分片Bernstein多项式的样条配点法求解四阶微分方程

Collocation Method Based on Piecewise Bernstein Polynomials for Solving Fourth-Order Differential Equations

本文对四阶微分方程边值问题,给出一种基于分片Bernstein多项式的样条配点法求解,该格式构造过程容易理解,形成的线代数方程组系数矩阵稀疏,可用迭代法求解.数值实验表明,该方法可有效求解一般四阶线性微分方程边值问题,结合非等距配置点亦可用于求解含小参数的扰动问题.

In this paper, a spline collocation method based on piecewise Bernstein polynomials is proposed for boundary values problems of fourth order differential equations. The construction of the scheme is easy to understand, and the coefficient matrix of the linear algebraic equations is sparse and can be solved by iterative method. Numerical experiments show that this method can effectively solve the boundary value problems of general fourth order linear differential equations. Combining non equidistant collocation points the method can also be used to solve the perturbational problem with small parameters.