一阶完全非线性微分方程f(x,u,u')=g(x)的初值问题的数值解法(英文)

在再生核空间讨论了一阶完全非线性微分方程f(x,u,u')=g(x)的初值问题的数值解。利用再生核空间中再生核的再生性质,采用升元手段将完全非线性常微分方程转化为线性偏微分方程进行求解。将初始条件齐次化后融入到二维再生核空间中,求得一个带有未知量的解的表达式,然后通过最小二乘法的技巧,获得非线性算子方程的近似解。数值算例表明这个方法是有效的。

Numerical solution to an initial value problem for first-order fully nonlinear differential equation f(x,u,u')=g(x)

Numerical solutions to the initial value problem for the fully nonlinear differential equa-tion f(x,u,u') =g(x) are discussed in the reproducing kernel space. In terms of the reproducing prop-erty of reproducing kernel spaces, ordinary differential equations are converted to partial differential equa-tions to solve by adding variables. By homogenizing initial conditions and putting them into two-dimen-sional reproducing kernel spaces, a presentation of solutions with an unknown parameters is obtained.Then by using the least square technique, the solutions to operator equations are given. A numerical ex-ample is given, and its results demonstrate that this method is feasible.