Two Types Of Bessel Transform Numerical Integration Method And Its Realization

Bessel functions arise in solving differential equations. The integration of systems containing Bessel functions is a central point in many practical problems in physics, chemistry and engineering. In most of the cases, these transformations cannot be done analytically and one has to rely on numerical methods. But Bessel function become highly oscillatory when the oscillatory parameters are large, and the classic integration formulas will suffer difficulty or even fail completely. In this paper we present an efficient Filon-type method for Bessel transform based on Luke and Watson's theorem about Bessel functions, and we also develop an efficient calculation tool based on MATLAB and Maple which is powerful in symbolic computation. The tool could be used by others as references.This paper is composed of four chapters.In Chapter one we sum up the history of Bessel functions and the story so far about highly oscillatory integrals. We do much care about the latest methods for Bessel transform.In Chapter two we present a Filon-type Method to calculate the integrals which contain Bessel functions based on the computation of the special functions and give the error analysis. More attentions are paid on the calculation of the moments.In chapter three we implement our method presented in this paper based on MATLAB7.0.1, and develop a system that would be used as a simply visual integral tool for others.In chapter four we apply these Filon-type methods to calculate some integrals and display the error with figures. Comparisons have been made with the software Maple to show the efficiency and accuracy of these methods.