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

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2024-03-22 DOI:10.1007/s13540-024-00264-6

引用次数: 0

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].

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具有空间依赖系数的扩散方程和分形考尔型网络

摘要在本文中，我们提出并解决了扩散方程的表示问题：给定扩散方程的拉普拉斯变换在一个由增量\(h>0\)决定的空间尺度上的空间离散化下的离散化，我们能否构造一个连续于h的Cauer梯形网络族，其构成方程与所有\(h>0\)的离散化相匹配。研究证明，对于均匀几何空间尺度上的有限差分离散化，当且仅当扩散系数是空间变量中的指数函数时，分形考尔网络的表示问题是可能的。这种扩散方程允许一个具有分数行为的（拉普拉斯）传递函数，其指数是明确的。这使我们能够证明 Sabatier 及其合作者[15, 16]和 Oustaloup 及其合作者[14]之前所做工作的正确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464