Exponential Dichotomies of Nonlinear Discrete Systems and Its Application to Numerical Analysis and Computation

Zhang Wei-jiang

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 1992 > 13(1): 81-88

Zhang Wei-jiang. Exponential Dichotomies of Nonlinear Discrete Systems and Its Application to Numerical Analysis and Computation[J]. Applied Mathematics and Mechanics, 1992, 13(1): 81-88.

Citation:

Zhang Wei-jiang. Exponential Dichotomies of Nonlinear Discrete Systems and Its Application to Numerical Analysis and Computation[J]. Applied Mathematics and Mechanics, 1992, 13(1): 81-88.

Citation:

Zhang Wei-jiang. Exponential Dichotomies of Nonlinear Discrete Systems and Its Application to Numerical Analysis and Computation[J]. Applied Mathematics and Mechanics, 1992, 13(1): 81-88.

PDF( 526 KB)

Exponential Dichotomies of Nonlinear Discrete Systems and Its Application to Numerical Analysis and Computation

Zhang Wei-jiang

Dept.of Applied Mathematics, Shanghai Jiaotong University, Shangai

Received Date: 1990-08-20
Publish Date: 1992-01-15

Abstract

Abstract

In this pepar we consider the upwind difference scheme of a kind of boundary value problems for nonlinear,second order,ordinary differential equations.Singular perturbation method is applied to construct the asymptotic approximation of the solution to the upwind difference equation.Using the theory of exponential dichotomies we show that the solution of an order-reduced equation is a good approximation of the solution to the upwind difference equation except near boundaries.We construct correctors which yield asymptotic approximations by adding them to the solution of the order-reduced equation.Finally,some numerical examples are illustrated.
- Nonlinear difference equation,
- singular perturbation,
- exponential dichotomy

FullText(HTML)

References(10)

References

[1]	Kopell,N.and G.B.Ermentrout,Symmetry and phase-locking in chains of weakly coupled scillators,Comm.Pure and Appl.Math,39(1986),623-660.
[2]	Dollan,E.P.,J.J.H.Miller and W.H.A.Schilders,Uniform Numerical Methods for Problems with Initial and Boundary Layers,Boole Press,Dublin(1980).
[3]	Miller,J.J.H.,(Ed.),An Introduction to Computational and Asymptotic Methods for Boundary and Interior Layers,Boole Press,Dublin(1982).
[4]	Kreiss,B.and H-O.Kreiss,Numerical methods for singular perturbation problems,SIAM J.Numer.Anal.,18(1981),262-276.
[5]	苏煌城、吴启光.椭圆一双抛物偏微分方程奇异摄动问题的差分解法.应用数学和力学,1(2)(1980),187-176.
[6]	Abrahamsson,L.and S.Osher,Monotone difference schemes for singular perturbation problems,SIAM J.Numer.Anal.,19(1982),979-992.
[7]	Reinhardt,H.J.,Singular perturbation of difference methods for linear ordinary differential equations,Applied Analysis,10(1980),53-70.
[8]	Kopell,N.,W.Zhang and G.B.Ermentrout,Multiple coupling in chains of oscillators,SIAM J.Math.Anal.,4(1990).
[9]	Palmer,K.J.,Exponential dichotomies,the shadowing lemma and transversal homoclinic points,Dynamical Reported,1(1988),265-306.
[10]	O'Malley,R.E.,Jr.,An Introduction to Singular Perturbations,Academic Press,New York(1974).