Starting from the Cattaneo constitutive relation with exponential kernel applied to mass diffusion the derivation of a new form the diffusion equation with a relaxation term expressed through the Caputo-Fabrizio time-fractional operator (derivative) has been developed. The developed equation reduces to the fractional Dodson equation for large relaxation times corresponding to low fractional order of the Caputo-Fabrizio derivative. The approach separates large time effects resulting in the classical Dodson equation with exponentially decaying in time diffusivity and the short time relaxation process modeled by Caputo-Fabrizio time fractional derivative. The solution developed allows seeing a new physical background of the Caputo-Fabrizio time-fractional operator (derivative) and to demonstrate a new interpretation of the Dodson equation incorporating fading memory effects. Moreover a new model with two memories corresponding to large and short time relation effects has been conceived. Defining the diffusion process parameters then the fractional order of the Caputo-Fabrizio time fractional derivative can be determined in a straightforward manner as a function of the Deborah number calculated as a ratio of the relaxation time to the characteristic diffusion time of the process.
Digital Object Identifier (DOI)
"Derivation of the Fractional Dodson Equation and Beyond: Transient Diffusion With a Non-Singular Memory and Exponentially Fading-Out Diffusivity,"
Progress in Fractional Differentiation & Applications: Vol. 3
, Article 2.
Available at: https://dc.naturalspublishing.com/pfda/vol3/iss4/2