DQAINF: an algorithm for automatic integration of infinite oscillating tails

Espelid, Terje O.; Overholt, Kristopher J.

doi:10.1007/bf02145697

Cited by 15 publications

(26 citation statements)

References 14 publications

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“…Namely, the spectral function arising in layered media problems has the asymptotic form (4) where is a constant and where , which is related to and , and are easily determined [1]. It is also well known that the Bessel function behaves for large arguments as [5, p. 364] (5) Hence, the simplest choice of break points are the equidistant points [6], [7] ( 6) where is the asymptotic half-period of the Bessel function and denotes the first break point greater than . The value of may be adjusted, for example, to coincide with the first zero of the Bessel function exceeding .…”

Section: Introductionmentioning

confidence: 99%

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Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

253

234

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A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e., semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (a a a, 1 1 1) of the integration path, where a a a > > > 0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.

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Section: Introductionmentioning

confidence: 99%

“…The computation of the tail integral in (2) has thus been reduced to finding the limit of a sequence of the partial sums (7) as . However, this sequence usually approaches slowly, i.e., the remainders (8) do not decay rapidly with increasing .…”

Section: Introductionmentioning

confidence: 99%

Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

253

234

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“…Thus, under the piecewise linear assumption, (5) and (4) are identical. Continuing with integration by parts in (5) and noting f (1) …”

Section: Higher Order Correction Terms and Error Estimatesmentioning

confidence: 89%

“…Otherwise, if f (1) (∞) = 0, integral (1) does not exist, which is evident from (5). Applying this argument recursively, all derivatives…”

Section: Higher Order Correction Terms and Error Estimatesmentioning

confidence: 96%

A short tale of long tail integration

Luo

Shevchenko

2010

Numer Algor

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, is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration over a finite domain (a, b), leaving a truncation error equal to the tail integrationin addition to the discretization error. This paper describes a very simple, perhaps the simplest, end-point correction to approximate the tail integration, which significantly reduces the truncation error and thus increases the overall accuracy of the numerical integration, with virtually no extra computational effort. Higher order correction terms and error estimates for the end-point correction formula are also derived. The effectiveness of this one-point correction formula is demonstrated through several examples.

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“…As shown in [6] and [21], in Step 1 we can also let the partition points t j be the extrema of cos(ωt) to get a faster convergence of the sequence of partial sums ∞ j=0 r j . 5.…”

Section: Computation Of I ±mentioning

confidence: 99%

An Algorithm for Melnikov Functions and Application to a Chaotic Rotor

Xu¹,

Yan²,

Zhang³

2005

SIAM J. Sci. Comput.

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Abstract. In this work we study a dynamical system with a complicated nonlinearity, which describes oscillation of a turbine rotor, and give an algorithm to compute Melnikov functions for analysis of its chaotic behavior. We first derive the rotor model whose nonlinear term brings difficulties to investigating the distribution and qualitative properties of its equilibria. This nonlinear model provides a typical example of a system for which the homoclinic and heteroclinic orbits cannot be analytically determined. In order to apply Melnikov's method to make clear the underlying conditions for chaotic motion, we present a generic algorithm that provides a systematic procedure to compute Melnikov functions numerically. Substantial analysis is done so that the numerical approximation precision at each phase of the computation can be guaranteed. Using the algorithm developed in this paper, it is straightforward to obtain a sufficient condition for chaotic motion under damping and periodic external excitation, whenever the rotor parameters are given.Key words. mathematical modeling, Melnikov function, chaos, modulus of continuity, oscillatory integrand AMS subject classifications. 34C28, 34C37, 34C45 DOI. 10.1137/S10648275034207261. Introduction. The oscillation phenomenon of the turbine rotor has long been observed in industry. Many efforts [7,8] have been made to analyze and control the mode and scale of such oscillations, which are caused by a number of factors, e.g., unbalanced centrifugal force. However, to eliminate such vibrations effectively, which are harmful to the quality and life span of the system, it is necessary to examine the underlying nature of those nonlinear oscillations and to find the relationship between the oscillatory behavior and the physical parameters of the real system.Consider the shaft of a rigid rotor supported symmetrically by four equal bearings at the ends on a low-speed balance platform, as shown in Figure 1. The elasticity in each bearing can be modeled equivalently as a spring. Let L denote the length of the shaft, and let c and K denote the length and stiffness of each relaxed spring, respectively. The following linear model is often used [3,4,18,27]:

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DQAINF: an algorithm for automatic integration of infinite oscillating tails

Cited by 15 publications

References 14 publications

Extrapolation methods for Sommerfeld integral tails

Extrapolation methods for Sommerfeld integral tails

A short tale of long tail integration

An Algorithm for Melnikov Functions and Application to a Chaotic Rotor

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