Trap-controlled fractal diffusion model of an atypical dielectric response

Ion trapping model of the ac conductivity in disordered solids

In this paper, we develop an approach based on ions trapping to describe the conductivity spectra in disordered solids. Applying the multiple trapping model and the diffusion equation for ions, we obtained a new expression for the ac conductivity, which allows us to describe the conductivity spectra in wide ranges of frequencies and temperatures. In the high-temperature region, the new expression for the ac conductivity reproduces the Jonscher behavior, and with decreasing temperatures transforms into an expression of the NCL-type. A new expression for the ac conductivity was successfully tested on the conductivity spectra of ion-conducting glasses.

Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.