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The different kinds of boundary conditions for standard and fractional diffusion and advection diffusion equations are analyzed. Near the interface between two phases there arises a transition region which state differs from the state of contacting media owing to the different material particle interaction conditions. Particular emphasis has been placed on the conditions of nonperfect diffusive contact for the time-fractional advection diffusion equation. When the reduced characteristics of the interfacial region are equal to zero, the conditions of perfect contact are obtained as a particular case.

In recent years considerable interest has been shown in fractional differential equations which describe important physical phenomena in amorphous, colloid, glassy and porous materials, in fractals and percolation clusters, comb structures, dielectric materials and semiconductors, biological systems, polymers, random and disordered media, geophysical and geological processes (see, for example, [

Different kinds of boundary conditions for time-fractional diffusion equation were analyzed in [

The standard theory of diffusion is based on the balance equation for mass:

Generally speaking, we can consider the balance equation for the transported quantity and a phenomenological law which states the proportionality of the flux to the gradient of this quantity (the Fourier law, the Fick law, the Darcy law,

When the diffusion _{S} being a point at the surface

If the surface of a body is under the given matter flux _{e} from the environment, then at the boundary we have:

The Newton condition of convective mass exchange between a body and the environment with the concentration

When the surfaces of two bodies are in perfect diffusive contact, the concentrations at the contact surfaces are equal:

The boundary conditions presented above are well known and can be found in every textbook on diffusion or heat conduction. We have recalled them here to facilitate obtaining the proper boundary conditions for generalized equations.

The following constitutive equation for the matter flux (see, for example, [

For the sake of simplicity we have restricted ourselves to the case

The convective mass exchange between a body and the environment provides:

The boundary conditions of the perfect diffusive contact have the following form:

The nonclassical theories in which the Fick law

The time-nonlocal dependence between the matter flux and the concentration gradient with the “long-tail” power kernel [

Here

Following [

The constitutive

The corresponding boundary conditions for the time-fractional diffusion-wave

The condition of convective mass exchange between a body and the environment is written as:

The conditions of perfect diffusive contact are the following:

The time-nonlocal generalization of the constitutive

The boundary conditions for

The convective mass exchanged between a medium and the environment yields:

The boundary conditions of the perfect diffusive contact are of the form:

Consider a composite body consisting of three parts: the domain 1, the domain 2, and the intermediate domain designated by the index 0. Matter transport is described by the time-fractional advection diffusion equations appropriate to each domain:

At the boundary surfaces _{1} and _{1} between the intermediate domain and the corresponding body, the conditions of perfect diffusive contact are fulfilled:

To investigate the transport processes in such a composite body in the general case is a very complicated problem. When the thickness 2_{1} and _{2} be the principal radii of curvature of the median surface. If

The time-fractional advection diffusion equation in the intermediate layer _{Σ} is the surface Laplace operator, ∇_{Σ} denotes the surface del operator taking effect along a surface.

Next we average

Introducing the averaged characteristics of concentration Θ_{1} and Θ_{2} is similar to introducing the stress resultants (forces and moments) in the theory of thin elastic shells. The interested reader is referred to the extended literature on this subject (see, for example, [

Furthermore, we use the conditions of perfect diffusive contact at the surfaces _{0} on the coordinate _{0} on the coordinate

When the drift terms are not considered,

For the classical diffusion equation (

When the reduced diffusive characteristics of the median surface are equal to zero (

We have analyzed different kinds of boundary conditions for the standard diffusion equation and advection diffusion equation as well for their fractional counterparts. It should be emphasized that due to the generalized constitutive equations for the matter flux the boundary conditions for the time-fractional diffusion equations have their particular traits in comparison with those for the standard ones. The proper physical boundary conditions should be formulated in terms of the matter flux, not in terms of the normal derivative of concentration alone. Specifying the boundary value of the matter flux in the case of the diffusion equation leads to the Neumann boundary condition, but in the case of the advection diffusion equation leads to the Robin boundary condition. The drift parameter

The author declares no conflict of interest.

Thin intermediate layer between two media.

A contact surface Σ having its own physical characteristics.