The Lorenzo–Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics

Abstract

Considering the applications and a great significance of the study pertaining to generalized functions in applied sciences, the author investigated the Lorenzo–Hartley’s function which is a generalization of classical functions widely used in fractional calculus. The Laplace transform pairs are derived and the requirements for the Lorenzo–Hartley’s function $G_{\nu , \mu ,\delta }(a,c,t)$ to be completely monotonic (for $t > 0$) are investigated. The author has shown the applications of this generalized function to describe the relaxation models, particularly in dielectrics. The Lorenzo–Hartley’s function $G_{\nu , \mu ,\delta }(a,c,t)$ of a real variable t is considered to investigate a computable mathematical framework for standard Debye and non-Debye relaxation processes in dielectric materials. The non-negative spectral distribution function is obtained for the corresponding response function. It is also demonstrated that the classical models like Cole–Cole (C–C), Davidson–Cole (D–C), and Havriliak–Negami (H–N) for non-Debye relaxation and standard Debye relaxation are the particular cases of the Lorenzo–Hartley’s function. Some of the study cases are also worked out to visualize the effects of variations of parameters on the response function and corresponding spectral distribution function. A generalized and unified fractional relaxation differential equation which governs response functions for classical dielectrics models pertaining to non-Debye relaxation (C–C, D–C, and H–N) is also established in the present investigation.