常差分方程奇异摄动问题的渐近方法

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Asymptotic Method for Singular Perturbation Problem of Ordinary Difference Equations

摘要: 在本文中,我们讨论如下差分方程问题(P_ε):(L.y)_k≡εy(k+1)+a(k,ε)y(k)+b(k,ε)y(k-1)=f(k,ε)(1≤k≤N-1)B₁y≡-y(0)+c₁y(1)=a,B₂y≡-c₂y(N-1)+y(N)=β这里ε是一个小参数,c₁,c₂,a,β为常数,a(k,ε),b(k,ε),f(k,ε)(1≤k≤N)是k和ε的函数.首先,我们讨论了常系数的情形;接着引进伸长变换对变系数的情形进行了讨论,给出了解的一致渐近展开式;最后给出了一个数值例子.

Abstract: This paper is taken up for the following difference equation problem (P_ε):(L.y)_k≡εy(k+1)+a(k,ε)y(k)+b(k,ε)y(k-1)=f(k,ε)(1≤k≤N-1)B₁y≡-y(0)+c₁y(1)=a,B₂y≡-c₂y(N-1)+y(N)=β where e is a small parameter, c₁, c₂,α,β constants and a(kε),b(kε),ƒ(kε)(1≤k≤N) functions of k and ε. Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P_a) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method.

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