Fractional Advection-Diffusion Equation and Associated Diffusive Stresses

The theory of diffusive stressesDiffusive stresses deals with mechanical and diffusive effects in elastic body. The conventional theory is based on the classical Fick lawFick law, which relates the matter flux to the concentration gradient. In combination with the balance equation for mass, this law leads to the classical diffusion equation. We study nonlocal generalizations of the diffusive flux governed by the Fick lawFick law and of the advection flux associated with the velocity field. The nonlocal constitutive equation with the long-tailLong-tail kernel power memory kernel results in the fractional advection-diffusion equationFractional advection-diffusion equation. The nonlocal constitutive equation with the middle-tailMiddle-tail kernel memory kernel expressed in terms of the Mittag-Leffler functionMittag-Leffler function leads to the fractional advection-diffusion equation of the Cattaneo type. The theory of diffusive stressesDiffusive stresses based on the fractional advection-diffusion equationFractional advection-diffusion equation is formulated. Fundamental solutions to the Cauchy and source problemsSource problem and associated diffusive stressesDiffusive stresses are studied. The numerical results are illustrated graphically.