求解一类非线性常微分方程的方法

A Method of Solving a Class of Nonlinear Ordinary Differential Equation

在再生核空间中,利用升元的方法将一类非线性常微分方程u″+N(u,u')=f(x)0≤x≤1 u(0)=0,u'(0)=1转化为二维线性算子方程Lv=f.通过构造零空间的一组标准正交基,得到了线性算子方程Lv=f的所有解的表达形式.如果该方程的解存在且唯一,文章给出了该方程的精确解的形式表示.并进一步给出了该方程的ε近似解.数值实验表明所给的方法是有效的.

In the paper,{u″＋N（u,u′）=f（x） 0≤x≤1 u（0）=0,u′（0）=1 is transformed into a two- dimension linear operator equation Lv = f in reproducing kernel spaces by using the method of increasing variable. The representation of all the solutions for the equation Lv = f is obtained through constructing a standard orthogonal basis of null space N （L）. If the solution of the equation is existent and unique, the exact solution of the equation is given. Further, the ε approximate solution is introduced of the equation. The final numerical experiment shows that our method is effective.