scholarly journals On the Filon and Levin methods for highly oscillatory integral ∫abf(x)eiωg(x)dx

2007 ◽  
Vol 208 (2) ◽  
pp. 434-439 ◽  
Author(s):  
Shuhuang Xiang
Keyword(s):  
Oscillatory Integral ◽  
Highly Oscillatory Integral ◽  
Highly Oscillatory

scholarly journals Differential equations with general highly oscillatory forcing terms

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130490 ◽  
Author(s):  
M. Condon ◽  
A. Iserles ◽  
S. P. Nørsett
Keyword(s):  
Differential Equations ◽  
Large Family ◽  
Oscillatory Integral ◽  
Powerful Method ◽  
Initial Value ◽  
Linear Combinations ◽  
Highly Oscillatory Integral ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Similar Theory

The concern of this paper is in expanding and computing initial-value problems of the form y ′= f ( y )+ h ω ( t ), where the function h ω oscillates rapidly for ω ≫1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators 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 e i ωg k ( t ) , it is possible to expand y ( t ) in a different manner. Each r th term in the expansion is for some ς >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.

Efficient evaluation of the earth return mutual impedance of overhead conductors over a horizontally multilayered soil

2014 ◽  
Vol 33 (4) ◽  
pp. 1379-1395 ◽  
Author(s):  
Jae-bok Lee ◽  
Jun Zou ◽  
Benliang Li ◽  
Munno Ju
Keyword(s):  
Unit Length ◽  
Oscillatory Integral ◽  
Electromagnetic Transients ◽  
Closed Form Expression ◽  
Computational Time ◽  
Horizontal Distance ◽  
Content Type ◽  
Mutual Impedance ◽  
The Earth ◽  
Highly Oscillatory

Purpose – The per-unit-length earth return mutual impedance of the overhead conductors plays an important role for analyzing electromagnetic transients or couplings of multi-conductor systems. It is impossible to have a closed-form expression to evaluate this kind of impedance. The purpose of this paper is to propose an efficient numerical approach to evaluate the earth return mutual impedance of the overhead conductors above horizontally multi-layered soils. Design/methodology/approach – The expression of the earth return mutual impedance, which contains a complex highly oscillatory semi-infinite integral, is divided into two parts intentionally, i.e. the definite and the tail integral, respectively. The definite integral is calculated using the proposed moment functions after fitting the integrand into the piecewise cubic spline functions, and the tail integral is replaced by exponential integrals with newly developed asymptotic integrands. Findings – The numerical examples show the proposed approach has a satisfactory accuracy for different parameter combinations. Compared to the direct quadrature approach, the computational time of the proposed approach is very competitive, especially, for the large horizontal distance and the low height of the conductors. Originality/value – The advantage of the proposed approach is that the calculation of the highly oscillatory integral is completely avoided due to the fact that the moment function can be evaluated analytically. The contribution of the tail integral is well included by means of the exponential integral, though in an asymptotic way. The proposed approach is completely general, and can be applied to calculate the earth return mutual impedance of overhead conductors above a soil structure with an arbitrary number of horizontal layers.

scholarly journals Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 728 ◽  
Author(s):  
SAIRA ◽  
Shuhuang Xiang
Keyword(s):  
Singular Integrals ◽  
Oscillatory Integral ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Cauchy Kernel ◽  
Cauchy Type ◽  
Highly Oscillatory ◽  
Logarithmic Singularities ◽  
Highly Oscillatory Integrals ◽  
Modified Moments

In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

scholarly journals Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 115
Author(s):  
Sakhi Zaman ◽  
Latif Ullah Khan ◽  
Irshad Hussain ◽  
Lucian Mihet-Popa
Keyword(s):  
Integral Equation ◽  
Initial Value Problems ◽  
Oscillatory Integral ◽  
Factor X ◽  
Initial Value ◽  
Forcing Term ◽  
Communication Circuits ◽  
Solution Equation ◽  
Highly Oscillatory ◽  
Nonlinear Form

The paper demonstrates symmetric integral operator and interpolation based numerical approximations for linear and nonlinear ordinary differential equations (ODEs) with oscillatory factor x′=ψ(x)+χω(t), where the function χω(t) is an oscillatory forcing term. These equations appear in a variety of computational problems occurring in Fourier analysis, computational harmonic analysis, fluid dynamics, electromagnetics, and quantum mechanics. Classical methods such as Runge–Kutta methods etc. fail to approximate the oscillatory ODEs due the existence of oscillatory term χω(t). Two types of methods are presented to approximate highly oscillatory ODEs. The first method uses radial basis function interpolation, and then quadrature rules are used to evaluate the integral part of the solution equation. The second approach is more generic and can approximate highly oscillatory and nonoscillatory initial value problems. Accordingly, the first-order initial value problem with oscillatory forcing term is transformed into highly oscillatory integral equation. The transformed symmetric oscillatory integral equation is then evaluated numerically by the Levin collocation method. Finally, the nonlinear form of the initial value problems with an oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the Levin method. Results of the proposed methods are more reliable and accurate than some state-of-the-art methods such as asymptotic method, etc. The improved results are shown in the numerical section.

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