## 1. Introduction

## 2. Different Kinds of Boundary Conditions

#### 2.1. The Classical Diffusion Equation

**J**to the concentration gradient:

**x**

_{S}being a point at the surface S.

**J**

_{e}from the environment, then at the boundary we have:

**n**is the outer unit normal to the surface S.

**n**is the common normal at the contact surface.

#### 2.2. The Standard Advection Diffusion Equation

**v**= const. Equation (13) can be interpreted in terms of diffusion or heat conduction with additional velocity field

**v**as well as in terms of Brownian motion, transport processes in porous media, groundwater hydrology, etc. [39–44]. The specification of boundary conditions for the advection diffusion Equation (13) with taking into account the constitutive Equation (12) for the matter flux gives the following kinds of conditions. The Dirichlet boundary condition (4) with the given value of the concentration at the surface of a body remains unchanged. The prescribed boundary value of the matter flux yields:

#### 2.3. The Time-Fractional Diffusion-Wave Equation

#### 2.4. The Time-Fractional Advection Diffusion Equation

## 3. Generalized Conditions of Nonperfect Contact

_{1}and S

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

_{1}and R

_{2}be the principal radii of curvature of the median surface. If $h/{R}_{1}<<1$ and $h/{R}_{2}<<1$, then a thin shell is obtained which allows us to reduce a three-dimensional problem in the intermediate layer to a two-dimensional one for the median surface. Thus we introduce the mixed coordinate system (ξ,η,z), where ξ and η are the curvilinear coordinates in the median surface and z is the normal coordinate ( $-h\le z\le h$), see Figure 1.

_{Σ}is the surface Laplace operator, ∇

_{Σ}denotes the surface del operator taking effect along a surface.

_{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, [54–57]). From Equation (41) we obtain:

_{0}on the coordinate z and proceed to the limit h→0 keeping ${\rho}_{\mathrm{\Sigma}}=2h{\rho}_{0}$, ${\kappa}_{\mathrm{\Sigma}}=2h{\kappa}_{0}$, ${R}_{\mathrm{\Sigma}}=2h/{\kappa}_{0}$ and ${v}_{\{n)\phantom{\rule{0.2em}{0ex}}\mathrm{\Sigma}}={v}_{(n)0}/(2h)$constant. It should be noted that more general polynomial dependence of the concentration c

_{0}on the coordinate z or the operator method can also be used (see [58,59]). As a result we obtain the generalized boundary conditions at the median surface Σ (see Figure 2):

## 4. Conclusions

**v**alters the convective mass transfer coefficient H. The transition region between two phases has been considered as a distinct phase having its own reduced characteristics (the reduced mass density ${\rho}_{\mathrm{\Sigma}}$, the reduced diffusive conductivity ${\kappa}_{\mathrm{\Sigma}}$, the reduced diffusive resistance ${R}_{\mathrm{\Sigma}}$, and the reduced drift parameter ${v}_{(n)\mathrm{\Sigma}}$ of the median surface), and the generalized conditions of nonperfect contact have been obtained.

## Conflict of Interest

## References

- Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] - Metzler, R.; Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen.
**2004**, 37, R161–R208. [Google Scholar] - Zaslavsky, G.M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep.
**2002**, 371, 461–580. [Google Scholar] - Magin, R.L. Fractional Calculus in Bioengineering; Begell House Publishers: Redding, CT, USA, 2006. [Google Scholar]
- Gafiychuk, V.; Datsko, B. Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl.
**2010**, 59, 1101–1107. [Google Scholar] - Uchaikin, V.V. Fractional Derivatives for Physicists and Engineers; Springer: Berlin, Germany, 2013. [Google Scholar]
- Povstenko, Y. Fractional Thermoelasticity; Springer: New York, NY, USA, 2015. [Google Scholar]
- Hoffmann, K.H.; Essex, C.; Schulzky, C. Fractional diffusion and entropy production. J. Non-Equilibr. Thermodyn.
**1998**, 23, 166–175. [Google Scholar] - Essex, C.; Schulzky, C.; Franz, A.; Hoffmann, K.H. Tsallis and Rényi entropies in fractional diffusion and entropy production. Physica A
**2000**, 284, 299–308. [Google Scholar] - Li, X.; Essex, C.; Davison, M.; Hoffmann, K.H.; Schulzky, C. Fractional diffusion, irreversibility and entropy. J. Non-Equilibr. Thermodyn
**2003**, 28, 279–291. [Google Scholar] - Cifani, S.; Jakobsen, E.R. Entropy solution theory for fractional degenerate convection-diffusion equations. Annales de l’Institut Henri Poincare (C) Nonlinear Anal
**2011**, 28, 413–441. [Google Scholar] - Magin, R.; Ingo, C. Entropy and information in a fractional order model of anomalous diffusion, Proceedings of the 16th IFAC Symposium on System Identification, Brussels, Belgium, 11–13 July 2011; pp. 428–433.
- Magin, R.; Ingo, C. Spectral entropy in a fractional order model of anomalous diffusion, Proceedings of the 13th International Carpathian Control Conference, High Tatras, Slovakia, 28–31 May 2012; pp. 458–463.
- Prehl, J.; Essex, C.; Hoffmann, K.H. Tsallis relative entropy and anomalous diffusion. Entropy
**2012**, 14, 701–706. [Google Scholar] - Prehl, J.; Boldt, F.; Essex, C.; Hoffmann, K.H. Time evolution of relative entropies for anomalous diffusion. Entropy
**2013**, 15, 2989–3006. [Google Scholar] - Magin, R.L.; Ingo, C.; Colon-Perez, L.; Triplett, W.; Mareci, T.H. Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Microporous Mesoporous Mater
**2013**, 178, 39–43. [Google Scholar] - Ingo, C.; Magin, R.L.; Colon-Perez, L.; Triplett, W.; Mareci, T.H. On random walks and entropy in diffusion-weighted magnetic resonance imaging studies of neural tissue. Magn. Reson. Med.
**2014**, 71, 617–627. [Google Scholar] - Ingo, C.; Magin, R.L.; Parrish, T.B. New insight into the fractional order diffusion equation using entropy and kurtosis. Entropy
**2014**, 16, 5838–5852. [Google Scholar] - Zhuang, P.; Liu, F.; Anh, V.; Turner, I. Numerical treatment for the fractional Fokker-Planck equation. ANZIAM J
**2007**, 48, C759–C774. [Google Scholar] - Chen, C.; Liu, F.; Turner, I.; Anh, V. Implicit difference approximation of the Galilei invariant fractional advection diffusion equation. ANZIAM J
**2007**, 48, C775–C789. [Google Scholar] - Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput.
**2007**, 191, 12–20. [Google Scholar] - Liu, F.; Zhuang, P.; Burrage, K. Numerical methods and analysis for a class of fractional advection-dispersion models. Comp. Math. Appl.
**2012**, 64, 2990–3007. [Google Scholar] - El-Sayed, A.M.A.; Behiry, S.H.; Raslan, W.E. Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation. Comp. Math. Appl.
**2010**, 59, 1759–1765. [Google Scholar] - Golbabai, A.; Sayevand, K. Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain. Math. Comp. Model.
**2011**, 53, 1708–1718. [Google Scholar] - Panday, R.K.; Singh, O.P.; Baranwal, V.K. An analytic algorithm for the space-time fractional advection-dispersion equation. Comp. Phys. Commun.
**2011**, 182, 1134–1144. [Google Scholar] - Saadatmandi, A.; Dehghan, M.; Azizi, M.-R. The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simulat.
**2012**, 17, 4125–4136. [Google Scholar] - Parvizi, M.; Eslahchi, M.R.; Dehghan, M. Numerical solution of fractional advection-diffusion equation with a nonlinear source term. Numer. Algor.
**2015**, 68, 601–629. [Google Scholar] - Zhang, X.; Huang, P.; Feng, X.; Wei, L. Finite element method for two-dimensional time-fractional Tricomi-type equations. Numer. Methods Partial Differ. Equ.
**2013**, 29, 1081–1096. [Google Scholar] - Yang, Q.; Moroney, T.; Burrage, K.; Turner, I.; Liu, F. Novel numerical methods for time-space fractional reaction diffusion equation in two dimensions. ANZIAM J
**2011**, 52, C395–C409. [Google Scholar] - Jiang, W.; Lin, Y. Approximate solution of the fractional advection-dispersion equation. Comp. Phys. Commun.
**2010**, 181, 557–561. [Google Scholar] - Fu, Z.-J.; Chen, W.; Yang, H.-T. Boundary particle method for Laplace transformed time-fractional diffusion equations. J. Comput. Phys.
**2013**, 235, 52–66. [Google Scholar] - Sun, H.G.; Zhang, Y.; Chen, W.; Reeves, D.M. Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contaminant Hydrol
**2014**, 157, 47–58. [Google Scholar] - Chen, W.; Sun, H.; Zhang, X.; Korošak, D. Anomalous diffusion modeling by fractal and fractional derivatives. Comp. Math. Appl.
**2010**, 59, 1754–1758. [Google Scholar] - Gafiychuk, V.; Datsko, B.; Meleshko, V.; Blackmore, D. Analysis of the solutions of coupled nonlinear fractional reaction-diffusion equations. Chaos Solitons Fractals
**2009**, 41, 1095–1104. [Google Scholar] - Povstenko, Y. Different kinds of boundary conditions for time-fractional heat conduction equation, Proceedings of the 13th International Carpathian Control Conference, High Tatras, Slovakia, 28–31 May 2012; pp. 588–591.
- Povstenko, Y.Z. Fractional heat conduction in infinite one-dimensional composite medium. J. Therm. Stresses
**2013**, 36, 351–363. [Google Scholar] - Povstenko, Y. Fractional heat conduction in an infinite medium with a spherical inclusion. Entropy
**2013**, 15, 4122–4133. [Google Scholar] - Podstrigach, Y.S.; Povstenko, Y.Z. Introduction to the Mechanics of Surface Phenomena in Deformable Solids; Naukova Dumka: Kiev, Ukraine, 1985; In Russian. [Google Scholar]
- Kaviany, M. Principles of Heat Transfer in Porous Media; Springer: NewYork, NY, USA, 1995. [Google Scholar]
- Feller, W. An Introduction to Probability Theory and Its Applications; John Wiley & Sons: New York, NY, USA, 1971. [Google Scholar]
- Scheidegger, A.E. The Physics of Flow through Porous Media; University of Toronto Press: Toronto, Canada, 1974. [Google Scholar]
- Van Kampen, N.G. Stochastic Processes in Physics and Chemistry; North-Holland: Amsterdam, The Netherlands, 1981. [Google Scholar]
- Risken, H. The Fokker-Planck Equation; Springer: Berlin, Germany, 1989. [Google Scholar]
- Nield, D.A.; Bejan, A. Convection in Porous Media; Springer: New York, NY, USA, 2006. [Google Scholar]
- Povstenko, Y.Z. Fractional heat conduction equation and associated thermal stresses. J. Therm. Stresses
**2005**, 28, 83–102. [Google Scholar] - Povstenko, Y.Z. Thermoelasticity which uses fractional heat conduction equation. J Math. Sci
**2009**, 162, 296–305. [Google Scholar] - Povstenko, Y.Z. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr.
**2009**, 136, 014017. [Google Scholar] - Povstenko, Y.Z. Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stresses
**2011**, 34, 97–114. [Google Scholar] - Povstenko, Y. Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal.
**2011**, 14, 418–435. [Google Scholar] - Gorenflo, R.; Mainardi, F. Fractional calculus: integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer: New York, NY, USA, 1997; pp. 223–276. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Povstenko, Y. Theory of diffusive stresses based on the fractional advection-diffusion equation. In Fractional Calculus: Applications; Abi Zeid Daou, R., Moreau, X., Eds.; NOVA Science Publishers: New York, NY, USA, 2015; pp. 227–241. [Google Scholar]
- Goldenveiser, A.L. Theory of Thin Shells; Pergamon Press: Oxford, UK, 1961. [Google Scholar]
- Naghdi, P.M. The theory of shells and plates. In Handbuch der Physik; Truesdell, C., Ed.; Springer: Berlin, Germany, 1972; pp. 425–640. [Google Scholar]
- Vekua, I.N. Some General Methods of Constructing Different Variants of Shell Theory; Nauka: Moscow, Russia, 1982; In Russian. [Google Scholar]
- Ventsel, E.; Krauthammer, T. Thin Plates and Shells: Theory, Analysis, and Applications; Marcel Dekker: New York, NY, USA, 2001. [Google Scholar]
- Podstrigach, Ya.S. Temperature field in a system of solids conjugated by a thin intermediate layer. Inzh.-Fiz. Zhurn
**1963**, 6, 129–136, In Russian. [Google Scholar] - Podstrigach, Ya.S.; Shvetz, R.N. Thermoelasicity of Thin Shells; Naukova Dumka: Kiev, Ukraine, 1978; In Russian. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).