Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem

Abstract

In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy″+ky=f(t), y′(a)=0, y′(b)=0, k>0, with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem.

Analysis of Finite Solution Spaces of Second-Order ODE with Dirac Delta Periodic Forcing

“…Specifically, in computer network modeling involving sporadic jitters, the constant factor is considered to be positive (i.e., k > 0). The main focus of the analysis of such differential equations with discontinuous forcing is to ascertain the convergence in the solution spaces and the rapidity of such convergence [19]. In general, the ODEs of perturbed systems consider uniform convergence of solutions within [a, b], where the solution intervals are finite in the set of reals R. Let us consider that the forcing factor f (.)…”

“…In general, the ODEs of perturbed systems consider uniform convergence of solutions within [a, b], where the solution intervals are finite in the set of reals R. Let us consider that the forcing factor f (.) is restricted to relative smoothness (i.e., the forcing f ∈ C 3 ([a, b]) for a singularly perturbed system [19]. This indicates that the governing equations of such systems do not consider discontinuities in the forcing factors.…”

“…Note that numerical solution approaches are frequently employed rather than complete analytical methods in order to avoid complexities. It has been observed that numerical solution approaches are prone to the effects of varying dimensionalities [14,19]. In such cases, the convergence analyses deal with the limiting value property δ ε → δ, ε > 0 through the regularization of the discrete delta function δ ε towards the Dirac delta forcing.…”