Abstract
In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.
Highlights
The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is discussed in detail
Ultraslow diffusion processes include a wide class of different stochastic processes characterised by a logarithmic growth of the mean squared displacement, namely h x2 (t)i ∼ logβ t, Citation: De Gregorio, A.; Garra, R
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Summary
Ultraslow diffusion processes include a wide class of different stochastic processes characterised by a logarithmic growth of the mean squared displacement, namely h x2 (t)i ∼ logβ t, Citation: De Gregorio, A.; Garra, R. A relevant example is the Sinai diffusion for which β = 4, which is related to a model in which a particle moves in a quenched random force field [1] Another interesting case is β = 4/3 in polymer physics (see [2]). In a series of relevant papers, the connection between distributed-order fractional differential equations and ultraslow anomalous diffusion has been shown; we refer for example to [4,5,6,7,8]. We exploit the theory of time-changed processes It is known (see, e.g., [14,15]) that the the solution of the time-fractional heat equation. We consider in detail the particular case of heat-type equations based on the Hadamard-type fractional derivative that can be directly related to ultra-slow diffusions. We provide some simple explicit results for the non-linear diffusive case
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