The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics. (English) Zbl 1401.26015
Summary: 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.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 33E12 | Mittag-Leffler functions and generalizations |
| 33E20 | Other functions defined by series and integrals |
| 44A10 | Laplace transform |
Keywords:
complete monotonicity; complex susceptibility; dielectric relaxation; Havriliak-Negami; Davidson-Cole; Cole-Cole; Laplace transform; Lorenzo-Hartley function; relaxation processes; response function; spectral distribution function; fractional relaxation equationReferences:
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