第一类Fredholm积分方程的Hermite数值解

The Hermitean Numerical Solution for Fredholm Integral Equations of First Kind

本文在Ｗ２２空间中讨论了第一类Ｆｒｅｄｈｏｌｍ积分方程的求解问题。利用Ｗ２２空间的再生核，构造了方程的Ｈｅｒｍｉｔｅ数值解ｕ２ｎ（ｘ），并证明了当节点无限加密时，ｕ２ｎ（ｘ）一致收敛于解析解ｕ（ｘ），ｕ′２ｎ（ｘ）一致收敛于ｕ′（ｘ）。且每增加一个节点，误差按Ｓｏｂｏｌｅｖ范数单调下降。

Abstract In this paper, the problem seeking solution to Fredholm integral equation of first kind is discussed in the space W 2 2 . By the reproducing kernel of the space W 2 2 , the hermitean numerical solution u 2n (x) of the equation is constructed, it is also proved that, as the desity of the node system increases infinitely, u 2n (x) uniformly corverge to the analytic solution u(x) and u ′ 2n (x) uniformly coverge to u′(x) . Furthermore, the error can be monotoically decreased in the sobolev norm as the nodes number n increases.