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Derivation of the Fractional Dodson Equation and Beyond: Transient Diffusion With a Non-Singular Memory and Exponentially Fading-Out Diffusivity |
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PP: 255-270 |
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doi:10.18576/pfda/030402
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Author(s) |
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Jordan Hristov,
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Abstract |
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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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