Asymptotic Quadrature Method And Combined Levin-asymptotic Quadrature Method For Highly Oscillatory Integral

Highly oscillatory problems have an important role in many practical issues. In practical, we meet with some highly oscillatory differential equation. When we solve the highly oscillatory differential equation, the key step is solving highly oscillatory integral. However, using the classical quadrature method solving highly oscillatory integrals is useless. Therefore, researching highly oscillatory integral method is of important theoretical and practical significance.This paper mainly deals with the numerical computation formula and computation method of highly oscillatory integral. The main contents may be summarized as the following.The paper presents some applications of highly oscillatory integral and the research history of highly oscillatory quadrature methods for recent seventy or eighty years in details. We give the introduction to the background of the study problems and the purpose besides the significance of this research.For some special integral regions of two and three dimension, the paper describes some special asymptotic expansion quadrature formulas with and without stationary point and analyses the error order of these formulas.For highly oscillatory integrals in two and three dimension, using Levin-type method and Stokes theorem gets two formulas of reduction of dimension. Moreover, we put forward to a new method of highly oscillatory integral, combined method of Levin-Asymptotic and give the error order of these methods. Some computational examples are given and prove the error order of the formula and method what we give.