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

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 $e_{\alpha ,\beta }^{-\gamma }(t,\lambda )$ 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.

References in corpus (4)

• The Prabhakar or three parameter Mittag--Leffler function: theory and application
• Hilfer-Prabhakar Derivatives and Some Applications
• Models of dielectric relaxation based on completely monotone functions
• Operator solutions for fractional Fokker-Planck equations

Cited by in corpus (6)

• A practical guide to Prabhakar fractional calculus
• General fractional calculus and Prabhakar's theory
• Stability of fractional-order systems with Prabhakar derivatives
• Non-Debye relaxations: smeared time evolution, memory effects, and the Laplace exponents
• The Volterra type equations related to the non-Debye relaxation
• Non-Debye relaxations: The characteristic exponent in the excess wings model