efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

complex gaussian quadrature of oscillatory integrals we construct and analyze gauss type quadrature rules with complex valued nodes and weights to approximate oscillatory integrals with stationary points of high order the method is based on substituting the original interval of integration by a set of contours in the complex plane corresponding to the paths of steepest descent each of these line integrals shows an exponentially decaying behaviour suitable for the application of gaussian rules with non standard weight functions the results differ from those in previous research in the sense that the constructed rules are asymptotically optimal i e among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence relative to the required number of evaluations of the integrand

scaling and long range dependence in option pricing iii a fractional version of the merton model with transaction costs abstract a model for option pricing of fractional version of the merton model with hurst exponent h being in 1 2 1 is established with transaction costs in particular for h 1 2 1 the minimal price c min t s t of an option under transaction costs is obtained which displays that the timestep t and the hurst exponent h play an important role in option pricing with transaction costs

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

error bounds for approximation in chebyshev points this paper improves error bounds for gauss clenshaw curtis and fejer s first quadrature by using new error estimates for polynomial interpolation in chebyshev points we also derive convergence rates of chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals preliminary numerical results show that the improved error bounds are reasonably sharp

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

the construction of cubature rules for multivariate highly oscillatory integrals we present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points a general method is presented to identify these points and to compute the weights of the corresponding rule the accuracy of the constructed rules increases with increasing frequency of the integrand for a fixed frequency the accuracy can be improved by incorporating more derivatives of the integrand the results are illustrated numerically for fourier integrals on a circle and on the unit sphere and for more general oscillators on a rectangular domain

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

optimal self tuning filtering prediction and smoothing for discrete multivariable processes an algorithm is proposed for self tuning optimal fixed lag smoothing or filtering for linear discrete time multivariable processes a z transfer function solution to the discrete multivariable estimation problem is first presented this solution involves spectral factorization of polynomial matrices and assumes knowledge of the process parameters and the noise statistics the assumption is then made that the signal generating process and noise statistics are unknown the problem is reformulated so that the model is in an innovations signal form and implicit self tuning estimation algorithms are proposed the parameters of the innovation model of the process can be estimated using an extended kalman filter or alternatively extended recursive least squares these estimated parameters are used directly in the calculation of the predicted smoothed or filtered estimates the approach is an attempt to generalize the work of hagander and wittenmark

on the stability of time domain magnetic field integral equation using laguerre functions the marching on in degree mod method has been proposed for solving time domain integral equations tdies and it uses the associated laguerre functions as temporal basis and testing functions although the mod scheme is assumed to be stable and accurate there exists little investigation into the accuracy of the integral operations involving the associated laguerre functions of high degree it is shown in this paper that the numerical quadrature for integrating the highly oscillatory laguerre functions may not be accurate and this may result in instability for the mod solvers in this paper the highly oscillatory nature of the high degree associated laguerre functions is investigated first based on that the numerical procedures are described which result in the inaccuracy of the numerical quadrature associated with the highly oscillatory nature for the higher order laguerre functions this can produce instability for the mod solvers when a solution requires a high degree of the laguerre functions hence a novel filon type quadrature method and a stabilized mod technique for time domain magnetic field integral equations td mfie smod are presented in this paper some numerical results are presented to illustrate the validity of the proposed filon type quadrature rule resulting in stable solution of the td mfie for transient scattering problems

h sub 2 inferential filtering prediction and smoothing with application to rolling mill gauge estimation a new minimum mean square error optimal linear estimation problem is considered where no direct measurement of the output to be estimated is available the optimal filter predictor and smoother are derived for this case where outputs must be inferred from available measurements the results cover the usual wiener or kalman filtering problems and also optimal deconvolution estimation problems however they also apply to the situation often found in industry where estimates of signals are required that can only be determined from secondary measurements a weighted h sub 2 cost function is minimized where the weighting function can be chosen to improve the robustness of the solution the optimal estimators are derived both for stable and for unstable signal source models a signal processing application is considered in detail to demonstrate the use of the optimal filter the gauge control problem in metal rolling mills is discussed where only force measurements are available

parabolic and hyperbolic contours for computing the bromwich integral some of the most eectiv e methods for the numerical inversion of the laplace transform are based on the approximation of the bromwich contour integral the accuracy of these methods often hinges on a good choice of contour and several such contours have been proposed in the literature here we analyze two recently proposed contours namely a parabola and a hyperbola using a representative model problem we determine estimates for the optimal parameters that dene these contours an application to a fractional diusion equation is presented

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

differential equations with general highly oscillatory forcing terms the concern of this paper is in expanding and computing initial value problems of the form y f y h t where the function h oscillates rapidly for 1 asymptotic expansions for such equations are well understood in the case of modulated fourier oscillators graphic 1 and they can be used as an organizing principle for very accurate and affordable numerical solvers however there is no similar theory for more general oscillators and there are sound reasons to believe that approximations of this kind are unsuitable in that setting we follow in this paper an alternative route demonstrating that for a much more general family of oscillators e g linear combinations of functions of the form ei g k t it is possible to expand y t in a different manner each r th term in the expansion is graphic 2 for some 0 and it can be represented as an r dimensional highly oscillatory integral because computation of multivariate highly oscillatory integrals is fairly well understood this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif

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 quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

technical communique descriptor wiener state estimators based on the modern time series analysis method a new time domain wiener filtering approach is presented asymptotically stable wiener state estimators are presented for discrete linear stochastic descriptor systems they can be implemented via the autoregressive moving average arma recursive filters they can handle the optimal state filtering smoothing and prediction problems in a unified framework and can simply be obtained based on the arma innovation model the solution of the diophantine equations and riccati equations is avoided so that the computational burden is reduced a simulation example shows the effectiveness of the new approach

numerical evaluation of airy type integrals arising in uniform asymptotic analysis we describe a method to evaluate integrals that arise in the asymptotic analysis when two saddle points may be close together these integrals which appear in problems from optics acoustics or quantum mechanics as well as in a wide class of special functions can be transformed into airy type integrals and we use the trapezoidal rule to compute these integrals numerically the quadrature method which remains valid when two saddle points coalesce is illustrated with numerical examples

new quadrature rules for highly oscillatory integrals with stationary points in this paper new algorithms are being proposed for evaluation of highly oscillatory integrals hois with stationary point s the algorithms are based on modified levin quadrature mlq with multiquadric radial basis functions rbfs coupled with quadrature rules based on hybrid functions of order 8 hfq8 and haar wavelets quadrature hwq aziz et al 2011 part of the new procedure presented in this paper is comprised of transplanting monomials which are used in the conventional levin method by the rbfs the linear and hermite polynomials based quadratures xiang 2007 are being replaced by the new methods based on hwq and hfq8 respectively both the methods are merged with mlq to obtain the numerical solution of highly oscillatory integrals having stationary points the accuracy of the new methods is neither dampened by presence of the stationary point s nor by the large value of frequency parameter theoretical facts about the error analysis of the new methods are analyzed and proved numerical examples are included to show efficiency and accuracy of the new methods a new framework for the numerical solution of highly oscillatory integrals is proposed the integrals are bifurcated in the neighborhood of stationary point the integral on the smaller subinterval is solved by hybrid functions and haar wavelets the integral on the longer subinterval is solved by the meshless method with mq radial basis functions convergence analysis of the proposed methods is performed

a closed form approximation for the fractional black scholes model with transaction costs abstract in this paper we investigate option valuation problems under the fractional black scholes model the aim is to propose a pricing formula for the european option with transaction costs where the costs structure contains fixed costs a cost propositional to the volume traded and a cost proportional to the value traded precisely we provide an approximate solution of the nonlinear hoggard whalley wilmott equation the comparison results reveal that our approximate solutions are close to the numerical computations moreover the comparison results demonstrate that the price of the european option decreases as the hurst exponent increases

a class of orthogonal polynomials on the radial rays in the complex plane abstract we introduce a class of polynomials orthogonal on some radial rays in the complex plane and investigate their existence and uniqueness a recurrence relation for these polynomials a representation and the connection with standard polynomials orthogonal on 0 1 are derived it is shown that their zeros are simple and distributed symmetrically on the radial rays with the possible exception of a multiple zero at the origin an analogue of the jacobi polynomials and the corresponding problem with the generalized laguerre polynomials are also treated

genetic algorithm approach to fixed order mixed h sub 2 h sub spl infin optimal deconvolution filter designs the mixed h sub 2 h sub spl infin optimal deconvolution filter is proposed to achieve the h sub 2 optimal reconstruction and a desired robustness against the effect of uncertainties in signal processing from the h sub spl infin norm perspective however the conventional mixed h sub 2 h sub spl infin optimal design filters are very complicated and are not practical for industrial applications for simplicity of implementation and conservation of operation time the fixed order mixed h sub 2 h sub spl infin optimal deconvolution filter design is interesting for engineers in signal processing from the practical design perspective in this study to avoid the trap of local minima a design method based on the genetic algorithm is introduced to treat the nonlinear optimization design problem of the fixed order mixed h sub 2 h sub spl infin deconvolution filter the convergence property of our design algorithm is also discussed finally an example is presented to illustrate the design procedure and confirm the robustness performance of the proposed method

a unified framework for asymptotic analysis and computation of finite hankel transform abstract in this paper we present a unified framework for asymptotic analysis of the finite hankel transform this framework enables us to derive asymptotic expansions of the transform including the cases where the oscillator has zeros and stationary points as a consequence two efficient and affordable methods for computing the transform numerically are discussed and a detailed analysis of their asymptotic error estimate is carried out numerical examples are provided to confirm our analysis

polynomial systems approach to optimal linear filtering and prediction the solution of the optimal linear estimation problem is considered using a polynomial matrix description for the discrete system the filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and new insights into the disturbance rejection properties of the filter are obtained the plant or signal model can be stable or unstable and allowance is made for both control and disturbance input subsystems and white and coloured measurement noise or an output disturbance subsystem the model structure was determined by the needs of several industrial filtering problems the polynomial form of filter may easily be included in a self tuning algorithm and a simple adaptive estimator is described

efficient evaluation of edge diffraction integrals using the numerical method of steepest descent for the problem of edge diffraction from an edge of finite length a frequency domain solution obtained from an analytical time domain solution has been presented by svensson et al acta acust acust 95 568 572 this formulation takes the form of a fourier type integral whose evaluation is expensive in the high frequency range this paper demonstrates that by using tailored highly oscillatory quadrature methods based on asymptotic properties of the integral accurate approximations in the high frequency case can be obtained with little computational effort

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