带自然边界条件多元多项式样条插值及微分方程数值解

Multivariate polynomial spline interpolation with natural boundary conditions and numerical solution of differential equation

文章考虑对d维散乱数据的一种带自然边界条件多元多项式样条插值问题,在使目标泛函极小的情况下,用Hilbert空间样条函数方法得出插值解可表示为一个多元多项式,其表示形式简单,且系数可由系数矩阵对称的线性方程组确定,最后将其应用于求微分方程数值解,并举例说明了方法的有效性。

A kind of interpolation for scattered data of d‐D by multivariate polynomial spline with natu‐ral boundary conditions is considered .Using the spline ways of Hilbert space ,the interpolation solu‐tion is constructed as a multivariate polynomial with simple expression .It makes the given objective function minimized and its coefficients be decided by a linear system with symmetry coefficient matrix . Finally ,the interpolation spline is used to solve differential equations and get the numerical solutions . Some examples are presented to illustrate the effectiveness of the method .