Numerical Methods Of Many Different Highly Oscillatory (Singular) Bessel Transforms

The problems of highly oscillatory integral are widely used in various fields of scientific engineering and computing,such as quantum chemistry,fluid mechanics,electromagnetic scattering and applied mathematics.The key of the problems is to construct a fast and stable numerical computation method for highly oscillatory integral.Traditional methods such as Newton-Cotes and Gauss methods are time-consuming for this kind of problem,so we must design the new and efficient calculation methods.There are many effective numerical approaches for computing highly oscillatory integral,such as asymptotic method,Filon-type method,Levin method and numerical steepest descent method.We mainly study the numerical computation methods for the highly oscillatory Bessel integral.Chapter 1 describes the background and research significance of oscillatory Bessel integral.Meanwhile,we introduce several commonly used numerical methods for highly oscillatory integral and summarize their respective advantages and disadvantages.In Chapter 2,by transforming the Bessel function into the confluent hypergeometric function and using the asymptotic expansion of the confluent hypergeometric function,we can transform the Bessel transform into the Fourier transform.At the same time,combining with the numerical steepest descent method,the numerical methods and error analysis of the finite and infinite Bessel transforms are proposed respectively.Chapter 3 presents the numerical calculation methods for the oscillatory Bessel transforms with algebraic singularity or logarithmic singularity.By the change of variable and converting the integration path to the complex plane,we design the corresponding complex integration method and the modified complex integration method.Furthermore,the error analysis in detail is shown here.The numerical methods and the corresponding error analysis for the oscillatory Bessel transform with a general oscillator g(x)are displayed in Chapter 4.We mainly discussed the problem including the two cases that g(x)with or without stationary point.Moreover,the later case can be divided into two case that g(x)with or without zero point.In Chapter 5,we summarize the methods proposed in Chapters 2-4,and provide areas for improvement and prospects.