the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

the discrete sloan iterate for the generalized airfoil equation abstract we have defined and established the superconvergence of the sloan iterate obtained from a polynomial galerkin approximation for solving the generalized airfoil equation when all the integrals involved are evaluated by using gaussian quadratures with the same number of nodes as basis elements the resulting discrete sloan iterate becomes the nystrom approximation obtained by using the same rules from this it follows that the discrete sloan iterate v n and the discrete galerkin approximation v n agree at the quadrature nodes thus v n converges no faster locally than v n for practical purposes computational superconvergence does not occur

discrete legendre spectral projection methods for fredholm hammerstein integral equations in this paper we discuss the discrete legendre galerkin and discrete legendre collocation methods for fredholm hammerstein integral equations with smooth kernel using sufficiently accurate numerical quadrature rule we obtain optimal convergence rates for both discrete legendre galerkin and discrete legendre collocation solutions in both infinity and l 2 norm numerical examples are given to illustrate the theoretical results

iterated discrete polynomially based galerkin methods golberg and bowman appl math comput 96 1998 237 have studied polynomially based discrete galerkin method for fredholm and singular integral equations in this paper we consider polynomially based iterated discrete galerkin method for solution of operator equations and for eigenvalue problem associated with an integral operator with a smooth kernel we show that the error in the infinity norm both for approximation of operator equation and of spectral subspace is of the order of n r where n is the degree of the polynomial approximation and r is the smoothness of the kernel thus the iterated discrete galerkin solution improves upon the discrete galerkin solution which was shown to be of order n r 1 by golberg and bowman appl math comput 96 1998 237 we also give a shorter proof of the result by golberg and bowman which states that the error in 2 norm in discrete galerkin method is of the order of n r

spurious solutions in a fredholm integral equation of the second kind with a cauchy kernel in this study we have addressed the appearance of a spurious solution generated when employing low order accurate schemes for a fredholm integral equation of the second kind in potential flow problems which is highly relevant to moving boundary problems it is shown that the numerical error resulting from discretization of the cauchy integral violates the fredholm solvability condition which is related to the conservation of the circulation at the discretization level in other words the numerical error is not orthogonal to the homogeneous solution of the adjoint equation and therefore does not comply with the fredholm compatibility condition the fredholm compatibility condition appears to be the total accumulated truncation error requiring circulation preservation at the discretized level therefore in order to obtain a solution the truncation error distribution must be redistributed to comply with the requirements imposed by the compatibility constraint the outcome of violating the solvability constraint is the spurious solution and it seems to be controlled by the numerical diffusion dispersion terms in the numerical scheme

the numerical evaluation of a class of logarithmically singular integral transforms abstract we consider the numerical evaluation of several logarithmically singular integral transforms which occur in the numerical solution of various types of integral equations with logarithmically singular kernels when the equations involve cauchy singular kernels as well the method is a substantial improvement in speed accuracy and ease of programming over the method introduced by fromme golberg and werth 5 convergence of the rule is proved under suitable differentiability conditions on the smooth part of the kernel and numerical results are presented substantiating this result

projection and iterated projection methods for nonliear integral equations consider the nonlinear operator equation x mathcal k x with mathcal k a completely continuous mapping of a domain in the banach space mathcal x into mathcal x and let x denote an isolated fixed point of mathcal k let mathcal k n n geqq 1 denote a sequence of finite dimensional approximating subspaces and let p n be a projection of mathcal x onto mathcal x n the projection method for solving x mathcal k x given by x n p n mathcal k x n and the iterated projection solution is defined as tilde x n mathcal k x n we analyze the convergence of x n and tilde x n to x giving a general analysis that includes both the galerkin and collocation methods a more detailed analysis is then given for a large class of urysohn integral operators in one variable showing the superconvergence of tilde x n to x

dpcm encoding of regenerative composite processes fixed nonadaptive and forward adaptive differential pulse code modulation of regenerative composite sources is investigated in the fixed code an approximate formula is given for the optimal value of the prediction coefficient this is then used as an initial guess to optimize the code predictor and quantizer through a numerical method in the forward adaptive scheme the state of the switch in the composite source is estimated using a map sequence estimation algorithm and the code is then matched to the mode process corresponding to the estimated switch state the performance of the two systems is evaluated with quantizers of 4 8 and 16 levels the results show that the forward adaptive scheme significantly outperforms optimized fixed dpcm in the sense of mean squared error stochastic stability of the code is also established for the fixed dpcm scheme as well as for an adaptive scheme which receives the switch state as side information

computation of kinematic paths and bifurcation points this article deals with the kinematic simulation of movable structures that go through special configurations of kinematic bifurcation as they move a series of algorithms are developed for structures that can be modelled using pin jointed bars and that admit a single parameter motion these algorithms are able to detect and locate any bifurcation points that exist along the path of the structure and at each bifurcation point can determine all possible motions of the structure the theory behind the algorithms is explained and the analysis of a simple example is discussed in detail then a simplified version of the particular problem that had motivated this work the simulation of the folding and deployment of a thin membrane structure forming a solar sail is analysed for the particular cases that are considered it is found that the entire process is inextensional but a detailed study of the simulation results shows that in more general cases it is likely that stretching or wrinkling will occur

analysis of a delayed delta modulator while delta modulation dm simply compares the current predictive estimate of the input with the current sample delayed delta modulation ddm also compares with the upcoming sample so as to detect and anticipate slope overloading since this future sample must be available before the present output is determined and the estimate updated delay is introduced at the encoding the performance of ddm with perfect integration and step function reconstruction is analyzed for each of three random input signals in every case the stochastic stability of the system is established for a discrete time independent and identically distributed input the limiting joint distribution of input and output is derived and the asymptotic mean square sample point error mse sp is computed when the input is gaussian for a wiener input the joint distribution of the sample point and prediction errors is derived and mse sp and the time averaged mse mse ta are computed for a stationary first order gauss markov input the joint distribution of input and output is derived and mse sp and mse ta computed graphs of the mse s illustrate the improvement attainable by using ddm instead of dm with optimal setting of parameters mse sp mse ta is reduced about 15 percent 35 percent

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

optimal convergence rates for some discrete projection methods using recent results of sloan on hyperinterpolation operators j approx theory 81 1995 238 we show how to obtain optimal convergence rates for a variety of discrete polynomially based galerkin and collocation methods for a variety of fredholm and singular integral equations these results improve on those given in golberg j integral equations appl 8 1996 307 and are analogous to those obtained by atkinson and bogomolny for spline based methods math comp 48 1987 596 these results are then applied to analyze double approximation methods where both numerical integration and data errors are considered here we show that the errors are dominated by those in the data hence it may not be possible in practice to obtain optimal convergence rates unless the data can be computed exactly

convergence of the projection method for an autoregressive process and a matched dpcm code the key step in the analysis of differential pulse code modulation dpcm is to find the steady state probability distribution of a random process in terms of which the code distortion can be evaluated for the case of an autoregressive source and a matched dpcm code a well known approximation technique has been used for the evaluation of the steady state distribution of the prediction error process however the validity of this approximation method has not been justified before a framework in which this approximation technique can be viewed as the projection method for the solution of integral equations is established sufficient conditions under which the approximation method can be rigorously justified are obtained

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

double layer potential scheme for dirichlet problems on smooth open arcs abstract in this paper we present a new numerical scheme for the dirichlet problem on the smooth open arcs in the plane by using the double layer potential to construct the solution of the problem we induce a fredholm integral equation of the second kind existence and uniqueness of the solution of this integral equation are proved in the case when the boundary arc is convex a numerical example is given to show the validity of the present scheme

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

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

a meshless galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions abstract this paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations these types of integral equations arise from boundary value problems of laplace s equations with nonlinear robin boundary conditions the discrete galerkin method together with the inverse multiquadric radial basis functions established on scattered points is utilized to approximate the solution the discrete galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method the proposed scheme uses a special accurate quadrature formula via the nonuniform gauss legendre integration rule to compute logarithm like singular integrals appeared in the scheme since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain it is meshless and does not depend to the domain form we also investigate the error analysis of the proposed method illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates

a chebyshev expansion of singular integrodifferential equations with a 2 in s t s t kernel singular integrodifferential equations of the first kind with a symmetric kernel having a 2 in s t s t singularity i e a derivative of a cauchy type kernel are studied an algorithm based on the expansion in chebyshev polynomials of the second kind is presented the method is applied to two dimensional wave scattering and convergence is demonstrated

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

the convergence of several algorithms for solving integral equations with finite part integrals ii we establish the uniform convergence and obtain convergence rates for several algorithms for solving a class of hadamard singular integral equations these results improve on the mean square convergence shown in 2

a chebyshev expansion of singular integral equations with a logarithmic kernel abstract singular integral equations of the first kind with a symmetric kernel having a logarithmic singularity are studied an algorithm based on the expansion in chebyshev polynomials of the first kind is presented the method is applied to two dimensional wave scattering and convergence is demonstrated

richardson extrapolation of iterated discrete galerkin solution for two dimensional fredholm integral equations in this paper we study the numerical solution of two dimensional fredholm integral equation by discrete galerkin and iterated discrete galerkin method we are able to derive an asymptotic error expansion of the iterated discrete galerkin solution this expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth the expansion gives rise to richardson type extrapolation schemes which rapidly improve the original rate of the convergence numerical experiments confirm our theoretical results

discrete projection methods for cauchy singular integral equations with constant coefficients we consider proving the mean square and uniform convergence of various discrete galerkin and collocation methods for cauchy singular integral equations with constant coefficients for galerkin s method the convergence rates improve those given in an earlier paper while for collocation methods the results appear to be new in particular we consider product quadrature methods for equations with discontinuous coefficients and develop and prove the convergence of a new algorithm for equations of the first kind with index zero and logarithmically singular kernels

a meshless discrete galerkin mdg method for the numerical solution of integral equations with logarithmic kernels abstract this paper describes a computational method for solving fredholm integral equations of the second kind with logarithmic kernels the method is based on the discrete galerkin method with the shape functions of the moving least squares mls approximation constructed on scattered points as basis the mls methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting the numerical scheme developed in the current paper utilizes the non uniform gauss legendre quadrature rule for approximating logarithm like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations the proposed method is meshless since it does not require any background mesh or domain elements the error analysis of the method is provided the scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of laplace s equation with linear robin boundary conditions finally numerical examples are included to show the validity and efficiency of the new technique

numerical solution of integral equations 1 a survey of boundary integral equation methods for the numerical solution of laplace s equation in three dimensions 2 superconvergence 3 perturbed projection methods for various classes of operator and integral equations 4 numerical solution of parallel processors of two point boundary value problems of astrodynamics 5 introduction to the numerical solution of cauchy singular integral equations 6 convergence theorems for singular integral equations 7 planing surfaces 8 abel integral equations

numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points an integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points acp as a first step a gaussian quadrature of order n hence of polynomial accuracy 2n 1 is employed for the numerical approximation of the integral until now the collocation which follows was performed on special points xk determined as roots of appropriate transcedental functions in order to retain the 2n 1 degree of polynomial accuracy of the gaussian quadrature in this paper an appropriate interpolatory technique is proposed so that xk may be arbitrary and yet the high 2n 1 accuracy of the gaussian quadrature is retained

the numerical solution of first kind logarithmic kernel integral equations on smooth open arcs consider solving the dirichlet problem au p o pem2 s u p h p pes sup u p co pes2 with s a smooth open curve in the plane we use single layer potentials to construct a solution u p this leads to the solution of equations of the form j g q og p q ds q h p pes this equation is reformulated using a special change of variable leading to a new first kind equation with a smooth solution function this new equation is split into a principal part which is explicitly invertible and a compact perturbation then a discrete galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically a complete convergence analysis is given numerical examples conclude the paper with s a smooth open contour in the plane this equation arises in a variety of contexts one being the study of elasticity crack problems in the plane we propose a new numerical method for solving 1 1 and then use it to solve some potential theory problems in the plane the method takes account of the expected singularities in g at the ends of the contour in an entirely natural way and the method is shown to converge rapidly when the curve 5 and the data are sufficiently smooth we limit the functions g and h to be real although the following development extends easily to the complex case let s have a parametrization

mismatched dpcm encoding of autoregressive processes a method for computing the mean squared error distortion of differential pulse code modulation dpcm applied to gaussian autoregressive sources is developed this extends previous work wherein the code predictor was matched to the source a two dimensional version of the projection method for the computation of the stationary distribution of the joint source state process is developed from which distortion can be readily evaluated an iterative algorithm is used to optimize the quantizer for a given source and predictor the results show that the matched predictor is very nearly optimal but not exactly so and that dpcm is fairly robust to mismatch of the prediction coefficient to the correlation coefficient of a first order autoregressive source

the stability of finite rank methods with applications to integral equations for mathcal l a compact linear operator on a normed linear space an approximate solution to lambda mathcal l z y is usually obtained by approximating mathcal l by an operator mathcal k of finite rank and solving the equation lambda mathcal k x y there is a system of linear equations lambda tilde mathcal k boldsymbol alpha boldsymbol beta the solution of which gives the solution of lambda mathcal k x y the stability of the solution x so obtained and this includes the condition number of lambda tilde mathcal k depends upon two theoretically defined numbers if either one of these numbers is small then the solution is stable this theory when applied to methods used in the numerical solution of fredholm integral equations of the second kind yields some known stability estimates some improved estimates and some estimates in cases which have not previously been analysed

numerical solution for the weakly singular fredholm integro differential equations using legendre multiwavelets an effective method based upon legendre multiwavelets is proposed for the solution of fredholm weakly singular integro differential equations the properties of legendre multiwavelets are first given and their operational matrices of integral are constructed these wavelets are utilized to reduce the solution of the given integro differential equation to the solution of a sparse linear system of algebraic equations in order to save memory requirement and computational time a threshold procedure is applied to obtain the solution to this system of algebraic equations through numerical examples performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

extended truss theory with simplex constraints this paper traces a way of generalization of the classical truss theory in addition to the kinematic constraint expressing the distance between two nodes connected by a bar element other similar constraints involving three and four nodes are introduced derived from energy principles a general tangent stiffness formulation is given possible mechanical interpretations as well as problems of pre stressing are also discussed

on the continuity of the stationary state distribution of dpcm continuity and singularity properties of the stationary state distribution of differential pulse code modulation dpcm are explored two level dpcm i e delta modulation operating on a first order autoregressive source is considered and it is shown that when the magnitude of the dpcm prediction coefficient is between zero and one half the stationary state distribution is singularly continuous i e it is not discrete but concentrates on an uncountable set with a lebesgue measure of zero consequently it cannot be represented with a probability density function for prediction coefficients with magnitude greater than or equal to one half the distribution is pure i e either absolutely continuous and representable with a density function or singular this problem is compared to the well known and still substantially unsolved problem of symmetric bernoulli convolutions

analysis of adaptive differential pcm of a stationary gauss markov input an adaptive matched differential pulse code modulator amdpcm is analyzed the adaptation of the symmetric uniform quantizer parameter delta n is performed by fixed multipliers assigned to the quantizer output levels the input is stationary first order gauss markov the correlation of the samples is used as the leakage parameter in the matched integrator with the predictive reconstruction similarly matched for a 4 level quantizer and multipliers gamma 1 gamma the limiting joint distribution of the prediction error and delta n is derived and the asymptotic sample point and time averaged mean square error rose and mean and variance of delta n as functions of gamma in 1 2 are computed and plotted it is found that the asymptotic performance of amdpcm does not depend on the choice of delta 0 that the increase in mse incurred by using a m dpcm instead of m dpcm with delta opt is small with mse a m dpcm downarrow min delta mse m dpcm as gamma downarrow 1 and that the signal to noise ratio of amdpcm does not depend on the input power

numerical solution of the two dimensional fredholm integral equations using gaussian radial basis function in this paper we introduce a numerical method for the solution of two dimensional fredholm integral equations the method is based on interpolation by gaussian radial basis function based on legendre gauss lobatto nodes and weights numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method

polynomially based multi projection methods for fredholm integral equations of the second kind in this paper we propose a multi projection and iterated multi projection methods for fredholm integral equations of the second kind with a smooth kernel using polynomial bases we obtain super convergence rates for the approximate solutions more precisely we prove that in m galerkin and m collocation methods not only iterative solution u n approximates the exact solution u in the supremum norm with the order of convergence n 4 k but also the derivatives of u n approximate the corresponding derivatives of u in the supremum norm with the same order of convergence n being the degree of polynomial approximation and k being the smoothness of the kernel

a movable pair of tetrahedra when we draw all the diagonals of all the faces of a cube we obtain the edges of two regular tetrahedra keeping one of these fixed we move the other under the condition that the originally intersecting edges of the two tetrahedra should still remain coplanar i e intersect are parallel or coincide we determine all such finite motions these will be seen to constitute one dimensional and two dimensional smooth manifolds we also deal with the infinitesimal degrees of freedom of the motions of our mechanism in several positions the number of infinitesimal degrees of freedom is not one or two but is three this is connected with the bifurcation phenomenon of the solutions

convergence and condition number of multi projection operators by legendre wavelets in this paper we use multi projection operators to solve the linear fredholm integral equation of the second kind with wavelets moreover a numerical convergence and computation of a bound for the condition number of the multi projection operators method is obtained