Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model
Abstract
:1. Introduction
2. Preliminaries
- (P1)
- The sets , , and are convex cones, closed under pointwise limits.
- (P2)
- if and only if .
- (P3)
- Let . Then, if and only if .
- (P4)
- Any function has an analytic continuation to the upper half-plane,
- (P5)
- Any function φ from the classes or admits an analytic extension to the complex plane cut along the negative real axis , such that
- (P6)
- Let φ be the analytic extension of a complete Bernstein function to . Then, the limit
- (P7)
- If , then
3. Model Formulation
4. First Integral Representation
5. Examples
5.1. Fractional Derivative of Uniformly Distributed Order
5.2. Convolutional Derivative with Multinomial Mittag-Leffler-Type Kernel
6. Second Integral Representation
7. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Bazhlekova, E.; Pshenichnov, S. Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model. Fractal Fract. 2023, 7, 636. https://doi.org/10.3390/fractalfract7080636
Bazhlekova E, Pshenichnov S. Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model. Fractal and Fractional. 2023; 7(8):636. https://doi.org/10.3390/fractalfract7080636
Chicago/Turabian StyleBazhlekova, Emilia, and Sergey Pshenichnov. 2023. "Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model" Fractal and Fractional 7, no. 8: 636. https://doi.org/10.3390/fractalfract7080636