The Volterra type equations related to the non-Debye relaxation

TL;DR

This paper models non-Debye relaxation processes using Volterra equations with Prabhakar function kernels, providing explicit solutions and connecting deterministic and stochastic approaches.

Contribution

It introduces a novel integro-differential equation framework with Prabhakar kernels for non-Debye relaxation, including explicit solutions and links to stochastic models.

Findings

Recovers the Cole-Cole model as a special case.

Provides explicit solutions using umbral calculus and Laplace transform.

Establishes equivalence with stochastic relaxation models.

Abstract

We investigate a possibility to describe the non-Debye relaxation processes using the Volterra-type equations with kernels given by the Prabhakar functions with the upper parameter $\nu$ being negative. Proposed integro-differential equations mimic the fading memory effects and are explicitly solved using the umbral calculus and the Laplace transform methods. Both approaches lead to the same results valid for admissible domain of the parameters $\alpha$, $\mu$ and $\nu$ characterizing the Prabhakar function. For the special case $\alpha\in (0,1]$, $\mu=0$ and $\nu=-1$ we recover the Cole-Cole model, in general having a residual polarization. We also show that our scheme gives results equivalent to those obtained using the stochastic approach to relaxation phenomena merged with integral equations involving kernels given by the Prabhakar functions with the positive upper parameter.