求解奇异摄动边值问题的精细积分法  

Precise Integration Method for Solving Singular Perturbation Problems

作　　者：富明慧[1] 张文志[1] S.V.薛申宁 

机构地区：[1]中山大学应用力学与工程系,广州510275 [2]莫斯科大学力学数学系

出　　处：《应用数学和力学》2010年第11期1382-1392,共11页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10672194);中俄NSFC-RFBR资助项目(10811120012)

摘　　要：提出了一种求解一端有边界层的奇异摄动边值问题的精细方法.首先将求解区域均匀离散,由状态参量在相邻节点间的精细积分关系式确定一组代数方程,并将其写成矩阵形式.代入边界条件后,该代数方程组的系数矩阵可化为块三对角形式,针对这一特性,给出了一种高效递推消元方法.由于在离散过程中,精细积分关系式不会引入离散误差,故所提出的方法具有极高的精度.数值算例充分证明了所提出方法的有效性.A precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end was presented.Firstly,the interval was divided evenly,then a set of algebraic equations in the form of matrix by the precise integration relationship of each segment was given.Substituting the boundary conditions into the algebraic equations,the coefficient matrix could be transformed to the form of block tridiagonal matrix.Combining the special nature of the problem,an efficient reduction method for singular perturbation problems was given.Since the precise integration relationship gives no discrete error in the discrete process,the present method has very high precision.Numerical examples show the validity of the present method.

关 键 词：奇异摄动问题 一阶常微分方程组 两点边值问题 精细积分法 递推方法 

分 类 号：O175.8[理学—数学] O241.81[理学—基础数学]