# The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Caputo DOFD

^{d}, we find

^{v}w(t), as well as asking which one would be more convenient, since both may be estimated numerically with roughly equal efficiency.

^{v}w(t) and the ${}_{c}{D}_{d}^{v}w(t)$. Simply put, the ${}_{c}{D}_{d}^{v}w(t)$ has two free parameters, c and d, while the D

^{v}w(t) has only v as a free parameter; this is the major formal difference. In our case, because of the apparent complexity of both the problem and, possibly, of its solution, we tentatively chose to use ${}_{c}{D}_{d}^{v}w(t)$.

## 3. Application to an Input-Output System with Memory Constitutive Equation

^{−1}s

^{−2}; in the case of electric phenomena, w(t) would be the applied electric field and g(t) the induction.

## 4. The Caputo-Fabrizio DOFD

## 5. Composite Materials by a DOFD

^{v}are defined by the Caputo-Fabrizio fractional derivative [13]

#### Dissipation Principle

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Caputo, M. Linear model of Dissipation whose Q is almost Frequency Independent-II. Geophys. J. Int.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Caputo, M. Mean Fractional-Order-Derivatives Differential Equations and Filters; Annali Università Ferrara: Ferrara, Italy, 1995; Volume 41, pp. 73–83. [Google Scholar]
- Caputo, M. Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal.
**2001**, 4, 421–442. [Google Scholar] - Bagley, R.L.; Torvik, P.J. On the existence of the order domain and the solution of distributed order equations—Part I. Int. J. Appl. Math.
**2000**, 2, 865–882. [Google Scholar] - Chechkin, A.V.; Gorenflo, R.; Sokolov, I.M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E
**2002**, 66, 046129. [Google Scholar] [CrossRef] [PubMed] - Lorenzo, C.F.; Hartley, T.T. Variable order and distributed order fractional operators. Nonlinear Dyn.
**2002**, 29, 1–4, 57–98. [Google Scholar] [CrossRef] - Chechkin, A.V.; Gorenflo, R.; Sokolov, I.M. Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal.
**2003**. [Google Scholar] [CrossRef] - Naber, M. Distributed order fractional sub-diffusion. Fractals
**2004**, 12, 23. [Google Scholar] [CrossRef] - Kochubei, A.N. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl.
**2008**, 340, 257–281. [Google Scholar] [CrossRef] - Mainardi, F.; Mura, A.; Pagnini, G. Time-fractional diffusion of distributed order. J. Vib. Control
**2008**, 14, 1267–1290. [Google Scholar] [CrossRef] - Jiao, Z.; Chen, Y.Q.; Podlubny, I. Distributed-Order Dynamic Systems Stability, Simulation, Applications and Perspectives; Springer: Berlin, Germany, 2012. [Google Scholar]
- Gorenflo, R.; Luchko, Y.; Stojanovic, M. Fundamental solution of a distributed order time-fractional diffusion-wave equations as probability density. Fract. Calc. Appl. Anal.
**2013**, 16, 297–316. [Google Scholar] [CrossRef] - Yang, X.J.; Baleanu, D.; Srivastava, H.M. Local Fractional Integral Transforms and Their Applications, 1st ed.; Academic Press: Cambridge, UK, 2015. [Google Scholar]
- Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl.
**2015**, 1, 73–85. [Google Scholar] - Jones, F.R. (Ed.) Handbook of Polymer-Fiber Composites; Longman Scientific and Technical: Essex, UK, 1994. [Google Scholar]
- Schwartz, M.M. Composite Materials Handbook; McGraw-Hill: New York, NY, USA, 1984. [Google Scholar]
- Vinson, J.R.; Sierakowski, R.L. The Behavior of Structures Composed of Composite Materials; Martinus Nighoff Publishers: Dordrecht, The Netherlands, 1986. [Google Scholar]
- Fabrizio, M. Fractional rheological models for thermomechanical systems. Dissipation and free energies. Fract. Calc. Appl. Anal.
**2014**, 17, 206–223. [Google Scholar] - Fabrizio, M.; Giorgi, C.; Morro, A. Minimum principles, convexity, and thermodynamics in viscoelasticity. Contin. Mech. Thermodyn.
**1989**, 1, 197–211. [Google Scholar] [CrossRef]

**Figure 1.**The form of the pseudo-Green function h(t) of the Caputo DOFD for different values of the couple (c, d) c = 0.8 ; d = 0.9 (triangles), c = 0.45; d = 0.55 (squares), c = 0.1; d = 0.2 (diamonds).

**Figure 2.**The time domain expression of the pseudo-Green function or kernel h(t) is shown graphically in the figure, with c = 0.8; d = 0.9; with c = 0.45; d = 0.55 and with c = 0.1; d = 0.2.

**Figure 3.**Comparison of the values of the kernels of Caputo DOFD and Caputo-Fabrizio DOFD when the order is distributed in the interval of integration (0.1–0.2).

**Figure 4.**Comparison of the values of the kernels of the Caputo DOFD and the Caputo-Fabrizio DOFD when the order is distributed in the interval of integration 0.45–0.55.

**Figure 5.**Comparison of the values of the kernels of the Caputo DOFD and the Caputo-Fabrizio DOFD when the order is distributed in the interval of integration 0.8–0.9.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Caputo, M.; Fabrizio, M.
The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials. *Fractal Fract.* **2017**, *1*, 13.
https://doi.org/10.3390/fractalfract1010013

**AMA Style**

Caputo M, Fabrizio M.
The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials. *Fractal and Fractional*. 2017; 1(1):13.
https://doi.org/10.3390/fractalfract1010013

**Chicago/Turabian Style**

Caputo, Michele, and Mauro Fabrizio.
2017. "The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials" *Fractal and Fractional* 1, no. 1: 13.
https://doi.org/10.3390/fractalfract1010013