A class of nonlinear perturbed difference equations

Abstract

In this paper we study a class of nonlinear singularly perturbed difference equations with boundary value conditions. Provide that the singular perturbation parameter is small enough, we give sufficient conditions to guarantee the existence and uniqueness of the solution. An iterative process is proposed to determine the asymptotic representation of the solution.

Boundary Value Problem for a Two-Time-Scale Nonlinear Discrete System

“…by using an algorithmic technique we developed firstly for perturbed difference equations (see [10], [11], [16], [17], [18], [19]). Recently, many researchers have studied discrete versions of boundary value problems (BVPs) (see [1], [4], [5], [8]), and applications of two-point BVP algorithms arise in pollution control problems, nuclear reactor heat transfer and vibration.…”

A class of multi-scales nonlinear difference equations

“…The different scales arrange the convenience to reduce order and separate time-scale by using the singular perturbation methodology to reduce the complexity of these systems. Recently in [9], we developed an iterative method that gives asymptotic solutions for difference equations with one parameter, this procedure was initially introduced in various linear problems, see [3,4,5,6,7,8]. In the present paper, we extend this procedure for a class of nonlinear equations containing several small multiscale parameters and we also allow variations for the boundary values.…”

Nonlinear perturbed difference equations

“…In [10,13], we have elucidated that we could define homogeneous asymptotic expansions for singularly perturbed difference equations bypassing the correction terms and we gave applications to control problems in [14][15][16][17]. Recently in [18,19], we used this homogeneous method for a class of nonlinear equations giving explicitly asymptotic solutions up to any order. The plan of this paper is to extend this procedure based on Faa Di Bruno formula [6] and contraction mapping principle, to a wide class of nonlinear singularly-perturbed difference equations.…”