TY - JOUR
TI - The Dirichlet problem of a conformable advection-diffusion equation
AU - Avci Derya
AU - Iskender Eroğlu Beyza Billur
AU - Özdemir Necati
JN - Thermal Science
PY - 2017
VL - 21
IS - 1
SP - 9
EP - 18
PT - Article
AB - 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.