用边界结点法(BKM)求解非齐次双调和方程的Cauchy问题  

Boundary Knot Method for the Cauchy Problem Associated with the Non-homogeneous Biharmonic Equation

作　　者：刘晓宇[1] 王小军[1] 杜亚楠[1] 

机构地区：[1]重庆大学数理学院,重庆400030

出　　处：《重庆工学院学报（自然科学版）》2009年第11期154-162,共9页Journal of Chongqing Institute of Technology

基　　金：国家"十一五"科技支撑计划重庆项目(2006BAJ02A09)

摘　　要：通过径向基函数和Laplace算子、重调和算子的基本解的线性组合来表示问题的解,并利用已知的边界条件来推导该线性组合的待定系数.当虚拟源点和边界结点数目不匹配时,需要用最小二乘法来求解超定线性方程组.由于边界条件给得不充分,Cauchy问题的解不唯一,故使用奇异值分解法求解该最小二乘问题.针对若干具有光滑边界或分段光滑边界的数值算例,验证了该方法的有效性,而且所得的数值计算结果是准确的,并随已知数据噪声的减小而收敛.In this paper,Boundary Knot Method is used to solve the Cauchy problem of non-homogeneous biharmonic equation.The main idea is to express the approximate solution of the problem by a linear combination of radial basis functions and fundamental solutions of Laplace and biharmonic operators,and to determine the coefficients by known data given on part of the boundary.A least square scheme is needed to solve the overdetermined linear equations when the number of boundary knots is not matched with the number of virtual sours points.Since the boundary condition given on part of the boundary is not sufficient enough,the solution of the Cauchy problem is not unique,so the method of singular value decomposition is used to solve the least square problem.The effectiveness of the proposed method is verified through several numerical examples with smooth or piecewise smooth boundaries,and the numerical results are accurate with respect to data noise,and convergent in pace with decreasing degree of noise in the known data.

关 键 词：边界结点法 双调和方程 Cauchy问题  最小二乘问题 奇异值分解法 

分 类 号：U473[机械工程—车辆工程]