The Second-order Quasilinear Boundary Value Problem Of Singularly Perturbed Difference-differential Equations

In recently years, solutions with internal layers are very hot in singularly per-turbed theory. Many good consequences about the formal asymptotic solution and the solution existence have been got yet. And the existence of the solution in those problems was proved mainly based on "differential inequality" method. However, in practicle, the variation of most problems is often not isolated in time, the solu-tion's stability of the system may be change because of the delayed affection. There-fore, considering the structure and the expression of the uniformly valid asymptotic solution is particularly important.In this paper, a kind of second-order quasilinear singularly perturbed difference-differential equations is considered. Under some hypothesis conditions, the origi-nal problems are patitioned into the two pure boundary layer singularly perturbed problems. Combinating the "fractional steps method" with the "boundary layer function method" in difference-differential equation, formal asymptotic solution is constructed. By means of sewing orbit smooth, we get the uniformly valid solution in the whole interval, and consider the error estimation between the true solution and the asymptotic solution, which proves the existence of solutions of the original problem. As to deal with the high dimension of the singularly perturbed problems with the internal layers, this method is also valid. Finally, a specific example was given to demonstrate the feasibility of the method.