Diffusion equations with spatially dependent coefficients and fractal Cauer-type networks

Abstract

In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment \(h>0\) , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all \(h>0\) the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15, 16] and Oustaloup and co-workers [14].