The double exponential formula for oscillatory functions over the half infinite interval

Ooura, Takuya; Mori, Miwako

doi:10.1016/0377-0427(91)90181-i

Cited by 88 publications

(61 citation statements)

References 3 publications

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“…Mori re-examined Ooura's analysis and carried out various sorts of numerical integration with Ooura's transformation and Mori, under Ooura's agreement, gave a joint talk at a RIMS symposium in November 1989. Their paper was published in 1991 [53]. Figure 5.…”

Section: −13mentioning

confidence: 99%

Discovery of the Double Exponential Transformation and Its Developments

Mori¹

2005

Publ. Res. Inst. Math. Sci.

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This article is mainly concerned how the double exponential formula for numerical integration was discovered and how it has been developed thereafter. For evaluation of I = If the trapezoidal formula is applied to the transformed integral and the resulting infinite summation is properly truncated a quadrature formula I ≈ h n k=−n f (φ(kh))φ (kh) is obtained. Its error is expressed as O(exp(−CN/ log N )) as a function of the number N ( = 2n + 1) of function evaluations. Since the integrand decays double exponentially after the transformation it is called the double exponential (DE) formula. It is also shown that the formula is optimal in the sense that there is no quadrature formula obtained by variable transformation whose error decays faster than O(exp(−CN/ log N )) as N becomes large. Since the paper by Takahasi and Mori was published the DE formula has gradually come to be used widely in various fields of science and engineering. In fact we can find papers in which the DE formula is successfully used in the fields of molecular physics, fluid dynamics, statistics, civil engineering, financial engineering, in particular in the field of the boundary element method, and so on. The DE transformation has turned out to be also useful for evaluation of indefinite integrals, for solution of integral equations and for solution of ordinary differential equations, so that the scope of its applications is expected to spread also in the future.

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Section: −13mentioning

confidence: 99%

Discovery of the Double Exponential Transformation and Its Developments

Mori¹

2005

Publ. Res. Inst. Math. Sci.

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“…It was instead evaluated numerically for different choices of the initial condition for the vortex core. A combination of the monte-carlo integration routine ''Vegas'' 11 and the double exponential routine of Ooura 12 was used. The double exponential routine was used to speed convergence of the oscillatory I c integral for large .…”

Section: B Contribution Of the Planar Velocitymentioning

confidence: 99%

The velocity-scalar cross spectrum of stretched spiral vortices

O’Gorman

Pullin

2003

Physics of Fluids

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The stretched-spiral vortex model is used to calculate the velocity-scalar cross spectrum for homogeneous, isotropic turbulence in the presence of a mean scalar gradient. The only nonzero component of the cospectrum is that contributed by the velocity component in the direction of the imposed scalar gradient while the quadrature spectrum is identically zero, in agreement with experiment. For the velocity field provided by the stretched-spiral vortex, the velocity-scalar spectrum can be divided into two additive components contributed by the velocity components along the vortex axis, and in the plane normal to this axis, respectively. For the axial velocity field, a new exact solution of the scalar convection-diffusion equation is found exhibiting scalar variation in the direction of the vortex tube axis. An asymptotic expression was found for the cospectrum contributed by this solution and the axial velocity, with the leading order term showing a k Ϫ5/3 range. This term is produced by the winding of the initial axial velocity field by the axisymmetric vortex core. The next order term gives a k Ϫ7/3 range, and arises from the lowest order effect of the nonaxisymmetric vorticity on the evolution of the axial velocity. Its coefficient can be of either sign or zero depending on the initial conditions. The contribution to the cospectrum from the velocity in the plane of the vortex is also calculated, but no universal high wave number asymptotic form is found. The integrals are evaluated numerically and it is found that the resulting cospectrum does not remain of one sign. Its form depends on the choice of the vortex core velocity profile and time cutoff in the spectral integrals. The one-dimensional cospectrum contributed by the axial velocity is compared with the experimental data of Mydlarski and Warhaft ͓J. Fluid Mech. 358, 135-175 ͑1998͔͒.

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“…(6) with the numerical method. 11) These results demonstrate that the terminal tails of f '(w )(∝w 2 ) and f "(w )(∝w ) known in the general phenomenological framework of linear relaxation (cf. Eq.…”

Section: Discussionmentioning

confidence: 53%

A Note for Kohlrausch-Williams-Watts Relaxation Function

Kawasaki

Watanabe

Uneyama

2011

J. Soc. Rheol., Jpn

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The Kohlrausch-Williams-Watts (KWW) relaxation function, F KWW (t ) = exp{-(t/t KWW ) b } with 0 < b ≤ 1 has been often utilized to describe relaxation processes in systems governed by cooperativity/coupling of molecules therein. Nevertheless, this function has not been well addressed in the general, phenomenological framework of linear relaxation phenomena, for which the relaxation function is expressed as F(t ) = Σ p g p exp{-t/t p } with g p and t p being the normalized intensity and characteristic time of pth (exponential) relaxation mode. In this study, the KWW function is analyzed to address this function in this general framework and derive analytical expressions of average relaxation times.

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The double exponential formula for oscillatory functions over the half infinite interval

Cited by 88 publications

References 3 publications

Discovery of the Double Exponential Transformation and Its Developments

Discovery of the Double Exponential Transformation and Its Developments

The velocity-scalar cross spectrum of stretched spiral vortices

A Note for Kohlrausch-Williams-Watts Relaxation Function

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