常差分方程奇异摄动问题的渐近方法

Asymptotic Method for Singular Perturbation Problem of Ordinary Difference Equations

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)B_1y≡-y(0)+c_1y(1)=a,B_2y≡-c_2y(N-1)+y(N)=β这里ε是一个小参数,c_1,c_2,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.

This paper is taken up for the following difference equation problem(P2):where is a small parameter, c1, c2, a,constants and functions of k and e. Firstly, the case with constant coefficients is considered. Secondly,a general method based on extended transformation is given to handle (P,), where the coefficients may be variable and uniform asymptotic expansions are obtained, Finally, a numerical example is provided to illustrate the proposed method.