A Cotangent Fractional Derivative with the Application

: In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville D σ , γ and Caputo cotangent fractional derivatives C D σ , γ , respectively, and their corresponding integral I σ , γ . The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if γ = 1 we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the D σ , γ , C D σ , γ and I σ , γ . Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.

In [17], the authors presented the conformable derivative (CD) of x of order γ ∈ [0, 1] is: where the drawback is that lim γ→0 + D γ x(t) = x(t). The author [18] presented some concepts of CD and raised an open problem about how to use CD to produce a more general FD. The general FD and fractional integrals (FI) proposed and studied [19,20] provided a response to this problem. In [21,22], Anderson hence improved the CD, i.e., lim γ→0 + D γ x(t) = x(t). In [23][24][25][26][27], the authors presented new types of FD that allow the appearance of the kernel (exponential function or the Mittag-Leffler (ML) function). Nevertheless, the new nonsingular kernel does not possess a semi-group property which makes it difficult to solve certain complicated fractional systems. Concurrently, remarkable efforts have been made to define different types of FD and integrals involving ML functions in their representations; see the papers [28][29][30]. Motivated by the above-mentioned background, we introduce a new type of FC.
Cotangent FD has three features that make it different and special: 1.
The kernel operator is the exponential of the cotangent function, 2.
If order γ = 1, we obtain the RL-FD, C-FD, and RL-FI. Contained in this paper, in Section 2, we provide preliminaries of FC. In Section 3, we present the Riemann-Liouville cotangent fractional derivatives and their corresponding integrals, giving the main results and studying their properties. In Section 4, we present the Laplace transforms for the cotangent Riemann-Liouville fractional derivative and use them to solve linear cotangent fractional differential equations of the Riemann-Liouville type. In Section 5, we present the Caputo cotangent fractional derivatives, the Laplace transforms for the cotangent, and the Caputo fractional derivative and use them to solve linear cotangent fractional differential equations of the Caputo type. Finally, in Section 6, we present the application.

Preliminaries of FC
In this section, we give some definitions for FD and FI that will be for the sake of comparison. Let δ ∈ C, Re(δ) > 0 and a function x : [t i , t f ] −→ R, we state the following definitions: 1.
The left RL-FI of x of order δ is: 6.

The Riemann-Liouville Cotangent Fractional Derivatives
Now, we present the Riemann-Liouville cotangent fractional derivatives and their corresponding integrals, giving the main results and studying their properties. The first time has been presented CD by Khalil et al. [17] as x of order γ is Equation (1).
Then, the proportional derivatives (PD) of x of order γ is: We will confine ourselves to an important special case when f (γ, t) = sin( π 2 γ) and g(γ, t) = cos( π 2 γ). Therefore, (14) becomes Notice that lim γ→0 + D γ x(t) = x(t) and lim γ→1 − D γ x(t) = x (t). We want to search for the integral associated with PD in (15). Let us use the following equation: the solution of (16) is: where cot is the cotangent function, defined by cot(t) = cos(t) sin(t) . The proportional integral (cotangent fractional integral) associated with D γ is defined by: where we accept that t i I 0,γ x(t) = x(t). 1]. The x(t) = e − cot( π 2 γ)t is a nonconstant function. However, (D γ x)(t) = 0.
Proof. We have For producing a general type of FI depending on the cotangent fractional integral Equation (17), we have From (18), we have the following definition.

Proof. 1.
We have and using the Beta function defined by, B(σ 1 , 2. Similar to 1.
, where E σ 1 ,σ 2 is the Mittag-Lefler function [3]. Then Proof. We have In the Theorem 1, we present the semi-group property for the Riemann-Liouville cotangent integral.
Proof. We have Proof. Using the definition and the Lemma 1.7 in [21] we continue l-times in this method until we reach (24).
Proof. From the help of the Theorems 1 and 2, we obtain Proof. From the definition and Theorem 1, we obtain In particular, if l = 1, then Proof. Observing that so we have where the Laplace transform starting from t i is: . We use Equation (29) in Theorem 5, we obtain .
Alternatively, since the Equation (26) can be proved by integration by parts.

Remark 3.
We have x(t i ).
• Lemma 3 is valid for any real σ.
Proof. By the Definition 3, we have from applying (25) in Lemma 3, we obtain and use of the first point in Remark 3, we have and the Theorem 1, we have , with the change of variable m = n σ − k has been used.

with X t i (p) = L t i {x(t)}(p). In particular, if x is continuous at t i then
Proof. By applying Theorems 5 and 6 we have Theorem 8. Let the linear cotangent fractional differential equation: Then, the solution of Equation (32) is: Proof. We apply L t i to (32) and make use of Theorem 7 with n σ = 1, then we obtain Using the inverse of L t i and the fact that (see Theorem 1.9.13 in [3]) Using the convolution formula we obtain (33).

The Caputo Cotangent Fractional Derivative
Now, we present the Caputo cotangent fractional derivatives with a solution of their linear cotangent fractional equations.
Proof. From the Theorem 4 where σ = n σ , we obtain Proof. By using Theorems 5 and 6, we obtain By using Theorems 7 and 10, we obtain the following proposition.
Theorem 11. Let the linear Caputo cotangent fractional differential equation: Then the solution of (40) is: Proof. Applying L t i to (40) and use the Theorem 10 where n σ = 1, we obtain We applying the inverse of L t i and using the Theorem (see Theorem 1.9.13 in [3]) and Using the convolution formula we obtain (41).

Application
Now, we present the application of the SIR model (see for example [42][43][44][45][46][47][48][49][50]). Let the following model: where x(t) the number of susceptible, y(t) the number of infected and z(t) the number of removed individuals at time t. The parameters µ, a and b represent the recruitment rate, the natural death rate, the infection rate, and the removal rate, respectively. Let T(t) be the total population. Then The exact solution of Equation (51) is: We replace the classical derivative by C t i D σ,γ , so from Equation (51), we obtain We are interested in solving Equation (53), which plays a significant role in virology as well as in epidemiology.
Now, we trace the impact of the order of the new type of FD on the dynamics behavior of the solution given by (64). We choose Λ = 10 cells µL −1 day −1 , µ = 0.0139 day −1 , and T(0) = 600 cells µL −1 .
In Figure 1, left T(t) in (64) for different values of σ with γ = 1; Right T(t) in (64) for different values of γ with σ = 0.5. When σ = γ = 1, the graph of (64) coincides with that of the ordinary differential equation given by (52).

Conclusions
We have presented the cotangent fractional derivatives D σ,γ (Riemann-Liouville type) and C D σ,γ (Caputo type) whose kernel contains exponential cotangent function. The advantage of the new type of FD is that they achieve a semi-group property, and we have special cases; if γ = 1 we obtain the RL-FD, C-FD, and RL-FI. We noticed that the function x(t) = e − cot( π 2 γ)t is a nonconstant function, however, C D σ,γ of x(t) is zero. Using the Laplace transform of cotangent derivatives and integrals and we give the exact solution for linear cotangent fractional differential equations. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.
Funding: This research received no external funding. Data Availability Statement: Data sharing is not applicable to this article, as no data sets were generated or analyzed during the current study.