高振荡Fredholm积分方程的数值计算在许多数学、物理领域有着广泛而重要的应用，是科学计算领域一个非常重要的研究课题，目前仍有许多挑战性的问题亟需解决。本项目集中研究高频散射问题和镭射理论问题中高振荡Fredholm积分方程的高效数值解法。主要研究内容：(1)，利用高频散射问题中的高振荡Fredholm积分方程的混合渐近方法，结合奇异Hankel型振荡积分的快速计算方法，改进已有的混合渐近方法。(2)，利用在Clenshaw-Curtis点上的Hermite插值，改进镭射理论问题中高振荡Fredholm积分方程的快速Chebyshev配置方法，并将其推广到奇异的情形。本项目的完成将会丰富和完善已有的研究工作，并为实际工程计算提供算法和理论依据。

The numerical methods of highly oscillatory Fredholm integral equations have extensive and important application in many mathematic and physic fields, which is a very important research topic in scientific computation, and there are still many challenging problems to be solved. This project focuses on research efficient numerical methods of highly oscillatory Fredholm integral equations in large frequency scattering problems and laser theory and its applications. The main research contents as follows: Firstly, by means of Hybrid-Asymptotic method in high frequency scattering problems, together with fast algorithm of singular Hankel-type oscillatory integrals, we shaw improve the Hybrid-Asymptotic method。Secondly, we want to, using the Hermite interpolation on Clenshaw-Curtis points, improve the fast Chebyshev collocation method of the highly oscillatory Fredholm integral equation in laser theory, and extend it for the case of singular kernel. The completion of this project will enrich and improve existing research work, and give algorithms and theoretical evidence for practical engineering computation