一类含导数条件的插值问题教学研究

Research on teaching of a class of interpolation problem with derivative conditions

从实际应用中常见的一类含导数插值条件的问题出发,推广余项校正法,构造了适合该类插值问题的Birkhoff插值多项式,并利用基于Birkhoff插值的Chebyshev谱配置点方法求解常微分方程边值问题的近似解.与常用的谱配置点方法相比,新的谱配置点方法有两个优点:导出的线性代数方程组的条件数与配置点个数无关;可以精确施加本质边界条件.数值实验表明,即使对于很大的配置点个数,新的谱配置点方法仍能得到稳定的解.

Generalized the conventional remainder correction approach in order to construct the Birkhoff interpolation polynomial to fit for the interpolation problem with derivative conditions,which originates in practice application.The Chebyshev collocation method based on the Birkhoff interpolation was used to solve the approximate solution of the boundary value problems of ordinary differential equations.Compared with the common collocation method,the novel collocation scheme has the following two advantages：the condition number of the linear system is independent of the number of the collocation points,and the essential boundary conditions are imposed exactly.Numerical experiments show that the stable solutions would be produce by the novel collocation method,even for the large number of the collocation points.