微分方程组的最小二乘解

Least square solution of differential equation system

提出了象凸微分方程组的概念,并用这一概念对一类微分方程组的边值问题提出了一种新的变分迭代解法,此迭代解的极限U*存在;在适当的条件下,U*为此微分方程组的广义解,应该指出:1.不同于[1-2]用有限维空间去逼近无穷维空间,本文空间是不变的.2.不同于[3-4]要求I(u)变分后得到Euler-Langerge方程即为微分方程组,本文的变分目标函数I(F1(U),…,Fq(U))是固定的,不取决于微分方程组的形状.

A concept of image set convex differential equation system is outlined. This new concept is then used to propose a new iterative variational algorithm for a class of boundary value problems of differential equation system, in which the limit of the iterative solution U＊ exists. In a proper condition, U＊ is a generalized solution of the differential equation system. It should be stated that the space dimension discussed is invariant in contrast to that discussed in references[1-2], in which finite dimensional space is used to approximate infinite dimensional space. Moreover, the variation aim function I （F1 （U） ,… ,Fq （U）） is fixed and not determined by the form of differential equation system in contrast to that shown in references [3-4] in which the Euler-Langerge equation obtained after variation of I（u）should be required to be the equation systems.