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
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摘要: 在本文中,我们讨论如下差分方程问题(Pε):(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 (Pε):(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 (Pa) 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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