eigenvalue analysis of doubly connected plates with different configurations this paper deals with the vibration and buckling analyses of the first order shear deformable rhombic and square plates using a p type variational method the formulation includes rotary inertia and transverse shear terms the full plate is modeled with a single domain defined by a set of geometric points in the plane of the plate square and rectangular plates with a circular hole at the center are treated as the numerical example for the doubly connected plate and are modeled using one quarter of the geometry and only two elements the symmetry about x and y axes with regards to load and boundary conditions are exploited buckling coefficients and natural frequencies are calculated and compared with results from the published sources and the finite element method

a comparison of some methods for the evaluation of highly oscillatory integrals abstract in part i the extended clenshaw curtis method for finite fourier integrals is discussed and a number of timed comparisons are made between the various implementations which appear in the literature part ii deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation their effectiveness is discussed in the light of a set of test examples

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

an improved algorithm for the evaluation of cauchy principal value integrals of oscillatory functions and its application abstract a new interpolatory type quadrature rule is proposed for the numerical evaluation of cauchy principal value integrals of oscillatory kind 1 1 f x x e i x d x where 1 1 the method is based on an interpolatory procedure at clenshaw curtis points and the singular point and the fast computation of the modified moments with cauchy type singularity based on this result a new method is presented for the computation of the oscillatory integrals with logarithmic singularities too these methods enjoy fast implementation and high accuracy convergence rates on are also provided numerical examples support the theoretical analyses

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

uniform approximation to finite hilbert transform of oscillatory functions and its algorithm abstract for the finite hilbert transform of oscillatory functions q f c 1 1 f x e i x x c d t with a smooth function f and real 0 for c 1 1 in the sense of cauchy principal value or for c 1 of hadamard finite part we present an approximation method of clenshaw curtis type and its algorithm interpolating f by a polynomial p n of degree n and expanding in terms of the chebyshev polynomials with o n log n operations by the fft we obtain an approximation q p n c q f c we write q p n c as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n 1 we efficiently evaluate the oscillatory integral with a combination of authors previous method and keller s method for f z analytic on the interval 1 1 in the complex plane z the error of q p n c is bounded uniformly with respect to c and numerical examples illustrate the performance of our method

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

free vibration of a point supported membrane stretched by inextensible strings abstract an analysis is presented for the free vibration of a point supported rectangular membrane with uniform tension stretched by inextensible strings along the edges the membrane is transformed into a square membrane of unit length by a transformation of variables the transverse deflection of the square membrane is expressed in a series of the products of the deflection functions of strings parallel to the edges and the frequency equation is derived by the ritz method this method is applied to point supported membranes symmetrical with respect to the center lines and the natural frequencies and the mode shapes are calculated numerically up to higher modes

free in plane vibration of rectangular plates basedonthekantorovich krylovvariationalmethod kantorovich l v andkrylov v i approximatemethods of higher analysis noordhoff international groningen the netherlands 1964 pp 241 357 we analytically solve for modal frequency and displacement mode shapes of in plane vibration of rectangular plates with free and clamped boundary conditions free and clamped boundary conditions do not allow closed form solutions from partial differential equations we develop analytical expressions of plate mode shapes consisting of a linear combination of progressive waves with unknown wave amplitudes we then use these to calculate the modal frequencies and the wave amplitudes with analytical mode shape expressions an iteration scheme is presented to calculate efe ciently the natural frequency and corresponding mode shape functions based on the kantorovich krylov method three cone gurations are considered a plate with four edges clamped a plate with three edges clamped and one edge free and a plate with two parallel edges clamped and the other two edges free the e rst six natural frequencies predicted by our approach are validated using nastran and other analyses from the literature improvements in accuracy of the e rst six predicted natural frequencies were achieved using the kantorovich krylov method when compared against other results in the literature a length of plate in x direction b length of plate in y direction e young s modulus of plate material h plate thickness k1 2 two wave numbers in the solution of mode shape functions m n number of half wavelengths for mode shapes

fast computation of a class of highly oscillatory integrals in this paper the problem of numerical evaluation of a class of highly oscillatory integrals a b f x x a b x k 1 n x k k e i x dx where 1 k 1 a k b and 1 has been discussed based on analytic continuation the integrals are transformed into the problems of integrating the forms on 0 with the integrand does not oscillate and decays exponentially fast which can be efficiently computed by using general gauss laguerre quadrature rule theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if is sufficiently large

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

a well conditioned levin method for calculation of highly oscillatory integrals and its application abstract this paper is devoted to studying efficient calculation of generalized fourier transform 1 x f t e i g t d t for the general phase function g t we develop a modified levin method by the spectral coefficient approach a sparse and well conditioned linear system is constructed to help accelerate calculation of highly oscillatory integrals numerical examples are included to show the convergence properties of the new method with respect to both quantities of collocation points and the frequency furthermore we apply this approach to solving oscillatory volterra integral equations

evaluation of cauchy principal value integrals of oscillatory kind abstract this paper studies the evaluation of cauchy principal value c p v integrals of oscillatory kind a b f x x e i x d x where a b the singularity of the c p v integral is transferred to an individual oscillatory integral that can be analytically calculated the remaining integral which is non singular can be well calculated with a stable and fast quadrature method namely the improved levin method numerical examples confirm the advantages of the proposed method

on the evaluation of cauchy principal value integrals of oscillatory functions the problem of the numerical evaluation of cauchy principal value integrals of oscillatory functions 1 1e i w xf x x tdx where 1 t 1 has been discussed based on analytic continuation if f is analytic in a sufficiently large complex region g containing 1 1 the integrals can be transformed into the problems of integrating two integrals on 0 with the integrand that does not oscillate and that decays exponentially fast which can be efficiently computed by using the gauss laguerre quadrature rule the validity of the method has been demonstrated in the provision of two numerical experiments and their results

quadrature formulae for cauchy principal value integrals of oscillatory kind etude de l evaluation numerique d une integrale 7b f 1 1 exp i x f x dx ou f possede un pole simple dans l intervalle 1 1 application de la formule d interpolation de lagrange estimation de l erreur et exemples numeriques

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

efficient integration for a class of highly oscillatory integrals abstract this paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval one is a filon type method based on the asymptotic expansion the other is a clenshaw curtis filon type method which is based on a special hermite interpolation polynomial and can be evaluated efficiently in o n log n operations where n 1 is the number of clenshaw curtis points in the interval of integration in addition we derive the corresponding error bound in inverse powers of the frequency for the clenshaw curtis filon type method for the class of highly oscillatory integrals the efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

on quadrature of highly oscillatory integrals with logarithmic singularities in this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities at the same time its error depends on the frequency and the computation of its moments are given the new rule is implemented by interpolating f at chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions numerical experiments conform the efficiency for obtaining the approximations

recursive estimation of simultaneous equation models abstract a recursive instrumental variable estimator is derived for simultaneous equation estimation the choice of the instruments is discussed a computationally simple and asymptotically efficient recursive estimator is proposed in this context

computational methods for modifying seemingly unrelated regressions models computational efficient methods for updating seemingly unrelated regressions models with new observations are proposed a recursive algorithm to solve a series of updating problems is developed the algorithm is based on orthogonal transformations and has as main computational tool the updated generalized qr decomposition ugqrd strategies to compute the orthogonal factorizations by exploiting the block sparse structure of the matrices are designed the problems of adding and deleting exogenous variables from the seemingly unrelated regressions model have also been investigated the solution of these problems utilize the strategies for computing the ugqrd

on quadrature for cauchy principal value integrals of oscillatory functions the authors develop an algorithm for the numerical evaluation of cauchy principal value integrals of oscillatory functions the method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a jacobi weight a numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results bounds of the error and of the amplification factor are also provided

natural vibration of circular and annular thin plates by hamiltonian approach abstract the present paper deals with the natural vibration of thin circular and annular plates using hamiltonian approach it is based on the conservation principle of mixed energy and is constructed in a new symplectic space a set of hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy the separation of variables is employed to solve hamiltonian dual equations of eigenvalue problem analytical frequency equations are obtained based on different cases of boundary conditions the natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q1 r three basic edge constraint cases for circular plates and nine edge constraint cases for annular plates are calculated and the results are compared well with existing ones

on the calculation of highly oscillatory integrals with an algebraic singularity abstract in this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t x r where r is a positive real number and the analytical continuation robust numerical results demonstrate the accuracy and effectiveness of the proposed approach

efficient evaluation of oscillatory bessel hilbert transforms in this paper we study the asymptotics and evaluation of the oscillatory bessel hilbert transform 0 f x email protected n wx dx with 0 t the singularity of the hilbert transform is transferred to an individual oscillatory integral independent of f x for this singular integral we present two methods one is the combination of a filon type method and a complex integration method the other is the combination of a filon type method and an adaptive clenshaw curtis quadrature the remaining integral which is nonsingular can be well calculated with a combination of a filon type method and a gauss laguerre quadrature the efficiency and accuracy of the proposed methods are illustrated by numerical examples

computing cauchy principal value integrals using a standard adaptive quadrature we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals a b f x x 1 d x a b using a standard adaptive quadrature in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

on the asymptotic order of filon type methods for highly oscillatory integrals with an algebraic singularity abstract filon type methods for computing highly oscillatory integrals with an algebraic singularity of the form 0 1 x f x e i x dx where 0 1 0 and f is a sufficiently smooth function on 0 1 and 1 has been proposed by hascelik a i hascelik suitable gauss and filon type methods for oscillatory integrals with an algebraic singularity appl numer math 59 2009 101 118 in this paper we first expand such integrals into asymptotic series in inverse powers of and then give the asymptotic order of the filon type methods numerical examples are provided to confirm our analysis

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

roundoff errors in the problem of computing cauchy principal value integrals we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals mathrm p int a b f x x tau 1 dx a tau b using standard adaptive quadratures in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

free in plane vibration of isotropic non rectangular plates abstract a numerical study for the free in plane vibration of non rectangular plate is presented in this paper the plate is defined by four curved boundaries using eight points and natural co ordinates to map the geometry the displacement nodes are introduced at the boundary of the plate as well as inside the perimeter each displacement node has two degrees of freedom and much higher order polynomials than the ones used for the geometric representation are used to represent the displacement fields using the energy functional the matrix equation is obtained and solved for natural frequencies and associated mode shapes numerical results are calculated first for rectangular plates with three sets of boundary conditions and then compared with the data published in the literature by other researchers additional results are also presented and discussed in this paper for the in plane vibration of rhombic and circular annular sector plates

efficient computation of highly oscillatory integrals with hankel kernel in this paper we consider the evaluation of two kinds of oscillatory integrals with a hankel function as kernel we first rewrite these integrals as the integrals of fourier type by analytic continuation these fourier type integrals can be transformed into the integrals on 0 the integrands of which are not oscillatory and decay exponentially fast consequently the transformed integrals can be efficiently computed by using the generalized gauss laguerre quadrature rule moreover the error analysis for the presented methods is given the efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results

a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels based on the filon clenshaw curtis method for highly oscillatory integrals and together with the sommariva s result sommariva 2013 for clenshaw curtis quadrature rule we present a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels whose unknown function is assumed to be less oscillatory than the kernel in the proposed method the filon clenshaw curtis method is used to compute the involved oscillatory integrals which makes the proposed method very precise by solving only a small system of linear equations we can obtain a very satisfactory numerical solution the performance of the presented method is illustrated by several numerical examples compared with the method proposed by li et al 2012 this method enjoys a lower computational complexity furthermore numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in li et al 2012

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

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

efficient quadrature of highly oscillatory integrals with algebraic singularities in this paper we are concerned with the numerical evaluation of a class of highly oscillatory integrals containing algebraic singularities first we expand such integrals derived by two transformations t x b b 0 t 21 z email protected email protected 1 into asymptotic series in inverse powers of the frequency w then based the asymptotic series two methods are presented one is the filon type method the other is the clenshaw curtis filon type method which is based on a special hermite interpolation polynomial in the clenshaw curtis points and can be evaluated efficiently in o nlogn operations where n 1 is the number of clenshaw curtis points in the interval of integration some error and convergence analysis and robust numerical examples are used to demonstrate the accuracy and effectiveness of the proposed approaches for approximating the class of highly oscillatory singular integrals

the uniform asymptotic swallowtail approximation practical methods for oscillating integrals with four coalescing saddle points practical methods for the numerical implementation of the uniform swallowtail approximation have been developed this approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points a complex contour quadrature technique has been used to evaluate the swallowtail canonical integral s x y z and its partial derivatives delta s delta x delta s delta y delta s delta z this method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained a comparison is made with other methods that have been reported in the literature for the evaluation of s x y z isometric plots of mod s x y z mod mod delta s delta x mod mod delta s delta y mod mod delta s delta z mod are presented and some properties of the zeros of s 0 y z that lie on the line y 0 are also discussed two methods for the evaluation of the mapping parameters x y z are described an iterative method that is valid when x y z is not close to the swallowtail caustic and an algebraic method valid for x y z on the caustic and for y 0 symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations in practice both methods for determining x y z are complementary an application of the uniform swallowtail approximation to the butterfly canonical integral has been made the uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points

a recursive three stage least squares method for large scale systems of simultaneous equations a new numerical method is proposed that uses the qr decomposition and its variants to derive recursively the three stage least squares 3sls estimator of large scale simultaneous equations models sem the 3sls estimator is obtained sequentially once the underlying model is modified by adding or deleting rows of data a new theoretical pseudo sem is developed which has a non positive definite dispersion matrix and is proved to yield the 3sls estimator that would be derived if the modified sem was estimated afresh in addition the computation of the iterative 3sls estimator of the updated observations sem is considered the new recursive method utilizes efficiently previous computations exploits sparsity in the pseudo sem and uses as main computational tool orthogonal and hyperbolic matrix factorizations this allows the estimation of large scale sems which previously could have been considered computationally infeasible to tackle numerical trials have confirmed the effectiveness of the new estimation procedures the new method is illustrated through a macroeconomic application

uniform approximations to cauchy principal value integrals of oscillatory functions this paper presents an interpolatory type integration rule for the numerical evaluation of cauchy principal value integrals of oscillatory integrands 1 1e i w xf x email protected where 1 t 1 for a given smooth function f x the proposed method is constructed by interpolating f x at practical chebyshev points and subtracting out the singularity a numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast fourier transform the validity of the method has been demonstrated by several numerical experiments

quadrature formulas for oscillatory integral transforms quadrature formulas are obtained for the fourier and bessel transforms which correspond to the well known gauss laguerre formula for the laplace transform these formulas provide effective asymptotic approximations complete with error bounds comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

on fast multipole methods for volterra integral equations with highly oscillatory kernels abstract this paper explores the fast multipole methods fmms to accelerate the approximation for weakly singular volterra integral equations with highly oscillatory trigonometric kernels by constructing the fast translation path the fmm is utilized to speed up the iterative method which reduces the complexity from o n 2 to o n especially we use the collocation method to discretize the volterra integral equation with constants and linear elements respectively then apply the gmres to solve the dense and non symmetric linear system in addition the highly oscillatory integrals derived from the algorithm are calculated effectively by the steepest descent method the proposed method shows that the numerical solutions become more accurate as the frequency increases both of the optimal convergence rates of truncation and the error bounds analysis are represented in the end

semi analytic solutions for the free in plane vibrations of confocal annular elliptic plates with elastically restrained edges abstract a two dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions based on the navier displacement equation of motion for a state of plane stress the model has been simplified by invoking the helmholtz decomposition theorem and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of even and odd angular and radial mathieu functions extensive numerical results are presented in an orderly fashion for the first three anti symmetric symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical clamped and free and flexible boundary conditions also the occurrences of frequency veering between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed moreover selected 2d deformed mode shapes are presented in vivid graphical form the accuracy of solutions is checked through appropriate convergence studies and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature the set of data reported herein is believed to be the first rigorous attempt to obtain the in plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

on the numerical approximation for fourier type highly oscillatory integrals with gauss type quadrature rules in this paper we present an improved numerical steepest descent method for the approximation of fourier type highly oscillatory integrals based on the previous numerical steepest descent method the new method used the integrand information at endpoints and stationary points the asymptotic order is given that is improved both for the case of stationary points and stationary points free several numerical examples are presented which show the high efficiency of the proposed method numerical results support our theoretical analyses