Numerical Integration Of Two Kinds Of Highly Oscillating Singular Integrals

High oscillation integral is widely used in various fields of science,engineering and calculation,such as medical imaging,fluid dynamics,signal processing,molecular dynamics,quantum chemistry,electromagnetic scattering and applied mathematics.As we all know,Newton-Cortes and Gauss integral rules are very expensive in calculating high oscillation integral.This paper mainly studies the effective calculation of high oscillation integral of Bessel type and Fourier type numerical calculation methods.Chapter 1 briefly introduces the background and research significance of high oscillatory integrals,and introduces several commonly used numerical methods for calculating high oscillatory integrals.In chapter 2,an efficient method for calculating the Cauchy principal value integral of the oscillating Bessel function is proposed and analyzed.For the two forms of a>0 and a=0,the integral rule is obtained by establishing a new steepest descent integral path and using the contour integral technique.According to the relationship between Meijer G-function and Bessel function,the explicit expression of the integral ∫0+∞xj/(x-c)Jm(ωx)dx is derived.In addition,the error analysis of the obtained integral formula is carried out in detail.Compared with the combination method given in the latest paper(J.Comput.Appl.Math.410(2022),Article number:114216),the proposed method takes less time and the results are more accurate.In chapter 3,two efficient numerical methods are given for the computation of highly oscillatory Fourier-type integrals with an oscillation factor xr.The first is a two-point Taylor interpolation Filon-type method using two-point Taylor polynomials instead of f(x)to construct the integral;the second is an improved complex integration method based on the analytic extension,using the Taylor polynomials of the function f(x)at x=0,and transforming the integration path to the complex plane by the additivity of the integration interval and variable substitution.Finally,a detailed error analysis is presented for both numerical methods.Chapter 4 presents numerical methods and corresponding error analysis for a class of highly oscillatory hyper-singular integrals of Fourier type.Chapter 5 summarizes and looks ahead to the methods given in chapters 2,3 and 4.