A Gaussian quadrature rule for Fourier-type highly oscillatory integrals in the presence of stationary points

2021 ◽  
pp. 113592
Author(s):  
H. Ranjbar ◽  
F. Ghoreishi
Keyword(s):  
Gaussian Quadrature ◽  
Quadrature Rule ◽  
Oscillatory Integrals ◽  
Stationary Points ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Fourier Type ◽  
Gaussian Quadrature Rule

scholarly journals Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1160 ◽  
Author(s):  
Sakhi Zaman ◽  
Irshad Hussain ◽  
Dhananjay Singh
Keyword(s):  
Collocation Method ◽  
Stationary Point ◽  
Numerical Evaluation ◽  
Oscillatory Integrals ◽  
Test Problems ◽  
Fast Computation ◽  
Splitting Algorithm ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Fourier Type

An adaptive splitting algorithm was implemented for numerical evaluation of Fourier-type highly oscillatory integrals involving stationary point. Accordingly, a modified Levin collocation method was coupled with multi-resolution quadratures in order to tackle the stationary point and irregular oscillations of the integrand caused by ω . Some test problems are included to verify the accuracy of the proposed methods.

scholarly journals Moment-free numerical approximation of highly oscillatory integrals with stationary points

2007 ◽  
Vol 18 (4) ◽  
pp. 435-447 ◽  
Author(s):  
SHEEHAN OLVER
Keyword(s):  
Asymptotic Expansion ◽  
Stationary Phase ◽  
Asymptotic Behaviour ◽  
Approximation Method ◽  
Numerical Approximation ◽  
Oscillatory Integrals ◽  
Numerical Quadrature ◽  
Stationary Points ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

scholarly journals A Gaussian quadrature rule for oscillatory integrals on a bounded interval

2014 ◽  
Vol 34 (3) ◽  
pp. 883-901 ◽  
Author(s):  
Andreas Asheim ◽  
◽  
Alfredo Deaño ◽  
Daan Huybrechs ◽  
Haiyong Wang
Keyword(s):  
Gaussian Quadrature ◽  
Quadrature Rule ◽  
Oscillatory Integrals ◽  
Bounded Interval ◽  
Gaussian Quadrature Rule
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