A Numerical Method For Computing Highly Oscillatory Integrals With A Bessel Kernel

The computation of highly oscillatory integrals is an important research topic in many fields like quantum mechanics, signal processing, electromagnetic scattering. It is inefficiently to integrate Bessel-type oscillatory integral with Gauss and Newton-Cotes methods, thus we need new methods for solving Highly oscillatory integrals.In recent years, Filon-type method, asymptotic method, numerical steepest descent method and so on have been proposed to calculate highly oscillatory integrals.The most important issue is how to calculate highly oscillatory integrals efficiently such as integral with Bessel-type kernels. Currently, Filon method, Levin method, the steepest descent method are three main methods to integrate Bessel-type oscillatory integral. This paper is devoted to propose a new numerical method to evaluate integrals with Bessel kernel more efficiently based on the study of above methods.In chapter one, we summarizes some efficient numerical methods of calculating highly oscillatory integrals. Besides we introduce the advantages and disadvantages and the relation of these methods respectively.Chapter two, we introduce the methods of Fourier-type integral in the finite and infinite interval. The steepest descent method and the Filon-type methods are highlighted.Finally, we combine these two methods together to integrate the Fourier type integral.Then we generalize the method to the Bessel-type and Airy-type oscillatory integrals.Finally, the corresponding error analysis is given.Chapter three, the method proposed in chapter two is used to calculate the Fourier type and Bessel-type integrals and the convergence and asymptotic order is verified by some numerical experiments. Finally, we illustrate that our proposed method is more efficiently than Filon method for solving Bessel-type integrals via numerical experiments.