2.1. Riemann-Liouville and Caputo fractional derivatives

Now, to understand a fractional derivative, we start by acknowledging that the n-fold integral of a generic function f tð Þ is given by the formula

$$\underbrace{\int\_{a}^{t} \int\_{a}^{t} \cdots \int\_{a}^{t} f(t) dt dt \dots dt}\_{n \text{ times}} = J\_{a}^{n} f(t) = \frac{1}{(n-1)!} \int\_{a}^{t} (t - t')^{n-1} f(t') dt'.\tag{8}$$

A generalization to non-integer values of n can be performed using the Euler Gamma function Γð Þx , leading to the Riemann-Liouville fractional integral

$$J\_a^n f(t) = \frac{1}{\Gamma(\alpha)} \int\_a^t \left(t - t'\right)^{\alpha - 1} f(t') \, dt',\tag{9}$$

where we have used α to represent the generalization of n to non-integer values. A fractional derivative of any order can then be obtained by manipulating the number of integrations and differentiations of the function f tð Þ. By performing the <sup>m</sup> � <sup>α</sup>-fold integration of the mth derivative of f tð Þ, J <sup>m</sup>� ð Þ with <sup>m</sup> <sup>¼</sup> <sup>α</sup> , we arrive at the generalized derivative formula (Caputo <sup>α</sup>Dmf t d e <sup>a</sup> fractional derivative [6]) of order m � 1 < α < m,

$$\frac{d^a f(t)}{dt^a} = \frac{1}{\Gamma(m-a)} \int\_a^t (t-t')^{-a+m-1} \frac{d^m f(t')}{dt'^m} dt', \quad m-1 < a < m,\tag{10}$$

This last fractional derivative is the one chosen to deal with physical processes due to the ease in handling initial and boundary conditions [7].

Next, we present two models that rely on the Mittag-Leffler function (a function closely related to fractional calculus) to improve their modeling and fitting capabilities when describing the behavior of viscoelastic materials. These are the fractional K-BKZ (integral) and the generalized Phan-Thien and Tanner (differential) models.
