Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

Abstract

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented.

General Fractional Calculus Operators of Distributed Order

“…In the papers [13][14][15][16][17], Tarasov suggested several non-local physical theories based on the GFI and GFDs with Sonin kernels, including general fractional dynamics, general non-Markovian quantum dynamics, general non-local electrodynamics, non-local classical theory of gravity, and non-local statistical mechanics. Furthermore, the GFI and GFDs with Sonin kernels were used in some mathematical models for anomalous diffusion and in linear viscoelasticity; see, e.g., [18][19][20][21][22].…”