Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel
Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel
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具有高振荡核的 Volterra 积分方程的渐近展开和 Filon 型方法
DOI：
10.1093/imanum/drp048
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发表时间：
2011-04-01
影响因子：
2.1
通讯作者：
Xiang, Shuhuang
中科院分区：
文献类型：
--
作者：
Wang, Haiyong;Xiang, Shuhuang
在本文中，我们感兴趣的是寻找近似解的沃尔泰拉积分方程的第一类具有高度振荡的贝塞尔核。我们开始我们的分析，显示之间的关系，一个高度振荡的贝塞尔函数的积分和阶的被积函数贝塞尔函数。然后，我们建立的沃尔泰拉积分方程的精确解的频率的倒数幂的渐近展开，并开发一个Filon型方法提供近似。数值算例验证了该方法的有效性。
In this paper we are interested in finding approximate solutions to a Volterra integral equation of the first kind with a highly oscillatory Bessel kernel. We begin our analysis by showing the relationship between the integral of a highly oscillatory Bessel function and the order of the integrand Bessel function. We then establish the asymptotic expansion of the exact solution of the Volterra integral equation in inverse powers of the frequency and develop a Filon-type method to provide approximations. Numerical examples are provided to demonstrate the effectiveness of the proposed method.