Finite-difference Method for Parameterized Singularly Perturbed Problem

ε > 0 is a small parameter and {u(x),λ} is a solution. For ε 1, the function u(x) has a boundary layer of thickness O(ε) near x = 0. Under the above conditions, there exists a unique solution to problem (1.1), (1.2) (see [7, 12]). An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [6, 7, 8, 9, 12, 13, 15, 16]. In [7, 9, 12], have also been considered some approximating aspects of this kind of problems. But designed in the above-mentioned papers, algorithms are only concerned with the regular cases (i.e., when the boundary layers are absent). The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problems undergo rapid