A note on paper "Anomalous relaxation model based on the fractional derivative with a Prabhakarlike kernel" [Z. Angew. Math. Phys. (2019) 70:42]

TL;DR

This paper provides a new solution to an anomalous relaxation model involving the Prabhakar kernel, extending the range of effective relaxation times and connecting to known relaxation patterns like Cole-Cole and Debye models.

Contribution

It offers a complementary solution to the fractional differential equation with the Prabhakar kernel, broadening the admissible relaxation times beyond previous limits.

Findings

Solution valid for all positive relaxation times

Includes special cases like Cole-Cole and Debye models

Extends previous theoretical results

Abstract

Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) which authors D. Zhao and HG. Sun studied the integro-differential equation with the kernel given by the Prabhakar function $e^{-\gamma}_{\alpha, \beta}(t, \lambda)$ we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times which admissible range extends the limits given in \cite[Theorem 3.1]{DZhao2019} to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel the solution comprises to known phenomenological relaxation patterns, e.g. to the Cole-Cole model (if $\gamma = 1, \beta=1-\alpha$) or to the standard Debye relaxation.