Abstract
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.
Highlights
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
It should be emphasized that entropy is used in the analysis of anomalous diffusion processes and fractional diffusion equations [8,9,10,11,12,13,14,15,16,17,18]
The spectral entropy for Entropy 2015, 17 the case of fractional diffusion equation was calculated by Magin and Ingo [12,13]
Summary
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 The authors of [31] considered the time-fractional advection-diffusion equation and used the Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation in the transform domain They employ the boundary particle method to solve the transformed problem and implement the Stehfest numerical inverse Laplace transform. Small thickness of the interface region allows us to consider it as a distinct two-dimensional phase and to formulate the corresponding two-dimensional equations for the interface In this approach, mathematical description of processes occurring in the bulk phases consists in formulation and solution of some system of differential (or more complicated) equations with certain boundary conditions being the two-dimensional analogue of the corresponding three-dimensional equations [38]
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