重心插值配点法求解初值问题

将计算区间采用第二类Chebyshev点离散,利用数值稳定性好、计算精度高的重心Lagrange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初值问题的重心插值配点法。采用重心插值配点法将微分方程及其初始条件离散为线性代数方程。将初始条件离散代数方程直接附加到微分方程离散代数方程组,得到n个变量n 2个方程的代数方程组,采用最小二乘法法求解线性代数方程,得到节点的函数值。进而利用微分矩阵直接计算得到未知函数在节点的一阶导数和二阶导数值。数值算例表明本文方法具有计算公式简单、程序实施方便和计算精度高的优点。

Barycentric interpolation collocation method for solving initial value problems of differential equation

Barycentric Lagrange interpolation has merits of small operations,good numerical stability and high precision.Discrete computational interval by second kind of Chebyshev points,the differentiation matrices of the unknown function are constructed by using barycentric Lagrange interpolation.The barycentric interpolation collocation method(BICM) for solving initial value problems of differential equation is presented.The BICM transforms differential equation and initial value conditions into a set of algebraic equations system and two algebraic equations,respectively.A new algebraic equations system of n variables and n 2 equations is obtained by attaching algebraic equations of initial value conditions to the algebraic equations system of governing equation.The obtained algebraic equations system is solved using least-square method.The numerical examples demonstrate that the proposed numerical method have advantages of simple formulations,easy programming and high precision.