Numerical Analysis Of The Singular Bessel Integral Transforms

The problems of highly oscillatory integral have a wide range of applications in science and engineering,such as quantum chemistry,molecular dynamics,sound wave scattering,medical image.High-frequency oscillatory problem is often considered difficult and challenging,and it is hard to perform e cient calculations directly through the traditional classical methods,including Newton-cotes and Gauss methods.Therefore,it is necessary to seek novel and e cient calculation methods.We particularly concern numerical computation methods for the highly oscillatory Bessel integral in this paper.In Chapter 1,the research value and application background of highly oscillatory integral are introduced.Moreover,we also list several common numerical methods for this kind of integrals,such as the asymptotic method,the Filon-type method,the numerical steepest descent method and so on.The Chapter 2 mainly studies the e cient numerical methods of weak singular Bessel transform on finite interval.Firstly,the modified Filon-type method is proposed based on the explicit formula of generalized moment derived from the Meijer G function and the two-point Taylor interpolation polynomial.Moreover,the homogeneous recursive relation of the modified moment is constructed through the Bessel equation,and the Clenshaw-Curtis-Filon-type method is proposed.Further,we also conduct the error analysis by asymptotic formula for generalized moments and derivatives of Bessel function.The e cient numerical methods are proposed to calculate the Cauchy principal integral with Bessel function in Chapter 3.We mainly consider two cases a = 0 and a > 0.After constructing the integrand and transforming the integral interval,we calculate integrals over finite interval by the Filon-type method,Clenshaw-Curtis-Filon method and Clenshaw-Curtis-Filon-type method.At the same time,the integrals on infinite interval are respectively calculated by combining the numerical steepest descent method with an explicit formula.Furthermore,the calculation method of integral and error analysis are presented in detail.The Cauchy principal value integral with more general oscillator x is considered in Chapter 4.Firstly,based on the Taylor polynomial of the integrand at the singularity,we propose hybrid method I by combining the numerical steepest descent method and the explicit formula.Secondly,after making appropriate transformations of the original integral,the hybrid method II is designed by combining the Filon-type method and the numerical steepest descent method.Moreover,the detailed error analysis is carried out.According to the whole paper,we make a summary and point out the shortcomings and the way forward in Chapter 5.