Asymptotic Expansion Method with Respect to Small Parameter for Ternary Diffusion Models

Marek Danielewski,Anna Szafrańska,Henryk Leszczyński

doi:10.1007/s12539-017-0228-5

Abstract

Ternary diffusion models lead to strongly coupled systems of PDEs. We choose the smallest diffusion coefficient as a small parameter in a power series expansion whose components fulfill relatively simple equations. Although this series is divergent, one can use its finite sums to derive feasible numerical approximations, e.g. finite difference methods (FDMs).

Highlights

The subject of divergent series dates back to the distant past and is linked with the names of outstanding mathematicians such as Euler, Poincaré, Borel, Padé, and Birkhoff.For more details and additional references we refer to articles by Tucciarone [1] and Ferraro [2]
The idea of the asymptotic series comes from Poincaré, who introduced the definition of an asymptotic expansion
The importance of methods which are based on asymptotic expansions of solutions in small or large parameters series has grown considerably in many branches of physical, chemical, biological and engineering sciences

Summary

Introduction

The subject of divergent series dates back to the distant past and is linked with the names of outstanding mathematicians such as Euler, Poincaré, Borel, Padé, and Birkhoff. The fundamental problem of reconstruction of physical observables, based on divergent power series expansions, has been considered in many areas of quantum physics. It was found that divergent perturbation expansions of quantum physics are very common [4, 5] This observation led to a substantial amount of research works in various areas of quantum physics, for example in quantum field theory [6,7,8] or in quantum mechanics [9, 10]. The asymptotic analysis of singularly perturbed problems is performed in the monograph [11], based on expansions of analytical solutions in the power series with respect to a small parameter. The coefficients depend on the unknown functions and the parabolicity of this system is conditional, namely it can be proved only if solutions remain in an admissible set The invariance of this set is connected to mass conservation laws. Interdiffusion becomes an important topic in electrochemistry and biochemistry, e.g. molecular channels (nano-channels) [15]

Preliminaries

Small Parameter Method

Numerical Analysis

Finite Difference Method

Numerical Simulations

Conclusions and Remarks