Title: Generalized diffusion equation with nonlocality of space-time. Memory function modelling
Author(s):

	P.P. Kostrobij (Lviv Polytechnic National University, 12 S. Bandera St., 79013 Lviv, Ukraine) ,
	B.M. Markovych (Lviv Polytechnic National University, 12 S. Bandera St., 79013 Lviv, Ukraine) ,
	M.V. Tokarchuk (Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine; Lviv Polytechnic National University, 12 S. Bandera St., 79013 Lviv, Ukraine)

We presented a general approach for obtaining the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and Zubarev's nonequilibrium statistical operator (NSO) method within Gibbs statistics. The new non-Markovian diffusion equations of ions in spatially heterogeneous environment with fractal structure and generalized Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality are obtained. Dispersion relations are found for the Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality in fractional derivatives. The frequency spectrum, phase and group velocities are calculated. It is shown that it has a wave behaviour with discontinuities, which are also manifested in the behaviour of the phase velocity.

Key words: Cattaneo equation, Cattaneo-Maxwell diffusion equation, Gibbs statistics, nonequilibrium statistical operator

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