求解边值问题的重心有理插值配点法

Barycentric rational interpolation method and its application in solving boundary value problems

将计算区间采用等距节点离散,利用重心有理插值近似未知函数,建立未知函数各阶导数在计算节点上的微分矩阵,提出数值求解微分方程边值问题的重心有理插值配点法。采用重心有理插值配点法将微分方程及其边值条件离散为线性代数方程,数值求解代数方程得到未知函数在节点的函数值,进而利用微分矩阵可以得到未知函数的各阶导数值。数值算例表明,重心有理插值配点法具有计算公式简单、程序实施方便和计算精度高的优点。

The authors discrete computational interval by uniformly spaced points, using barycentric rational interpolation method to approximate the unknown function and constructing the differentiation matrices which is each derivative of the unknown function on the computational points. The barycentric rational interpolation collocation method （BRICM） is presented for solving boundary value problems of differential equation. The BRICM transforms differential equation and boundary value conditions into a set of algebraic equations system and uses numerical method solving the algebraic equation for the value of the unknown function at the points. The differentiation matrices are useful to have each derivative of the unknown function. The numerical examples demonstrate that the proposed numerical method have advantages of simple formulations, easy programming and high precision.