Of Bessel Transform Numerical Integration Algorithm

In applied area such as physics, chemistry and engineering science, the quadrature involving Bessel function is a hot issue. However, Bessel function J_m(rx) will behave oscillatory when r is very large, which will bring us difficulties to compute the quadrature. Exploring methods to integrate highly oscillatory functions in recent years and presenting some new efficient methods to evaluate the quadrature is the main goal of the dissertation.In the first chapter, we review both the background and development on the quadrature of highly oscillatory function, especially on the quadrature involving Bessel function. Moreover, we analyzed the advantages and disadvantages of different numerical methods.In chapter two, we present two kinds of new method-asymptotic method and Levin-type method, analyze the error estimations for both methods .Some numerical examples are also given to prove the efficiency of the new methods.In chapter three, an algorithms and a user interface based Matlab are designed for the quadarture involving Bessel function. In addition, we generalize the quadrature from finite interval to infinite interval and present a new method to evaluate it.