非线性常微分方程转向点问题的数值解
Numerical Solution of Nonlinear Ordinary Differential Equation for a Turning Point Problem
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摘要: 本文利用文[3]的技巧得到了具转向点的非线性常微分方程边值问题的导数估计,再结合文[4]的方法,证明了所构造的差分格式关于小参数ε的一致收敛性.我们给出了数值例子,数值结果与理论分析完全符合.Abstract: By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained.With the help of the technique in [4], the uniform convergence on the small parameter ε for a difference scheme is proved.At the end of this paper, a numerical example is given.The numerical result coincides with theoretical analysis.
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[1] 江福汝,关于具转向点的一类常微分方程的边值问题,应用数学和力学,1,2 (1980), 201-203. [2] 林鹏程、颜鹏翔,无共振转向点问题的数值解,应用数学和力学,8, 6 (1986), 423-429. [3] Liseikin,V.D.,Numerical solution for a singular perturbation equation with turning point,J.Comp.Math,and Math.Phys.,24,12(1984),1812-1818.(in Russian) [4] Niijima,K.,A uniformly convergent difference scheme for a semilihear singular perturbation problem,Numer.Math.,43,2(1984),175-198. [5] Dorr,F.W.,Singular perturbation of nonlinear boundary value problems with turning points,J.Math.Anal.Appl.,29,2(1970),273-293. [6] Farrell,P.A.,A uniformly convergent difference scheme for turning point problems,The First Conference on Boundary and Interior Layers-Computational and Asymptotic Methods,Dublin,Ireland(1980). [7] Stynes,M.and E O'Riordan,L1 and L∞ uniform convergence of a difference scheme for a semilinear singular perturbation problem,Numer.Math.,50,4(1987),519-531. [8] Berger,A.Z.,H.D.Han and R.E.Kellogg,A priori estimates and analysis of a numerical method for a turning point problem,Math.Comp.,42,166(1984),465-492. [9] Niijima,K.,Error estimates for an exponentially fitted difference scheme for a singular perturbation problem Ⅰ,Computational and Asymptotic Methods for Boundary and Interior Layers,Proc.BAIL Ⅱ Short Course(1982),63-67.
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