Analysis, Modeling and Stability of Fractional Order Differential Systems 1

4Fractional Modeling of the Diffusive Interface

4.1. Introduction

Many physical and chemical processes, such as heat transfer in materials [BAT 01], eddy currents in electrical machines [RET 99], electrochemistry in batteries [SAB 06] and so on, are governed by diffusive partial derivative equations.

Practically, we are mainly concerned by the modeling of the diffusive interface at the boundary of the considered process, such as the voltage/current relationship in an accumulator [LIN 00] or the temperature/heat flux for heat transfer [BEN 06].

It has been demonstrated long ago [OLD 74] that the diffusive interface is characterized by a fractional order behavior in the frequency domain with n = 0.5. It is for this reason that a great number of research works have been devoted to the fractional modeling of diffusive interfaces, using various fractional order models [COI 01, LIN 01a]. Some of these models are motivated by physical properties [BAT 02], but many others are only identification (or black box) models motivated by the efficiency of the approximation.

Therefore, in this chapter, our purpose is to investigate why fractional models are adapted to the modeling of diffusive interfaces, and whether any fractional model can provide a correct physical response to this problem [MAA 15].

In order to formulate an objective response, we use a prototype interface, i.e. heat transfer in a plane wall.

Two families of fractional models are investigated: commensurate order models, ...