THERMAL SCIENCE

International Scientific Journal

THE DIRICHLET PROBLEM OF A CONFORMABLE ADVECTION-DIFFUSION EQUATION

ABSTRACT
The fractional advection-diffusion equations are obtained from a fractional power law for the matter flux. Diffusion processes in special types of porous media which has fractal geometry can be modelled accurately by using these equations. However, the existing nonlocal fractional derivatives seem complicated and also lose some basic properties satisfied by usual derivatives. For these reasons, local fractional calculus has recently been emerged to simplify the complexities of fractional models defined by nonlocal fractional operators. In this work, the conformable, a local, well-behaved and limit-based definition, is used to obtain a local generalized form of advection-diffusion equation. In addition, this study is devoted to give a local generalized description to the combination of diffusive flux governed by Fick’s law and the advection flux associated with the velocity field. As a result, the constitutive conformable advection-diffusion equation can be easily achieved. A Dirichlet problem for conformable advection-diffusion equation is derived by applying fractional Laplace transform with respect to time t and finite sin-Fourier transform with respect to spatial coordinate x. Two illustrative examples are presented to show the behaviours of this new local generalized model. The dependence of the solution on the fractional order of conformable derivative and the changing values of problem parameters are validated using graphics held by MATLcodes.
KEYWORDS
PAPER SUBMITTED: 2016-04-21
PAPER REVISED: 2016-05-30
PAPER ACCEPTED: 2016-06-15
PUBLISHED ONLINE: 2016-10-01
DOI REFERENCE: https://doi.org/10.2298/TSCI160421235A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE 1, PAGES [9 - 18]
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© 2017 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence