Constructive fractional models through Mittag-Leffler functions

Abstract

In recent decades, there has been a growing interest in generalizing differential equations of arbitrary order by replacing integer derivatives with non-integer derivatives. This allows for the construction of new models that can replicate memory effects due to the non-locality of fractional operators. However, this approach can lead to models that suffer from physical misinterpretation, do not maintain mass balance, and alter the original units of model parameters. This paper introduces an approach based on the generalization of exponential by Mittag-Leffler decays, which is associated with waiting time distributions. The model remains physically interpretable and ensures that the dimensions are constructively corrected. We discuss the mean waiting time, the asymptotic behavior, and apply this theory to develop a novel fractional SIRS model. Additionally, we present numerical results.

A Preliminary definitions and results

Let [a, b] be a finite real interval and $\alpha$ a real number, such that $0\le n-1<\alpha <n$, with n integer. We rewrite basic definitions and results with convenient notation, based in classic references as Oldham and Spanier (1974), Miller and Ross (1993), Samko et al. (1993), Podlubny (1998), Kilbas et al. (2006), and Diethelm (2004).

Definition 1

(Riemann–Liouville integral in finite intervals) The Riemann–Liouville integral of an arbitrary-order $\alpha$ is set to $t\in [a,b]$ by

$$\begin{aligned} I^\alpha _{a+} f(t)=\frac{1}{\Gamma (\alpha )} \displaystyle \int _a^t(t-\theta )^{\alpha -1}f(\theta )\textrm{d}\theta . \end{aligned}$$

In the following, we introduce the definitions of fractional derivatives that are utilized in this study. There are various definitions of these derivatives, each constructed from a specific perspective. In this study, we will focus on the two most common fractional derivatives: the Riemann–Liouville and the Caputo, also known as the Djrbashian–Caputo.

Definition 2

(Riemann–Liouville derivative in finite intervals). The Riemann–Liouville derivative of an arbitrary-order $\alpha$ is set to $t\in [a,b]$ by

$$\begin{aligned} \small D^\alpha _{a+}f(t)=D^n[I^{n-\alpha }_{a+}f(t)]= \frac{1}{\Gamma (n-\alpha )}\left( \frac{\hbox {d}^n}{\hbox {d}t^n}\right) \displaystyle \int _a^t(t-\theta )^{n-\alpha -1} f(\theta )\textrm{d}\theta , \end{aligned}$$

with $D^n$ representing the integer-order derivative.

Definition 3

(Caputo derivative in finite intervals). The Caputo derivative of an arbitrary-order $\alpha$ is set to $t\in [a,b]$ by

$$\begin{aligned} ^CD^\alpha _{a+}f(t)=I^{n-\alpha }_{a+}[D^nf(t)]= \frac{1}{\Gamma (n-\alpha )} \displaystyle \int _a^t(t-\theta )^{n-\alpha -1}\frac{\textrm{d}^n}{\textrm{d} \theta ^n} f(\theta )\textrm{d}\theta \,. \end{aligned}$$

We will now introduce the Mittag-Leffler functions with one, two, and three parameters. The classic one-parameter Mittag-Leffler function is a generalization of the exponential function. It is often referred to as the “queen of special functions” in Fractional Calculus, with related functions being her “court” (Gorenflo et al. 2014). Its significance in Fractional Calculus is similar to the importance of the exponential function in classical Calculus. In fact, the fractional derivative according to Caputo of the Mittag-Leffler function is also a multiple of the function itself, maintaining the main property of calculus. The following definition is presented.

Definition 4

(Mittag-Leffler function with one, two, and three parameters). Let $z\in \mathbb {C}$, and three parameters $\alpha ,\beta \in \mathbb {C}$, $\rho \in \mathbb {R}$, such that $\text {Re} (\alpha )> 0,\text {Re} (\beta )> 0,\rho > 0$. We define the Mittag-Leffler function with three parameters through the power series

$$\begin{aligned} E_{\alpha ,\beta }^\rho (z)=\sum _{k=0}^\infty \dfrac{(\rho )_k}{\Gamma (\alpha k+\beta )}\dfrac{z^ k}{k!}, \end{aligned}$$

where $(\rho )_k$ is the Pochhammer symbol, defined by $(\rho )_k={\Gamma (\rho +k)}/{\Gamma (\rho )}.$

The three-parameter Mittag-Leffler function is also known as the Prabhakar function. Particularly, when $\rho = 1$, we have $(\rho )_k=k!$. In this case, the definition recovers the two-parameter Mittag-Leffler function, denoted $E_{\alpha ,\beta }^1(t)=E _ {\alpha , \beta } (t)$. When $\rho =\beta =1$, we obtain the classic Mittag-Leffler function, denoted $E_{\alpha ,1}^1(t)=E_{\alpha ,1}(t)=E_ {\alpha }(t)$. Its behavior for negative argument is presented in Fig. 12. We recover the exponential function when $\alpha = \beta =\rho = 1$.

Fig. 12

The arrow shows the growth direction of $\alpha$ from 0.2 to 1 in increments of 0.2. The dark blue line represents the exponential function. It is important to note that when $\alpha$ is less than 1, the decay is faster in the beginning but slower in the end

Following these definitions, we present the main Laplace Transform formulas.

Proposition 9

(Laplace transform of the Riemann–Liouville integral) The Laplace transform of the arbitrary-order Riemann–Liouville integral is given by

$$\begin{aligned} \mathcal {L}[I^{\alpha }_{0+}f(t)](s)=s^{-\alpha }\mathcal {L}[f(t)](s). \end{aligned}$$

Proposition 10

(Laplace transform of the Riemann–Liouville derivative) The Laplace transform of the arbitrary-order Riemann–Liouville derivative is given by

$$\begin{aligned} \mathcal {L}[D^{\alpha }_{0+}f(t)](s)=s^{\alpha }\mathcal {L}[f(t)](s)-\displaystyle \sum _{k=0}^{n-1}s^{n-1-k}g^{(k)}(0), \end{aligned}$$

where $g(t)=I^{n-\alpha }_{0+}f (t)$. If f(t) is continuous, so $g^{(k)}(0)=0$ (Li and Deng 2007), and the summation vanishes.

Proposition 11

(Laplace transform of the Caputo derivative) The Laplace transform of the arbitrary-order Caputo derivative is given by

$$\begin{aligned} \mathcal {L}[^CD^{\alpha }_{0+}f(t)](s)=s^{\alpha }\mathcal {L}[f(t)](s)-\displaystyle \sum _{k=0}^{n-1}s^{\alpha -1-k}f^{(k)}(0). \end{aligned}$$

Proposition 12

(Laplace transform of Mittag-Leffler function of real variable) For positive $\lambda$ and $s>\lambda ^{1/\alpha }$, the Laplace transform of the function $t^{\beta -1} E_{\alpha ,\beta }(-\lambda t^\alpha )$, with $t\in [0,\infty ]$ and $0\le \alpha \le \beta \le 1$, is given by

$$\begin{aligned} \mathcal {L}[t^{\beta -1} E_{\alpha ,\beta }(-\lambda t^\alpha )](s)=\dfrac{s^{\alpha -\beta }}{s^{\alpha }+\lambda }. \end{aligned}$$

We finish this section with three Lemmas that will be useful.

Lemma 13

(Integer derivative of three-parameter Mittag-Leffler function) The following identity is valid:

$$\begin{aligned} \dfrac{\textrm{d}^k}{\textrm{d}z^k}E^\rho _{\alpha ,\beta }(z)=(\rho )_kE^{\rho +k}_{\alpha ,\beta +\alpha k} (z). \end{aligned}$$

Particularly

$$\begin{aligned} \dfrac{\textrm{d}}{\textrm{d}z}E_{\alpha }(z)=E^{2}_{\alpha ,1+\alpha } (z). \end{aligned}$$

Lemma 14

(Riemann–Liouville derivative of Mittag-Leffler function) The Riemann–Liouville derivative and the Mittag-Leffler function follow the relation:

$$\begin{aligned} D_{0+}^\alpha [t^{\beta -1}E_{\mu , \beta }(\lambda t^\mu )](x)={t^{\beta -\alpha -1}}E_{\mu , \beta -\alpha }(\lambda x^\mu ). \end{aligned}$$

(30)

Particularly

$$\begin{aligned} D_{0+}^{1-\alpha }[E_{\alpha }(\lambda t^\alpha )](x)={t^{\alpha -1}}E_{\alpha ,\alpha }(\lambda x^\alpha ). \end{aligned}$$

Lemma 15

(Composition of operators) Let $\alpha >0$. If there exists some $\psi \in L_1[a,b]$ such that $f(t)=I^\alpha _{a+}\psi (t)$, then

$$\begin{aligned} I^\alpha _{a+} ~D^\alpha _{a+} f(t)=D^\alpha _{a+}~I^\alpha _{a+}f(t)=f(t) \end{aligned}$$

almost everywhere.