Anomalous relaxation in dielectrics with Hilfer fractional derivative

Abstract

We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order ξ, with 0<ξ<1 and type η, with 0<η<1. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. Finally, the empirical models are recovered as particular cases.

arXiv comment20 pages

References in corpus (6)

• No Violation of the Leibniz Rule. No Fractional Derivative
• No Nonlocality. No Fractional Derivative
• Models of dielectric relaxation based on completely monotone functions
• Prabhakar-like fractional viscoelasticity
• On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics
• A new Generalized fractional derivative and integral