Research PaperNo Access

Variational principle of the one-dimensional convection–dispersion equation with fractal derivatives

Pin-Xia Wu

https://orcid.org/0000-0002-4958-8541

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China

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Wei-Wei Ling

School of Social Management, Jiangxi College of Applied Technology, Ganzhou, Jiangxi 341000, P. R. China

E-mail Address: weiweiling@csu.edu.cn

Corresponding author.

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Xiu-Mei Li

School of Social Management, Jiangxi College of Applied Technology, Ganzhou, Jiangxi 341000, P. R. China

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, and

Liang-Jin Xie

School of Social Management, Jiangxi College of Applied Technology, Ganzhou, Jiangxi 341000, P. R. China

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https://doi.org/10.1142/S0217979221501952Cited by:1 (Source: Crossref)

Abstract

The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform.

Keywords:

PACS: 02.30.Xx, 02.60.−x, 02.70.Wz

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