A class of singularly perturbed semi-linear boundary value problems with discontinuous functions is examined in this article. Using the boundary layer function method, the asymptotic solution of such a problem is given and shown to be uniformly effective. The existence and uniqueness of the solution for the system is also proved. Numerical result is presented as an illustration to the theoretical result.
We study a time delay equation for the lossless transmission line model. Under suitable conditions, by using the continuation theorem of the coincidence degree theory, the existence of the periodic solution for the nonlinear functional differential equation is obtained.
In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = a. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.
