High Oscillation Problems And Some Efficient Algorithms

In many application fields we encounter the problems of computing highly oscillatory integrals and integral equations.Basing on research of efficient numerical methods for highly oscillatory functions in recent years, first, we calculate a highly oscillatory integral in engineering, second, find numerical solutions to a integral equation with a highly oscillatory kernel based on moment-free Filon-type method.In the first chapter of this paper, we summarize efficient numerical methods appeared in recent years for highly oscillatory functions.We introduce the advantages and disadvantages and adaptive range of these methods respectively. At the same time we point out the their relation.In the second chapter, we study the highly oscillatory integral of Fourier transformation in the random vibration analysis model of arch dam-foundation interaction. And numerical examples discuss the effectiv-eness and present comparing of data and graphics for the methods.In the third chapter, we present a method to find numerical solutions to a Volterra integral equation of the first kind with a highly oscillatory Fourier kernel based on moment-free Filon-type method. Numerical examples are provided to discuss the effectiveness of the proposed method.