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Anomalous relaxation in dielectrics with Hilfer fractional derivative

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 Publication date 2020
  fields Physics
and research's language is English




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We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order {xi}, with 0<{xi}<1 and type {eta}, with 0<{eta}<1. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. Finally, the empirical models are recovered as particular cases.



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We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $frac{partial}{partial t}P(x,t)=D frac{partial^{gamma}}{partial x^{gamma}}[P(x,t) ]^{ u}$. Exact time-dependent solutions are found for $ u = frac{2-gamma}{1+ gamma}$ ($-infty<gamma leq 2$). By considering the long-distance {it asymptotic} behavior of these solutions, a connection is established, namely $q=frac{gamma+3}{gamma+1}$ ($0<gamma le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., $ u=1$ and $0<gamma le 2$). Finally, for $(gamma, u)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $gamma=2$ solutions available in the literature ($ u<1$ porous medium equation), thus exhibiting nonuniform convergence.
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
65 - Trifce Sandev , Weihua Deng , 2018
Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.
The stochastic solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < beta < alpha le 2$, where $0 < beta le 1$ and $0 < alpha le 2$ are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuos time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Levy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.
For the following semilinear equation with Hilfer- Hadamard fractional derivative begin{equation*} mathcal{D}^{alpha_1,beta}_{a^+} u-Deltamathcal{D}^{alpha_2,beta}_{a^+} u-Delta u =vert uvert^p, qquad t>a>0, qquad xinOmega, end{equation*} where $Omegasubset mathbb{R}^N$ $(Ngeqslant 1)$, $p>1$, $0<alpha _{2}<alpha _{1}<1$ and $0<beta <1$. $mathcal{D}^{alpha_i,beta}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $alpha_i$ and of type $beta$, we establish the necessary conditions for the existence of global solutions.
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