Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures

Milovanović, Gradimir V.

doi:10.1016/s0898-1221(98)00180-1

Cited by 64 publications

(18 citation statements)

References 30 publications

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“…The rule is uniformly convergent with respect to the pole x when f is analytic in a neighborhood of the interval [−1, 1], and it can be implemented by means of the fast Fourier transform (FFT). If the function f is analytic in a sufficiently large region of the complex plane containing [−1, 1], then the complex integration method [23] and the numerical steepest descent method [14] can be extended to compute such integrals efficiently, and the accuracy improves greatly as ω increases; see [34] for details. In this paper, we are concerned with one-sided oscillatory Hilbert transforms on the positive real axis:…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

2012

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In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the formwhere 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 ω are derived for each fixed x ≥ 0, which clarify the large ω 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 = O(1) or x ≫ 1, 0 < x ≪ 1 and x = 0. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency ω increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.

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Section: Introductionmentioning

confidence: 99%

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

2012

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“…Downloaded 11/18/14 to 129.93.206.33. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php In order to calculate the integral in (3.12) (or in (3.11)) we use an idea from complex integration (see [16,Thm. 2.2]).…”

Section: Numerical Computation Of Müntz-legendre Polynomialsmentioning

confidence: 99%

Gaussian-type Quadrature Rules for Müntz Systems

Milovanović¹,

Cvetković²

2005

SIAM J. Sci. Comput.

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A method for constructing the generalized Gaussian quadrature rules for Müntz polynomials on (0, 1) is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Müntz polynomials introduced by Badalyan [Akad. Nauk Armyan. SSR. Armenian summary)] and studied intensively by Borwein, Erdélyi, and Zhang [Trans. Amer. Math. Soc., 342 (1994), pp. 523-542], as well as by Milovanović on a numerical procedure for evaluation of such polynomials with high precision [Müntz orthogonal polynomials and their numerical evaluation, in

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“…where the nondecreasing sequence λ k , k¨1" 1W 2. A calculation of the inverse Laplace transform for the presented function F is motivated from the problem of the evaluation of the Müntz polynomials (see [1] and [27]) and is quite essential for the construction of the generalized Gaussian quadrature rules for the Müntz function systems (see [2]). The value of n is typically required to be around 100, while the numbers λ k , k¨1" 1W 2" 100 § .…”

Section: Numerical Examplementioning

confidence: 99%

Numerical inversion of the Laplace transform

Milovanović

Cvetković

2005

FACTA U EE

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We give a short account on the methods for numerical inversion of the Laplace transform and also propose a new method. Our method is inspired and motivated from a problem of the evaluation of the Müntz polynomials (see [1]), as well as the construction of the generalized Gaussian quadrature rules for the Müntz systems (see [2]). As an illustration of our method we consider an example with 100 real poles distributed uniformly on ¢ ¡ 1£ 2¤ 100¥. A numerical investigation shows the efficiency of the proposed method.

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Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures

Cited by 64 publications

References 30 publications

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

Gaussian-type Quadrature Rules for Müntz Systems

Numerical inversion of the Laplace transform

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