence relation explicitly when r 2

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

evaluation of supersingular integrals second order boundary derivatives the boundary integral representation of second order derivatives of the primary function involves second order hypersingular and third order supersingular derivatives of the green s function by defining these highly singular integrals as a difference of boundary limits interior minus exterior the limiting values are shown to exist with a galerkin formulation coincident and edge adjacent supersingular integrals are separately divergent but the sum is finite while the individual hypersingular integrals are finite moreover the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential and there is no continuity requirement on the surface flux the algorithm is efficient the non singular integrals vanish and the singular integrals are computed entirely analytically and accurate values are obtained for smooth surfaces however it is shown that a continuous linear interpolation is not appropriate for evaluation at boundary corners

hypersingular shape sensitivity boundary integral equation for crack identification under harmonic elastodynamic excitation abstract model based nondestructive testing ndt requires fast and accurate solutions of the response of the mechanical model including the defect as well as the sensitivity of this response to the variation of the parameters describing the defect for modelling crack type defects under dynamic conditions like vibration analysis or ultrasonics the boundary element method bem is especially well suited in particular due to the hypersingular formulation the present work presents the stress sensitivity boundary integral equation q bie and its use for the solution of the inverse problem when coupled to gradient based minimization algorithms the capability of solving numerically a ndt problem such as the location and characterization of cracks by measuring the dynamic response at an accessible boundary of the specimen is evaluated for that the accuracy and convergence of the sensitivity from the q bie is verified then comprehensive convergence tests are made for the initial guess the amount of supplied measurements and simulated errors on measurements geometry elastic constants and frequency

a meshless singular boundary method for three dimensional elasticity problems summary this study documents the first attempt to extend the singular boundary method sbm a novel meshless boundary collocation method for the solution of three dimensional 3d elasticity problems the sbm involves a coupling between the regularized boundary element method bem and the method of fundamental solutions mfs the main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration free attributes of the later this makes it particularly attractive for problems in complex geometries and three dimensions four benchmark 3d problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method the advantages disadvantages and potential applications of the proposed method as compared with the fem bem and mfs are also examined and discussed this article is protected by copyright all rights reserved

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

gaussian quadrature for matrix valued functions on the real line abstract parallel versions of the block chebyshev algorithm to generate the recursion coefficients of orthonormal matrix polynomials and the gaussian quadrature algorithm to approximate matrix integrals on the real line are implemented on an sp1

a new method for numerical evaluation of nearly singular integrals over high order geometry elements in 3d bem this work presents a new method for numerical computation of the two dimensional nearly singular integrals by using the eight node second order quadrilateral surface elements in 3d bem a new indirect regularized boundary element formulation excluding the cpv cauchy principal value and hfp hadamard finite part integrals is proposed based on this a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly and then the exponential transformation which has been widely employed in 2d bem is extended to 3d bem to remove the near singularities of integrands for considered integrals several numerical examples are given to verify the high efficiency and the stability of the proposed scheme

a simple local smoothing scheme in strongly singular boundary integral representation of potential gradient abstract a new approach for computation of potential gradient at and near boundary is introduced a strongly singular boundary integral representation of potential gradient whose integral density is the potential gradient is derived and analysed applying the concept of the osculating circle a local smoothing procedure which computes a continuous approximation of potential gradient from the results of a 2d boundary element method bem analysis using linear elements is proposed and evaluated this approximation is used in the integral representation derived as an integral density which fulfills the continuity requirements numerical experiments demonstrate for quasiuniform meshes an o h 2 accuracy of potential gradient computed by both the local smoothing procedure on smooth parts of the boundary and by the integral representation on smooth boundary parts and near smooth boundary parts for points inside the domain a consequence of the latter result is that no significant increase in the error appears near the boundary boundary layer effect thus being eliminated in this approach

explicit expressions for three dimensional boundary integrals in linear elasticity on employing isoparametric piecewise linear shape functions over a flat triangle exact formulae are derived for all surface potentials involved in the numerical treatment of three dimensional singular and hyper singular boundary integral equations in linear elasticity these formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle they can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a galerkin technique a numerical example involving a unit triangle and a source point located at various distances above it as well as sample problems solved by a collocation boundary element method for the lame equation are included to validate the proposed formulae

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

an improved exponential transformation for nearly singular boundary element integrals in elasticity problems abstract this paper presents an improved exponential transformation for nearly singular boundary element integrals in elasticity problems the new transformation is less sensitive to the position of the projection point compared with the original transformation in our work the conventional distance function is modified into a new form in the polar coordinate system based on the refined distance function an improved exponential transformation is proposed in the polar coordinate system moreover to perform integrations on irregular elements an adaptive integration scheme considering both the element shape and the projection point associated with the improved transformation is proposed furthermore when the projection point is located outside the integration element another nearest point is introduced to subdivide the integration elements into triangular or quadrilateral patches of fine shapes numerical examples are presented to verify the proposed method results demonstrate the accuracy and efficiency of our method

asymptotic zero distribution for a class of multiple orthogonal polynomials we establish the asymptotic zero distribution for polynomials generated by a four term recurrence relation with varying recurrence coefficients having a particular limiting behavior the proof is based on ratio asymptotics for these polynomials we can apply this result to three examples of multiple orthogonal polynomials in particular jacobi pineiro laguerre i and the example associated with modified bessel functions we also discuss an application to toeplitz matrices

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

an investigation into integral equation methods involving nearly singular kernels for acoustic scattering this work presents a numerical means of investigating the acoustic scattering by disk shaped bodies on the basis of burton and miller s method the singularity free formulations of helmholtz integral equation and its normal derivative are used to form a composite equation a triangle polar co ordinate transformation method is further applied to treat the nearly singular kernels arising from a situation in which the field points and source points are very close together numerical simulations consist of the acoustic scattering by a short circular cylinder and a thin circular disk respectively for the latter case with zero thickness the corresponding analytical solutions involving angular and radial oblate spheroidal wave functions are evaluated as well comparing the numerical results with the experimental data and analytical solutions demonstrates the effectiveness of the proposed method

on a class of gauss like quadrature rules we consider a problem that arises in the evaluation of computer graphics illumination models in particular there is a need to find a finite set of wavelengths at which the illumination model should be evaluated the result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene these values are then used to determine the rgb coordinates of the light by evaluating three definite integrals each with a common integrand the spd and interval of integration but with distinct weight functions we develop a method for selecting the sample wavelengths in an optimal manner more abstractly we examine the problem of numerically evaluating a set of m definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration it is shown that when m geq 3 it is not efficient to use a set of m gauss rules because valuable information is wasted we go on to extend the notions used in gaussian quadrature to find an optimal set of shared abcissas that maximize precision in a well defined sense the classical gauss rules come out as the special case m 1 and some analysis is given concerning the existence of these rules when m 1 in particular we give conditions on the weight functions that are sufficient to guarantee that the shared abcissas are real distinct and lie in the interval of integration finally we examine some computational strategies for constructing these rules

regularization of hypersingular and nearly singular integrals in the potential theory and elasticity both the hypersingular and nearly singular integrals which appear in the hypersingular boundary integralequations and integral representations of the secondary fields respectively are regularized by the application of the superposition principle two kinds of the non singular formulations namely those with the strongly singular and weakly singular kernels are presented in this paper the formulations are given in terms of the relevant boundary quantities and the collocation at element junctions is possible two and three dimensional problems are analysed simultaneously in a unique way for either internal or external problems of the potential theory and elasticity

hermitian cubic boundary elements for two dimensional potential problems a hermite interpolation based formulation is presented for the boundary element analysis of two dimensional potential problems two three noded hermitian cubic elements hce are introduced for the modelling of corners or points with non unique tangents on the boundary these elements along with the usual two noded hce are used in numerical examples the results obtained show that faster convergence can be achieved using hce compared with using lagrange interpolation type quadratic elements qe for about the same amount of computing resources

a general algorithm for evaluating nearly singular integrals in anisotropic three dimensional boundary element analysis abstract this paper presents a new general method for the evaluation of nearly singular boundary element integrals arising in anisotropic three dimensional 3d boundary element analysis it is shown that the original nearly singular integrals can be transformed into using a sinh function an element by element sum of regular integrals each one expressed in terms of intrinsic local coordinates as a consequence the actual computation can be performed by using standard n n gaussian quadrature and the procedure can be easily included in any existing computer code this new method has full generality and therefore can be applied to a wide class of integrals the numerical results demonstrate the accuracy and efficiency of the method along with its insensitivity to the location of the nearly singular points it is shown that several orders of magnitude improvement in relative errors can be obtained using this transformation when compared to a straightforward implementation of gaussian quadrature

formulation of a new error indicator for adaptive meshing with hermite boundary elements the main characteristic of hermite elements is that they include both the nodal values and their tangential derivatives in the functional representation of the field variables in the present work this feature is exploited to obtain an inexpensive and reliable error indicator based on the tangential derivatives of the field variables rather than the field variables themselves this new error indicator is developed together with an h adaptive strategy in the context of laplace s equation the numerical implementation shows satisfactory and very promising results

analytical integration of weakly singular integrals in boundary element analysis of helmholtz and advection diffusion equations abstract this paper presents analytical evaluation of weakly singular integrals arising in the boundary element analysis of helmholtz modified helmholtz and advection diffusion equations expressions for these boundary integrals are presented in terms of elementary integrals for straight line elements of arbitrary order which are applicable not only to the singular integrals but also to the regular integrals when the collocation point is collinear with the integration element analytical expressions have been derived for the elementary integrals using integrals of bessel functions analytical approximations of the zeroth order elementary integrals for the modified helmholtz and advection diffusion equations suitable for single as well as double precision computations have been proposed which would be specially useful for large wave numbers or peclet numbers for the respective problems

a hermite cubic collocation scheme for plane strain hydraulic fractures we describe a novel cubic hermite collocation scheme for the solution of the coupled integro partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain special blended cubic hermite power law basis functions with arbitrary index 0 1 are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture the implementation of blended infinite elements to model semi infinite crack problems is also described explicit formulae for the integrated kernels associated with the cubic hermite and blended basis functions are provided the cubic hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions the convergence rate of the cubic hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the o h4 error due to the hermite cubic interpolation the errors due to these two approximations need to be matched in order to achieve optimal convergence backward euler time stepping yields a robust algorithm that along with geometric increments in the time step can be used to explore the transition between propagation regimes over many orders of magnitude in time

a novel boundary element approach for solving the 2d elasticity problems abstract the presentation is mainly devoted to the research on the regularized bem formulations with indirect unknowns for two dimensional 2d elasticity problems a novel regularization technique in which regularized forms of the gradient equations without involving the direct calculation of cpv and hfp integrals are derived and shown to be independent of displacement equations is pursued in this paper after that a numerically systematic scheme with generality is established by adopting quadratic lagrange s elements moreover considering the special geometric domain such as circular or elliptic arcs a new boundary geometric approximate technique named as exact elements is presented and thus by the utilization of these elements the error of the results will arise mainly from the approximation of boundary quantities numerical examples show that a better precision and high computational efficiency can be achieved

a regularized boundary integral method in potential theory abstract a global method for solving the boundary integral equation in potential theory is presented with subtracting and adding back treatment several auxiliary functions are employed to reduce the singularities of boundary integrals after that all the integrals can be directly integrated by arbitrary numerical quadrature without any geometric approximation the advantage of this method is that high accuracy of numerical quadrature can be maintained with minimal computational nodes in boundary integral since collocation points can be arranged exactly as quadrature points the proposed auxiliary functions can also smooth the peaks of nearly singular integrals if they exist in such a case little extra work is needed to separate singular and nearly singular integrals into different elements after separation both kinds of integrals can be partially transformed into line integrals along their boundaries for the three dimensional case or several simple formulas for the two dimensional case

the traction boundary contour method for linear elasticity this paper presents a further development of the boundary contour method the boundary contour method is extended to cover the traction boundary integral equation a traction boundary contour method is proposed for linear elastostatics the formulation of traction boundary contour method is regular for points except the ends of the boundary element and corners the present approach only requires line integrals for three dimensional problems and function evaluations at the ends of boundary elements for two dimensional cases the implementation of the traction boundary contour method with quadratic boundary elements is presented for two dimensional problems numerical results are given for some two dimensional examples and these are compared with analytical solutions this method is shown to give excellent results for illustrative examples copyright 1999 john wiley sons ltd

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

the iterated sinh transformation a sinh transformation has recently been proposed to improve the numerical accuracy of evaluating nearly singular integrals using gauss legendre quadrature it was shown that the transformation could improve the accuracy of evaluating such integrals which arise in the boundary element method by several orders of magnitude here this transformation is extended in an iterative fashion to allow the accurate evaluation of similar types of integrals that have more spiked integrands results show that one iteration of this sinh transformation is preferred for nearly weakly singular integrals whereas two iterations lead to several orders of magnitude improvement in the evaluation of nearly strongly singular integrals the same observation applies when considering integrals of derivatives of the two dimensional boundary element kernel however for these integrals more iterations are required as the distance from the source point to the boundary element decreases

numerical solutions of the schrodinger equation with source terms or time dependent potentials we develop an approach to solving numerically the time dependent schr odinger equation when it includes source terms and time dependent potentials the approach is based on the generalized crank nicolson method supplemented with an euler maclaurin expansion for the time integrated nonhomogeneous term by comparing the numerical results with exact solutions of analytically solvable models we find that the method leads to precision comparable to that of the generalized crank nicolson method applied to homogeneous equations furthermore the systematic increase in precision generally permits making estimates of the error

application of tangent derivative boundary integral equations to the formulation of higher order boundary elements abstract higher order elements with additional degrees of freedom are implemented for three dimensional elastostatic problems through the coincident collocation of regularized forms of the displacement and the tangent derivative boundary integral equations the nodal values of the displacements the tractions and their tangential derivatives with respect to two orthogonal directions are used as the degrees of freedom associated with the functional representation of the boundary variables since the surface gradients of the displacements at the collocation points are immediately recovered from the boundary solution all the stress components are directly obtained at those locations to demonstrate the accuracy and utility of this approach a test case is presented in which the results provided by incomplete quartic elements are compared to the ones obtained using the quadratic elements of the serendipity family the methodology presented here is general and can be easily extended to other problems amenable to a boundary integral formulation

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized