Asymptotic Method for Singular Perturbation Problem of Ordinary Difference Equations

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.