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

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.


I. INTRODUCTION
In the recently published article 1 its authors Dazhi Zhao and HongGuang Sun studied the linear integro-differential equation where the kernel k(t; α) = e −γ α,β (t; λ) is given by the Prabhakar function which parameters satisfy 0 < γ ≤ 1 and α, β > 0, α + β = 1. For this range of parameters recall that the Laplace transform of k(t; α), namely K(s, α) = s −αγ−β (s α − λ) γ , satisfies the condition lim s→∞ [sK(s, α)] −1 = 0, which according to 1  showed that the model extends the Cole-Cole relaxation pattern and contains as the limiting case α → 1 the standard Debye relaxation. Here we would like to emphasize that just mentioned two cases do not exhaust possible mutual relations which link the relaxation phenomena and using the Eq. (I.1) for modeling their time behavior. An instructive example is an application of Eq. (I.1)like equation to describe the Havriliak-Negami relaxation, the most widely used "asymmetric" generalization of the Debye and Cole-Cole approaches. In the review paper 2 the authors presented a detailed analysis of equations describing the time behavior of the Havriliak-Negami relaxation function Ψ α,γ (t). They came to the conclusion that it is governed by a non-homogenous equation where the pseudo-differential operator C ( 0 D α t + τ −α ) γ is a Caputo-like counterpart of the operator ( 0 D α t + τ −α ) γ , the latter understood as an infinite binomial series of the Riemann-Liouville fractional derivatives ? . Next, using results of 3 , they argued that the operator C ( 0 D t α + τ −α ) γ may be represented in terms of an integro-differential operator involving the Prabhakar function in the kernel, the object usually nick-named the Prabhakar derivative. Adjusted to our notation the suitable equations 2 (Eq. (B.23)) read where ⋆ denotes the convolution operator. This justifies the condition β = 1 − αγ to appear in Eq. (I.1) as meaningful for understanding properties of physically admissible relaxation models.
In 4 it has been also shown that the nonlinear heat conduction equations with memory involving Prabhakar derivative can be characterized by Eq. (I.1) in which β = 1 − αγ.
The Laplace transform method applied to Eq.
will be used throughout, since this constraint neither harms nor restricts our further considerations.
In 1 the authors used the fact that the inverse Laplace transform of the geometric series (which results after pulling out K(s, α) in the nominator and denominator of Eq. (I.2) and subsequently reducing it) may be performed termwise. This leads to their main result formulated as 1 (Theorem

IV. CONCLUSION
We would like to point out that our result is complementary to the result given in 1 (Theorem 3.1) and extends it to the full range of τ > 0. This places it within the general scheme developed by A. N. Kochubei 5 who investigated the Cauchy problem for evolution equations governed by the integro-differential operator In addition some requirements are put on the Laplace transform K(s, α) of the kernel k(t, α).
which is the Debye relaxation function in time domain.