This paper focuses on the boundary layer phenomenon arising in the study of singularly perturbed differential equations. Our tools include the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear equations under consideration. Key words and phrases: singularly perturbed systems, three–point boundary value problem, method of l...

An upwind difference approximation is used for a singularly perturbed problem in material science. Based on the discrete Green’s function theory, the error estimate in maximum norm is achieved, which is first-order uniformly convergent with respect to the perturbation parameter. The numerical experimental result is verified the valid of the theoretical analysis. Keywords—singularly perturbed, u...

In this paper, we have proposed an ε-uniform initial value technique for singularly perturbed convection-diffusion problems in which an asymptotic expansion approximation of the solution of boundary value problem is constructed using the basic idea of WKB method. In this computational technique, the original problem reduces to combination of an initial value problem and a terminal value problem...

We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary co...

In this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly pertur...