Engineering Applications, The High Oscillation Of The Function Of Efficient Algorithms

In many areas of applied mathematics are often encountered the problems of computing rapidly oscillatory integrals. For two kinds of highly oscillatory integrals in engineering, we present methods which have more efficiency and precision than the former methods base on research of efficient numerical methods for highly oscillatory functions in recent years.In the first chapter of this paper, we summarize classical methods and new efficient numerical methods appeared in recent years for highly oscillatory functions. We introduce the characteristic and adaptive range of these methods respectively. At the same time we point out the relation between them.In the second chapter, we study the cosine transform and sine transform in transient electromagnetism sound. These two transformations are highly oscillatory integrals on [0,+oo) .In this paper we firstly present complex integration method[55] to calculate the highly oscillatory integrals. And numerical examples are illustrated that the method is more efficient than the former methods. So we improve the calculation of these highly o-oscillatory integrals.In the third chapter, we research the singular oscillating integrals appear in a boundary integral equation of plane magneto-quasistatic eddy current problem[59]. At first we use three methods to calculate the singular oscillating integrals and compare these three methods to the method which use in [59]. We find that the composite Levin method has higher efficiency than the method in [59] to calculate this kind of singular oscillating integrals. Finally We get efficient formula which include special function Si(x) through integration by parts in this paper. So we settle the calculation of these highly oscillatory integrals.