We apply a multiple–time version of the reductive perturbation method to study long waves as governed by the Boussinesq model equation. By requiring the absence of secular producing terms in each order of the perturbative scheme, we show that the solitary–wave of the Boussinesq equation can be written as a solitary–wave satisfying simultaneously all equations of the KdV hierarchy, each one in a...

This lecture is a survey of the theory of stability and error bounds for numerical methods for stii initial value problems. We begin with problems in singular perturbation form, where the distinction between \fast" and \slow" variables is clearest. Next we explain Kreiss's characterization of stii problems that includes singular perturbation problems as a special case. The requirement of unifor...

This paper deals with a class of chemical process with measurable time-varying disturbances, which is modeled within the framework of singular perturbation in non-standard form. The results in singular perturbation theory consider the systems in a standard form, therefore, a transformation to change a system representation from non-standard to standard form should be found. After this transform...

Here Ω is the unit disk centered at the origin, containing a small hole of radius ε centered at x = 0, i.e. Ωε = {x | |x| ≤ ε}. Calculate the exact solution, and from it determine an approximation to the solution in the outer region |x| ≫ O(ε). Can you re-derive this result from singular perturbation theory in the limit ε → 0? (Hint: the leading-order outer problem for Case I is different from ...

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. W...

In this contribution we study some topics of KAM theory on the so-called standard map. We consider the invariant curve of rotation number the golden mean obtained by analytical continuation, with respect the parameter ", of the invariant circle of such rotation number corresponding to the case " = 0. We show that if we consider the parameterization that conjugates the dynamics of this curve to ...

in chemical engineering, several processes are represented by singular boundary value problems. in general, classical numerical methods fail to produce good approximations for the singular boundary value problems. in this paper, chebyshev finite difference (chfd) method and dtm-pad´e method, which is a combination of differential transform method (dtm) and pad´e approximant, are applied for sol...

Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The onedimensional steady-state Poisson-Nernst-Planck (PNP) system is a useful representation of these devices but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of ...