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

Abstract

In this work, an algorithmic procedure is given to implement the solution of a two-point boundary value problem for a nonlinear two-timescale discrete-time system.

A class of nonlinear perturbed difference equations

“…The method of asymptotic expansions has been widely used when solving singularly perturbed differential equations, but has not yet been developed for singularly perturbed difference equations which remains an open field of research. In previous papers [5,6,7,8,9,10], we studied linear models, we noticed that there is no need to add correction terms in the asymptotic expansion, as for the singularly perturbed differential equations. In this work, we extend results obtained in [5,6], to a class of nonlinear difference equations containing a small parameter.…”

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

“…Kelley [4] studied a general second order model with a right-end perturbation; his method is given for zeroth and first order of approximation but higher order approximations have not been made explicitly. 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.…”