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 ν 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 α, μ and ν characterizing the Prabhakar function. For the special case α ∈ (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.

我们研究了使用Volterra型方程描述非Debye松弛过程的可能性，该方程的内核由Prabhakar函数给出，且上限参数ν为负。拟议的积分微分方程模拟了衰落记忆效应，并使用本影演算和Laplace变换方法明确求解。两种方法均得出相同的结果，这些结果对于表征Prabhakar函数的参数α，μ和ν的允许域有效。对于特殊情况α  ＆Element;（0,1]，$\mu =0$ 和 $\nu =-\mathrm{1个}$我们恢复了Cole-Cole模型，通常具有残留极化。我们还表明，我们的方案给出的结果与使用松弛现象的随机方法与与包含正上参数的Prabhakar函数给出的核的积分方程合并的积分方程所获得的结果相同。