双调和方程的有限积分方法

Finite Integration Method for Biharmonic Equations

利用有限积分法求解平面矩形区域双调和方程边值问题。首先,对双调和方程以及边界条件分别进行积分,得到一带有任意函数的线性常微分方程组;其次,将积分产生的任意函数分别进行插值估计,进而转化成为一可求解的线性代数方程组;最后,利用正则化方法求解奇异线性方程组,获得近似解误差估计。通过Matlab进行数值模拟实验获得数值结果,并进行误差分析。数值结果表明,与有限差分法、有限元法以及广义有限差分法相比较,有限积分法具有更高精度。

In this paper,a finite integration method is applied to deal with boundary value problem of biharmonic equations on rectangle domains.Firstly,biharmonic equation and boundary conditions are integrated to gain a system of linear ordinary differential equations equipped with some arbitrary functions.Secondly,interpolation estimation is conducted for arbitrary function generated from integration.The boundary value problem is thus transformed to the system of linear algebraic equations which can be solved.Finally,regularization method is used to solve singular system of linear equations and to gain the error estimate of approximate solution.Numerical simulation experiment is carried out by Matlab software,and error analysis is conducted.Compared with finite difference method,finite element method and generalized finite difference,the results indicate that finite integration method presents higher precision.