Progress in Fractional Differentiation & Applications

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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.

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