The Cotangent Derivative with Respect to Another Function: Theory, Methods and Applications

Abstract

This paper introduces a generalization of the Riemann–Liouville and Caputo cotangent derivatives and their corresponding integrals, known as the Riemann–Liouville and Caputo cotangent derivatives with respect to another function (RAF). These fractional derivatives possess the advantageous property of forming a semigroup. The paper also presents a collection of theorems and lemmas, providing solutions to linear cotangent differential equations using the generalized Laplace transform. Moreover, we present the numerical approach, the application for solving the Caputo cotangent fractional Cauchy problem, and two examples for testing this approach.

1. Introduction

Fractional calculus (FC) deals with integral and differential operators of arbitrary orders, much like conventional calculus, which focuses on integrals and derivatives of non-negative integer orders. However, due to the limitations of traditional calculus operators in modeling certain real-world phenomena, researchers have sought generalizations of these operators. Fractional operators have emerged as powerful tools for modeling long-memory processes and a wide range of phenomena encountered in fields such as chemistry, physics, electricity, mechanics, and various other disciplines. For further exploration, we invite readers to refer to [1,2,3,4] and the references therein. However, to enhance understanding and achieve more accurate modeling of real-world problems, researchers required fractional operators beyond the scope of Riemann–Liouville fractional operators. The literature abounds with numerous works proposing new fractional operators, such as those found in [5,6,7,8]. Nevertheless, the fractional integrals and derivatives introduced in these works were specific cases of fractional integrals/derivatives (FIs/FDs) incorporating a kernel function dependency [9,10,11]. It is worth noting that there exist other types of fractional operators suggested in the literature.

In contrast, to address the challenges posed by singularities present in traditional fractional operators, several researchers have recently put forth new types of nonsingular fractional operators. Some of these operators involve exponential kernels, while others incorporate Mittag–Leffler functions. For further exploration of these fractional operators, we recommend referring to [12,13,14,15,16]. It is important to note that all the fractional operators mentioned in the preceding paragraphs are non-local in nature. However, the literature also encompasses numerous local operators that enable differentiation to non-integer orders, known as local fractional operators. In [17], the authors introduced the concept of what they referred to as conformable (fractional) derivatives. Additionally, in [18], other fundamental concepts related to conformable derivatives were proposed. It is worth mentioning that the fractional operators proposed in [7] are non-local fractional counterparts of the local operators suggested in [17]. Moreover, the non-local fractional versions of those proposed in [18] can be found in [6]. Conventionally, it is expected that a derivative of order 0 applied to a function would yield the function itself. However, conformable derivatives lack this fundamental property. Nevertheless, to address this limitation, a new definition of the conformable derivative was introduced in [19,20]. This revised definition ensures that the function itself is obtained when the order of the local derivative tends toward 0. Furthermore, in [21], Sadek extended the notion by presenting non-local fractional operators derived from the iteration of the aforementioned conformable derivative.

This article expands upon the findings presented in [21] by introducing novel fractional operators based on the proportional derivatives of a function with respect to another function. These operators can be defined in a manner analogous to the definitions discussed in [19]. The resulting fractional operators incorporate an exponential function within their kernel, which is dependent on the functions involved. Additionally, the properties of these operators as semigroups will be examined and discussed.

This paper is structured as follows: Section 2 provides the necessary preliminaries on FC. Section 3 focuses on the cotangent derivatives of a function with RAF, along with their corresponding integrals. In this section, we present the key results and analyze the properties of these derivatives. Moving on to Section 4, we introduce the v-Laplace transforms for the cotangent Riemann–Liouville derivative. This transform is then utilized to solve linear cotangent differential equations of the Riemann–Liouville type. Similarly, in Section 5, we delve into the Caputo cotangent derivatives with RAF. The v-Laplace transforms for the cotangent Caputo derivative are derived, enabling the solution of linear cotangent differential equations of the Caputo type. In Section 6, we present the numerical approach and the application for solving the Caputo cotangent fractional Cauchy problem, and in Section 7, we provide two examples for testing this approach.

2. Preliminaries

In this section, we introduce several key definitions of fractional operators to facilitate comparison and analysis.

2.1. The Traditional Fractional Operators and Their Generalized Expressions

In this subsection, we provide definitions of FD (fractional derivative) and FI (fractional integral) for the purpose of comparison. Consider $r\in \mathbb{C}$, $\mathrm{Re}\left(r\right)>0$, $n_r=\left[r\right]+1$, and a function $x:[{\ell }_i,{\ell }_f]⟶\mathbb{R}$. The following definitions are presented:

The left RL-I (Riemann–Liouville integral) of x with order r is defined as

$$\left({}_{{\ell }_i}\mathit{I}^{\mathrm{r}}x\right)(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{(\ell -\rho )}^{r-1}x\left(\rho \right)d\rho .$$

(1)

The right RL-I of x with order r is defined as

$$\left(I_{{\ell }_f}^{r}x\right)(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{(\rho -\ell )}^{r-1}x\left(\rho \right)d\rho .$$

(2)

The left RL-D (Riemann–Liouville derivative) of x of order r is

$$\left({}_{{\ell }_i}\mathit{D}^{\mathrm{r}}x\right)(\ell )={\left({\displaystyle \frac{d}{d\ell }}\right)}^{n_r}\left({}_{{\ell }_i}\mathit{I}^{n_r-r}x\right)(\ell ).$$

(3)

The right RL-D of x with order r is defined as

$$\left(D_{{\ell }_f}^{r}x\right)(\ell )={\left(-{\displaystyle \frac{d}{d\ell }}\right)}^{n_r}\left(I_{{\ell }_f}^{n_r-r}x\right)(\ell ).$$

(4)

The left C-D (Caputo derivative) of x of order r is

$$\left({}_{{\ell }_i}{}^C\mathit{D}^r\mathrm{x}\right)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{n_r-r}{x}^{\left(n_r\right)}\right)(\ell ).$$

(5)

The right C-D of x with order r is defined as

$$\left({}^C\mathit{D}_{{\ell }_f}^{r}x\right)(\ell )=\left(I_{{\ell }_f}^{n_r-r}{(-1)}^{n_r}{x}^{\left(n_r\right)}\right)(\ell ).$$

(6)

The left K-I (Katugampola integral) [7] is given by

$$\left({}_{{\ell }_i}\mathbf{I}^{r,\gamma }x\right)(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{\left({\displaystyle \frac{{\ell }^{\gamma }-{\rho }^{\gamma }}{\gamma }}\right)}^{r-1}x\left(\rho \right){\displaystyle \frac{d\rho }{{\rho }^{1-\gamma }}},$$

(7)

and the right K-I by

$$\left({\mathbf{I}}_{{\ell }_f}^{r,\gamma }x\right)(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{\left({\displaystyle \frac{{\rho }^{\gamma }-{\ell }^{\gamma }}{\gamma }}\right)}^{r-1}x\left(\rho \right){\displaystyle \frac{d\rho }{{\ell }^{1-\gamma }}}.$$

(8)

The left K-D (Katugampola derivative) [7] of x of order r is

$$\left({}_{{\ell }_i}\mathbf{D}^{r,\gamma }x\right)(\ell )={\kappa }^{n_r}\left({}_{{\ell }_i}\mathbf{I}^{n_r-r,\gamma }x\right)(\ell ),$$

(9)

and right K-D is

$$\left({\mathbf{D}}_{{\ell }_f}^{r,\gamma }x\right)(\ell )={(-\kappa )}^{n_r}\left({\mathbf{I}}_{{\ell }_f}^{n_r-r,\gamma }x\right)(\ell ),$$

(10)

where $\gamma \in [0,1]$ and $\kappa ={\ell }^{1-\gamma }{\displaystyle \frac{d}{d\ell }}$. The left CK-D (Caputo–Katugampola derivative) [8] of x of order r is

$$\left({}_{{\ell }_i}{}^C\mathbf{D}^{r,\gamma }x\right)(\ell )=\left({}_{{\ell }_i}\mathbf{I}^{n_r-r,\gamma }{\kappa }^{n_r}x\right)(\ell ),$$

(11)

and the right CK-D is

$$\left({}^C\mathbf{D}_{{\ell }_f}^{r,\gamma }x\right)(\ell )=\left({\mathbf{I}}_{{\ell }_f}^{n_r-r,\gamma }{(-\kappa )}^{n_r}x\right)(\ell ).$$

(12)

Let the function v be continuously differentiable and increasing. The left RL-I of order r of x with respect to v [9,10] is defined as

$${}_{{\ell }_i}\mathit{I}^{r,v}x(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{(v(\ell )-v\left(\rho \right))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,$$

(13)

and the right RL-I of order r of x with respect to v [3,9] is defined as

$$I_{{\ell }_f}^{r,v}x(\ell )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{(v\left(\rho \right)-v(\ell ))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .$$

(14)

The generalized left RL-D of order r of x with respect to v [9,10] is defined as

$${}_{{\ell }_i}\mathit{D}^{r,v}x(\ell )={\displaystyle \frac{{\left(\frac{1}{v^{\prime }(\ell )}\frac{d}{d\ell }\right)}^{n_r}}{\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{(v(\ell )-v\left(\rho \right))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .$$

(15)

and the generalized right RL-I of order r of a function x with respect to v is defined as

$$D_{{\ell }_f}^{r,v}x(\ell )={\displaystyle \frac{{\left(-\frac{1}{v^{\prime }(\ell )}\frac{d}{d\ell }\right)}^{n_r}}{\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{(v\left(\rho \right)-v(\ell ))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .$$

(16)

It is evident that by selecting $v(\ell )=\ell$, the integrals in (13) and (14) correspond to the left and right RL-Is, respectively, while (15) and (16) represent the left and right RL-Ds. The choice $v(\ell )=\ln (\ell )$ yields the Hadamard operators [9,10]. Moreover, considering $v(\ell )={\displaystyle \frac{{\ell }^r}{r}}$ leads to the derivation of the fractional operators in the framework proposed by Katugampola [5,7].

The generalized left C-D of order r of a function x with respect to v [11] is defined as

$${}_{{\ell }_i}{}^C\mathit{D}^{r,v}x(\ell )=\left({}_{{\ell }_i}\mathit{I}^{n_r-r,v}{x}^{\left[n_r\right]}\right)(\ell ),$$

(17)

and the generalized right C-D of order r of a function x with respect to v [11] is defined as

$${}^C\mathit{D}_{{\ell }_f}^{r,v}x(\ell )=\left(I_{{\ell }_f}^{n_r-r,v}{(-1)}^{n_r}{x}^{\left[n_r\right]}\right)(\ell ),$$

(18)

where $x^{\left[n_r\right]}(\ell )={\left({\displaystyle \frac{1}{v^{\prime }(\ell )}}{\displaystyle \frac{d}{d\ell }}\right)}^{n_r}x(\ell )$.

2.2. The Cotangent Derivatives and Their Integrals

The Riemann–Liouville cotangent integrals are defined as follows:

Definition 1

([21]). Consider $\gamma \in ]0,1]$ and $r\in \mathbb{C}$ such that $Re\left(r\right)>0$. The left Riemann–Liouville cotangent integral of x is

$${(}_{{\ell }_i}{I}^{r,\gamma }x)(\ell )={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot ({\displaystyle \frac{\pi }{2}}\gamma )(\ell -\rho )}{(\ell -\rho )}^{r-1}x\left(\rho \right)d\rho ,$$

(19)

and the right Riemann–Liouville cotangent integral of x is

$$\left(I_{{\ell }_f}^{r,\gamma }x\right)(\ell )={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot ({\displaystyle \frac{\pi }{2}}\gamma )(\rho -\ell )}{(\rho -\ell )}^{r-1}x\left(\rho \right)d\rho .$$

(20)

Definition 2

([21]). Consider $\gamma \in ]0,1]$ and $Re\left(r\right)\ge 0$. The left Riemann–Liouville cotangent derivative of x is given as

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{D}^{r,\gamma }x)(\ell )& ={\displaystyle \frac{D^{n_r,\gamma }}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(\ell -\rho )}\hfill \\ \hfill & \times {(\ell -\rho )}^{n_r-r-1}x\left(\rho \right)d\rho ,\hfill \end{array}$$

(21)

and the right Riemann–Liouville cotangent derivative of x is given as

$$\begin{array}{cc}\hfill \left(D_{{\ell }_f}^{r,\gamma }x\right)(\ell )& ={\displaystyle \frac{D^{n_r,\gamma }}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(\rho -\ell )}\hfill \\ \hfill & \times {(\rho -\ell )}^{n_r-r-1}x\left(\rho \right)d\rho ,\hfill \end{array}$$

(22)

where $n_r=\left[Re\left(r\right)\right]+1$ and $\left(D^{n_r,\gamma }x\right)(\ell )=\left(\underset{\underset{n_{r}times}{︸}}{D^{\gamma }{D}^{\gamma }\cdots D^{\gamma }}x\right)(\ell ).$

The left Caputo cotangent derivative is

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}^C{D}^{r,\gamma }x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(\ell -\rho )}{(\ell -\rho )}^{n_r-r-1}\hfill \\ \hfill & \times \left(D^{n_r,\gamma }x\right)\left(\rho \right)d\rho ,\hfill \end{array}$$

(23)

and the right Caputo cotangent derivative is

$$\begin{array}{cc}\hfill {(}^C{D}_{{\ell }_f}^{r,\gamma }x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(\ell -\ell )}{(\ell -\rho )}^{n_r-r-1}\hfill \\ \hfill & \times \left(D^{n_r,\gamma }x\right)\left(\rho \right)d\rho .\hfill \end{array}$$

(24)

2.3. Two-Scale Fractal Derivative and Its Properties

• The two-scale fractal derivative [22,23] of order $\gamma$ is

  $$D_{TS}^{\gamma }x(\ell )=\underset{{\ell }^{\prime }\to \ell }{\lim }{\displaystyle \frac{x\left({\ell }^{\prime }\right)-x(\ell )}{{\ell }^{\prime \gamma }-{\ell }^{\gamma }}}.$$
• Key Properties:

  -

  Linearity: $D_{TS}^{\gamma }[af(\ell )+bg(\ell )]=a{D}_{TS}^{\gamma }f(\ell )+b{D}_{TS}^{\gamma }g(\ell )$.

  -

  Scaling Behavior: $D_{TS}^{\gamma }\left[x(\lambda \ell )\right]={\lambda }^{1-\gamma }{D}_{TS}^{\gamma }x(\lambda \ell )$.

  -

  Compatibility: Relation to Riemann–Liouville and Caputo derivatives.

3. The Cotangent Derivatives with RAF

Definition 3

([24]). Consider $\gamma \in [0,1]$ and functions $f,g:[0,1]\times \mathbb{R}\to [0,+\infty ]$ that are continuous and satisfy the following conditions:

$$\underset{\gamma \to 1^{-}}{\lim }g(\gamma ,\ell )=0,\underset{\gamma \to 1^{-}}{\lim }f(\gamma ,\ell )=1,\underset{\gamma \to 0^{+}}{\lim }g(\gamma ,\ell )=1,\underset{\gamma \to 0^{+}}{\lim }f(\gamma ,\ell )=0,$$

Let also $v(\ell )$ be a strictly increasing continuous function. Under these conditions, the PD of x of order γ with respect to v is defined as

$$D^{\gamma ,v}x(\ell )=g(\gamma ,\ell )x(\ell )+f(\gamma ,\ell ){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}.$$

(25)

In this particular case, the expression (25) can be simplified to

$$D^{\gamma ,v}x(\ell )=\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}.$$

(26)

The corresponding integral of (26) is

$${}_{{\ell }_i}\mathit{I}^{1,\gamma ,v}x(\ell )={\displaystyle \frac{1}{\sin \left(\frac{\pi }{2}\gamma \right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,$$

(27)

where $\cot (\ell )={\displaystyle \frac{\cos (\ell )}{\sin (\ell )}}$, and we accept that ${}_{{\ell }_i}\mathit{I}^{0,\gamma ,v}x(\ell )=x(\ell )$.

We use induction and change the order of integrals to show that with Equation (27), we have

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{n,\gamma ,v}x\right)(\ell )& ={\displaystyle \frac{1}{\sin \left(\frac{\pi }{2}\gamma \right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\rho }_1\right))}{v}^{\prime }\left({\rho }_1\right)d{\rho }_1{\displaystyle \frac{1}{\sin \left(\frac{\pi }{2}\gamma \right)}}{\int }_{{\ell }_i}^{{\rho }_1}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\rho }_1\right)-v\left({\rho }_2\right))}\hfill \\ \hfill & \times v^{\prime }\left({\rho }_2\right)d{\rho }_2\cdots {\displaystyle \frac{1}{\sin \left(\frac{\pi }{2}\gamma \right)}}{\int }_{{\ell }_i}^{{\rho }_{n-1}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\rho }_{n-1}\right)-v\left({\rho }_n\right))}x\left({\rho }_n\right)v^{\prime }\left({\rho }_n\right)d{\rho }_n\hfill \\ \hfill & ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^n\mathrm{\Gamma }\left(n\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{n-1}{v}^{\prime }\left(\rho \right)x\left(\rho \right)d\rho .\hfill \end{array}$$

(28)

From (28), we have the following definition.

Definition 4.

Consider $\gamma \in ]0,1]$ and $r\in \mathbb{C}$ such that $Re\left(r\right)>0$. The left Riemann–Liouville cotangent integral of x with respect to v is

$${(}_{{\ell }_i}{I}^{r,\gamma ,v}x)(\ell )={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,$$

(29)

and the right Riemann–Liouville cotangent integral of x with respect to v is

$$\left(I_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left(\rho \right)-v(\ell ))}{(v\left(\rho \right)-v(\ell ))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .$$

(30)

Remark 1.

We have the following:

• If $\gamma =1$ and $v(\ell )=\ell$, the integrals in (29) and (30) coincide with the integrals (1) and (2).
• If $v(\ell )=\ell$, the integrals in (29) and (30) coincide with the integrals (19) and (20).

Definition 5.

Let $\gamma \in ]0,1]$ and $Re\left(r\right)\ge 0$ and 0 and $v\in \mathcal{C}[{\ell }_i,{\ell }_f]$, where $v^{\prime }(\ell )>0$. The left Riemann–Liouville cotangent derivative of x with respect to v is given as

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{D}^{r,\gamma ,v}x)(\ell )& =\left(D^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)(\ell )\hfill \\ \hfill & ={\displaystyle \frac{D^{n_r,\gamma ,v}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}\hfill \\ \hfill & \times {(v(\ell )-v\left(\rho \right))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(31)

and the right Riemann–Liouville cotangent derivative of x with respect to v is given as

$$\begin{array}{cc}\hfill \left(D_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )& =\left(D^{n_r,\gamma ,v}{I}_{{\ell }_f}^{n_r-r,\gamma ,v}x\right)(\ell )\hfill \\ \hfill & ={\displaystyle \frac{D^{n_r,\gamma ,v}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left(\rho \right)-v(\ell ))}\hfill \\ \hfill & \times {(v\left(\rho \right)-v(\ell ))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(32)

where $n_r=\left[Re\left(r\right)\right]+1$ and $\left(D^{n_r,\gamma ,v}x\right)(\ell )=\left(\underset{\underset{n_r\mathit{times}}{︸}}{D^{\gamma ,v}{D}^{\gamma ,v}\cdots D^{\gamma ,v}}x\right)(\ell ).$

Remark 2.

If we let $\gamma =1$ in Definitions 4 and 5, we get

• The Riemann–Liouville operators (1)–(4) if $v(\ell )=\ell$;
• The fractional operators in the Katugampola setting (7)–(10) if $v(\ell )=\ell$
• The Hadamard operators [10] if $v(\ell )=\ln (\ell )$;
• The fractional operators mentioned in [8] if $v(\ell )={\displaystyle \frac{{(\ell -{\ell }_i)}^r}{r}}.$

Remark 3.

If $v(\ell )=\ell$, the fractional derivatives in (31) and (32) coincide with the cotangent derivatives (21) and (22).

We present the following lemma that will help us in the following.

Lemma 1.

Consider function $y(\ell )$. We have

$$D^{\gamma ,v}\left(y(\ell )e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)v(\ell )}\right)=\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right){\displaystyle \frac{y^{\prime }(\ell )}{v^{\prime }(\ell )}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)v(\ell )}.$$

Proposition 1.

Let $\gamma \in ]0,1]$ and $\mathrm{Re}\left(r_1\right)>0,\mathrm{Re}\left(r_2\right)>0$. Let

$$f(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1},$$

and

$$g(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^{r_2-1}.$$

We get

1.

${}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}(f(\ell ))={\displaystyle \frac{\mathrm{\Gamma }\left(r_2\right)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }(r_1+r_2)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_1+r_2-1}$.

2.

$I_{{\ell }_f}^{r_1,\gamma ,v}(g(\ell ))={\displaystyle \frac{\mathrm{\Gamma }\left(r_2\right)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }(r_1+r_2)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^{r_1+r_2-1}$.

3.

${}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}(f(\ell ))={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1-r_1}$.

4.

$D_{{\ell }_f}^{r_1,\gamma ,v}(g(\ell ))={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^{r_2-1-r_1}$.

Proof. 

1.

We have

$$\begin{array}{ccc}& & {}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}(f(\ell ))={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}\int {{\ell }_i}^{\ell }{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(\rho \right))}\hfill \\ & & \times {(v(\ell )-v\left(\rho \right))}^{r_1-1}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}v\left(\rho \right)-v\left({\ell }_i\right)}{(v\left(\rho \right)-v\left({\ell }_i\right))}^{r_2-1}{v}^{\prime }\left(\rho \right)d\rho \hfill \\ & =& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\hfill \\ & & \times {(v(\ell )-v\left(\rho \right))}^{r_1-1}{(v\left(\rho \right)-v\left({\ell }_i\right))}^{r_2-1}{v}^{\prime }\left(\rho \right)d\rho \hfill \\ & =& {\displaystyle \frac{e^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}{\int }_{{\ell }_i}^{\ell }{(v(\ell )-v\left(\rho \right))}^{r_1-1}{(v\left(\rho \right)-v\left({\ell }_i\right))}^{r_2-1}{v}^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

and, making the change in variable $y={\displaystyle \frac{v(\ell )-v(\ell )}{v(\ell )-v\left({\ell }_i\right)}}$, we get

$$\begin{array}{ccc}& & {}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}v(\ell )-v\left({\ell }_i\right)}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}\right)\hfill \\ & & ={\displaystyle \frac{e^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}{\int }_{{\ell }_i}^{\ell }{(v(\ell )-v\left(\rho \right))}^{r_1-1}{(v\left(\rho \right)-v\left({\ell }_i\right))}^{r_2-1}{v}^{\prime }\left(\rho \right)d\rho \hfill \\ & & ={\displaystyle \frac{e^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}{(v(\ell )-v\left({\ell }_i\right))}^{r_1+r_2-1}\hfill \\ & & \times {\int }_0^1{y}^{r_1-1}{(1-y)}^{r_2-1}dy.\hfill \end{array}$$

By utilizing the Beta function, denoted as $B(r_1,r_2)={\int }_0^1{\ell }^{r_1-1}{(1-\ell )}^{r_2-1}d\ell$, and the relationship $B(r_1,r_2)={\displaystyle \frac{\mathrm{\Gamma }\left(r_1\right)\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_1+r_2)}}$, we can conclude that

$$\begin{array}{ccc}& & {}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}v(\ell )-v\left({\ell }_i\right)}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}\right)\hfill \\ & & ={\displaystyle \frac{e^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_1\right)}}{(v(\ell )-v\left({\ell }_i\right))}^{r_1+r_2-1}{\displaystyle \frac{\mathrm{\Gamma }\left(r_1\right)\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_1+r_2)}}\hfill \\ & & {\displaystyle \frac{\mathrm{\Gamma }\left(r_2\right)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }(r_1+r_2)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_1+r_2-1}.\hfill \end{array}$$

2.

In a similar manner to 1, we can apply the same approach.

3.

Consider $x(\ell )=e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}v(\ell )-v\left({\ell }_i\right)}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}$. Using Lemma 1, we have

$$\begin{array}{ccc}& & {}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}(x(\ell ))=D_t^{n_{r_1},\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_{r_1}-r_1,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}\right)\hfill \\ & & =D_t^{n_{r_1},\gamma ,v}\left({\displaystyle \frac{\mathrm{\Gamma }\left(r_2\right)e^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_{r_1}-r_1}\mathrm{\Gamma }(n_{r_1}-r_1+r_2)}}{(v(\ell )-v\left({\ell }_i\right))}^{n_{r_1}-r_1+r_2-1}\right)\hfill \\ & & ={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_{r_1}}\mathrm{\Gamma }\left(r_2\right)(n_{r_1}-r_1+r_2-1)(n_{r_1}-r_1+r_2-1)\cdots (r_2-r_1)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_{r_1}-r_1}\mathrm{\Gamma }(n_{r_1}-r_1+r_2)}}\hfill \\ & & \times e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1-r_1}\hfill \\ & & ={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1-r_1}.\hfill \end{array}$$

4.

Similar to 3.

□

Lemma 2.

Consider $\lambda \in \mathbb{R}$, and let the cotangent Mittag–Leffler function be defined by ${\cot }_{r_1,r_2}^{\gamma }({\ell }_1,{\ell }_2)=e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right){\ell }_1}{E}_{r_1,r_2}\left({\ell }_2\right)$. Then

$${}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}{\cot }_{r_1,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r_1}\lambda (v(\ell )-v{\left({\ell }_i\right)}^{r_1})$$

$$=\lambda {\cot }_{r_1,1}^{\gamma }(v(\ell )-v({\ell }_i,\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r_1}(v(\ell )-v{\left({\ell }_i\right)}^{r_1}).$$

Proof. 

We have

$$\begin{array}{ccc}& & {}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}{\cot }_{r_1,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r_1}\lambda (v(\ell )-v{\left({\ell }_i\right)}^{r_1})\hfill \\ & & ={}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r_1,1}\left(\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r_1}{(v(\ell )-v\left({\ell }_i\right))}^{r_1}\right)\right)\hfill \\ & & ={}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}\left(\sum _{k=0}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-k{r}_1}{\lambda }^k}{\mathrm{\Gamma }(r_{1}k+1)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{k{r}_1}\right)\hfill \\ & & =\sum _{k=0}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-k{r}_1}{\lambda }^k}{\mathrm{\Gamma }(r_{1}k+1)}}{}_{{\ell }_i}\mathit{D}^{r_1,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{k{r}_1}\right)\hfill \\ & & =\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-k{r}_1}{\lambda }^k}{\mathrm{\Gamma }(r_{1}k+1)}}{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }(r_{1}k+1)}{\mathrm{\Gamma }(r_{1}k+1-r_1)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_{1}k-r_1}\hfill \\ & & =e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r_1(k-1)}{\lambda }^k}{\mathrm{\Gamma }(r_1(k-1)+1)}}{(v(\ell )-v\left({\ell }_i\right))}^{r_1(k-1)}\hfill \\ & & =\lambda e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r_1(k-1)}{\lambda }^{k-1}}{\mathrm{\Gamma }(r_1(k-1)+1)}}{(v(\ell )-v\left({\ell }_i\right))}^{r_1(k-1)}\hfill \\ & & =\lambda e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r_1,1}\left(\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r_1}{(v(\ell )-v\left({\ell }_i\right))}^{r_1}\right).\hfill \end{array}$$

□

The semigroup property for the Riemann–Liouville cotangent integral is presented in Theorem 1.

Theorem 1.

Consider $\gamma \in ]0,1],\mathrm{Re}\left(r_1\right)>0$, $\mathrm{Re}\left(r_2\right)>0$, and let x be continuous and defined for $\ell \ge {\ell }_i$ or $\ell \le {\ell }_f$. Then,

$${(}_{{\ell }_i}{I}^{r_1,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_2,\gamma ,v}x)(\ell )={(}_{{\ell }_i}{I}^{r_2,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}x)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{r_1+r_2,\gamma ,v}x\right)(\ell ),$$

(33)

and

$$\left(I_{{\ell }_f}^{r_1,\gamma ,v}{I}_{{\ell }_f}^{r_2,\gamma ,v}x\right)(\ell )=\left(I_{{\ell }_f}^{r_2,\gamma ,v}{I}_{{\ell }_f}^{r_1,\gamma ,v}x\right)(\ell )=\left(I_{{\ell }_f}^{r_1+r_2,\gamma ,v}x\right)(\ell ).$$

(34)

Proof. 

We have

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{I}^{r_1,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_2,\gamma ,v}x)(\ell )& ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1+r_2}\mathrm{\Gamma }\left(r_1\right)\mathrm{\Gamma }\left(r_2\right)}}\hfill \\ & \times {\int }_{{\ell }_i}^{\ell }{\int }_{{\ell }_i}^u{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(u\right))}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v\left(u\right)-v\left(\rho \right))}\hfill \\ & \times {(v(\ell )-v\left(u\right))}^{r_1-1}{(v\left(u\right)-v\left(\rho \right))}^{r_2-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho v^{\prime }\left(u\right)du\hfill \\ & ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1+r_2}\mathrm{\Gamma }\left(r_1\right)\mathrm{\Gamma }\left(r_2\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(\rho \right))}x\left(\rho \right)\hfill \\ & \times {\int }_s^{\ell }{(v(\ell )-v\left(u\right))}^{r_1-1}{(v\left(u\right)-v\left(\rho \right))}^{r_2-1}{v}^{\prime }\left(u\right)du{v}^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

and, making the change in variable $y={\displaystyle \frac{v\left(u\right)-v\left(\rho \right)}{v(\ell )-v\left(\rho \right)}}$, we get

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{I}^{r_1,\gamma }{}_{{\ell }_i}\mathit{I}^{r_2,\gamma ,v}x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1+r_2}\mathrm{\Gamma }\left(r_1\right)\mathrm{\Gamma }\left(r_2\right)}}\hfill \\ & \times {\int }_{{\ell }_i}^{\ell }{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r_1+r_2-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho \hfill \\ & \times {\int }_0^1{(1-y)}^{r_1-1}{y}^{r_2-1}dy\hfill \\ \hfill =& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1+r_2}\mathrm{\Gamma }(r_1+r_2)}}{\int }_{{\ell }_i}^{\ell }{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(\rho \right))}\hfill \\ & \times {(v(\ell )-v\left(\rho \right))}^{r_1+r_2-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho \hfill \\ \hfill =& \left({}_{{\ell }_i}\mathit{I}^{r_1+r_2,\gamma ,v}x\right)(\ell ).\hfill \end{array}$$

For relation (34), the analysis is similar. □

Theorem 2.

Let $0\le l<\left[\mathrm{Re}\right(r\left)\right]+1$. Then

$$D^{l,\gamma ,v}\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{r-l,\gamma ,v}x\right)(\ell ),$$

(35)

and

$$D^{l,\gamma ,v}\left(I_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )=\left(I_{{\ell }_f}^{r-l,\gamma ,v}x\right)(\ell ).$$

(36)

Proof. 

Using the definition and $D^{1,\gamma ,v}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left(\rho \right))}=0$, we have

$$\begin{array}{cc}\hfill D^{l,\gamma ,v}\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )& =D^{l-1,\gamma ,v}\left({\mathit{D}}^{1,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )\hfill \\ & =D^{l-1,\gamma ,v}{\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}\mathrm{\Gamma }(r-1)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}\hfill \\ & \times {(v(\ell )-v\left(\rho \right))}^{r-2}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

By repeating this method l times, we arrive at (35). Relation (36) follows in a similar manner. □

Corollary 1.

Consider $0<\mathrm{Re}\left(r_2\right)<\mathrm{Re}\left(r_1\right)$, $l-1<\mathrm{Re}\left(r_2\right)\le l$. We get

$$\left({}_{{\ell }_i}\mathit{D}^{r_2,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}x\right)(\ell )={}_{{\ell }_i}\mathit{I}^{r_1-r_2,\gamma ,v}x(\ell ),$$

and

$$\left(D_{{\ell }_f}^{r_2,\gamma ,v}{I}_{{\ell }_f}^{r_1,\gamma ,v}x\right)(\ell )=I_{{\ell }_f}^{r_1-r_2,\gamma ,v}x(\ell ).$$

(37)

Proof. 

With the help of Theorems 1 and 2, we get

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{D}^{r_2,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}x\right)(\ell )& =D^{l,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{l-r_2,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r_1,\gamma ,v}x(\ell )\hfill \\ & =D^{l,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{l-r_2+r_1,\gamma ,v}x(\ell )\hfill \\ & =\left({}_{{\ell }_i}\mathit{I}^{r_1-r_2,\gamma ,v}x\right)(\ell ).\hfill \end{array}$$

One can prove (37) in a similar way. □

Theorem 3.

Consider $\mathrm{Re}\left[r\right]>0,\gamma \in [0,1],n_r=\left[\mathrm{Re}\left(r\right)\right]+1$. Then, we get

$$\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )=x(\ell ),$$

and

$$\left(D_{{\ell }_f}^{r,\gamma ,v}{I}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )=x(\ell ).$$

Proof. 

Based on the definition and the findings presented in Theorem 1, we obtain the following result:

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )& =D^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x(\ell )\hfill \\ & =D^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r,\gamma ,v}x(\ell )\hfill \\ & =x(\ell ).\hfill \end{array}$$

□

4. The $\mathit{v}$-Laplace Transform

Definition 6

([11]). Let $x,v:[{\ell }_i,\infty )\to \mathbb{R}$ be real-valued functions such that $v(\ell )$ is continuous and $v^{\prime }(\ell )>0$ on $[{\ell }_i,\infty )$. The v-Laplace transform of x is defined by

$${\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)={\int }_{{\ell }_i}^{\infty }{e}^{-p(v(\ell )-v\left({\ell }_i\right))}x(\ell )v^{\prime }(\ell )d\ell ,$$

(38)

and for all values of p, the integral is valid.

The Generalized Laplace transform is:

$$\mathcal{G}\left\{x(\ell )\right\}\left(s\right)={\int }_0^{\infty }K(s,\ell )x(\ell )d\ell ,$$

where $K(s,\ell )$ represents different kernel functions:

$$\begin{array}{cc}\hfill \mathrm{Laplace}:& e^{-s\ell }.\hfill \\ \hfill \mathrm{Sumudu}:& e^{-\ell }.\hfill \\ \hfill \mathrm{Aboodh}:& {\displaystyle \frac{e^{-s\ell }}{s^2}}.\hfill \\ \hfill \mathrm{Mohand}:& \ell e^{-s\ell }.\hfill \end{array}$$

The Application to Cotangent Derivatives are:

• Step-by-step derivation of transform formulas.
• Comparison of convergence properties.
• Specific advantages for singular problems.

In the subsequent lemma, we provided the v-Laplace transforms of various elementary functions.

Lemma 3

([11]).

1.

${\mathcal{L}}_v\left\{1\right\}\left(p\right)={\displaystyle \frac{1}{p}},s>0$.

2.

${\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{\beta }\right\}\left(p\right)={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{p^{\beta +1}}},\Re \left(\beta \right)>0,p>0$.

3.

${\mathcal{L}}_v\left\{e^{\lambda (v(\ell )-v\left({\ell }_i\right))}\right\}\left(p\right)={\displaystyle \frac{1}{p-\lambda }},p>\lambda$.

4.

$${\mathcal{L}}_v\left\{e^{\lambda (v(\ell )-v\left({\ell }_i\right))}x(\ell )\right\}\left(p\right)={\mathcal{L}}_v\left\{x\right\}(p-\lambda ),$$

(39)

5.

$${\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)={\displaystyle \frac{1}{p^r-\lambda }},$$

(40)

6.

$${\mathcal{L}}_v\left\{E_r\left(\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)={\displaystyle \frac{p^{r-1}}{p^r-\lambda }},$$

(41)

The v-Laplace transform of the derivative of x with respect to v is given as follows.

Theorem 4

([11]). Suppose that $x(\ell )\in {\mathcal{C}}_v[{\ell }_i,T]$ and has $v(\ell )$-exponential order, with $x^{\left[1\right]}(\ell )$ being piecewise continuous over every finite interval $[{\ell }_i,T]$. Under these conditions, the v-Laplace transform of $x^{\left[1\right]}(\ell )={\displaystyle \frac{x^{\prime }}{v^{\prime }}}(\ell )$ exists and can be expressed as follows:

$${\mathcal{L}}_v\left\{x^{\left[1\right]}(\ell )\right\}\left(s\right)=s{\mathcal{L}}_v\left\{x(\ell )\right\}\left(s\right)-x\left({\ell }_i\right).$$

(42)

Definition 7

([11]). Consider two functions, x and y, that are piecewise continuous over each interval $[{\ell }_i,T]$ and possess v-exponential order. In this context, we define the v-convolution of x and y as follows:

$$\left(x{\ast }_{v}y\right)(\ell )={\int }_{{\ell }_i}^{\ell }x\left(\rho \right)y\left(v^{-1}(v(\ell )+v\left({\ell }_i\right)-v\left(\rho \right))\right)v^{\prime }\left(\rho \right)d\rho .$$

(43)

The v-convolution of two functions is commutative.

Lemma 4

([11]). Let x and y be two functions that are piecewise continuous at each interval $[{\ell }_i,T]$ and of exponential order. Then

$$x{\ast }_{v}y=y{\ast }_{v}x.$$

(44)

Theorem 5

([11]). Let x and y be two functions that are piecewise continuous at each interval $[{\ell }_i,T]$ and of v-exponential order. Then

$${\mathcal{L}}_v\left\{x{\ast }_{v}y\right\}\left(p\right)={\mathcal{L}}_v\left\{x\right\}\left(p\right){\mathcal{L}}_v\left\{y\right\}\left(p\right).$$

(45)

Lemma 5.

Let $r>0$ and $\gamma \in ]0,1]$; then

$$\begin{array}{cc}\hfill & \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{D}^{1,\gamma ,v}x\right)(\ell )=\left(D^{1,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )-{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }\left(r\right)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}}}x\left({\ell }_i\right),\hfill \end{array}$$

(46)

and

$$\begin{array}{cc}\hfill & \left(I_{{\ell }_f}^{r,\gamma ,v}{D}^{1,\gamma ,v}x\right)(\ell )=\left(D^{1,\gamma ,v}{I}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )-{\displaystyle \frac{{(v\left({\ell }_f\right)-v\left(t\right))}^{r-1}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v\left({\ell }_f\right)-v\left(t\right))}}{\mathrm{\Gamma }\left(r\right)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}}}x\left({\ell }_f\right).\hfill \end{array}$$

(47)

Proof. 

Using the Leibniz rule, we can prove that

$${\displaystyle \frac{r-1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r-2}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho =\left(D^{1,\gamma ,v}{I}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell ).$$

Now, by using Definition 4,

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{D}^{1,\gamma ,v}x\right)(\ell )=& \cos \left({\displaystyle \frac{\pi }{2}}\gamma \right){}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x(\ell )\hfill \\ & +{\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r-1}{x}^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

By employing the integration by parts formula, we can derive the following expression:

$$\begin{array}{cc}& \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{D}^{1,\gamma ,v}x\right)(\ell )=\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right){}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x(\ell )-{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }\left(r\right)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}}}x\left({\ell }_i\right)\hfill \\ & -\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right){}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x(\ell )+{\displaystyle \frac{r-1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r-2}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho \hfill \\ & =\left(D^{1,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )-{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }\left(r\right)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-1}}}x\left({\ell }_i\right).\hfill \end{array}$$

The validity of (47) can be established in a similar manner. □

We can generalize Lemma 5 as follows.

Corollary 2.

Consider $r>0$, $\gamma \in ]0,1]$, and $l\in \mathbb{N}$. Then,

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{D}^{l,\gamma ,v}x\right)(\ell )=& \left(D^{l,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )\hfill \\ \hfill & -\sum _{k=0}^{l-1}{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-l+k}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }(r+k-l+1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-l+k}}}\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right),\hfill \end{array}$$

(48)

and

$$\begin{array}{cc}\hfill \left(I_{{\ell }_f}^{r,\gamma ,v}{D}^{l,\gamma ,v}x\right)(\ell )& =\left(D^{l,\gamma ,v}{I}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )\hfill \\ & -\sum _{k=0}^{l-1}{\displaystyle \frac{{(v\left({\ell }_f\right)-v(\ell ))}^{r-l+k}{e}^{-\cot \left(\frac{\pi }{2}\gamma \right)(v\left({\ell }_i\right)-v(\ell ))}}{\mathrm{\Gamma }(r+k-l+1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-l+k}}}\left(D^{k,\gamma ,v}x\right)\left({\ell }_f\right).\hfill \end{array}$$

Proof. 

Mathematical induction can be employed to prove the statement. □

Remark 4.

We have

• $\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{-r,\gamma ,v}x\right)(\ell )$.
• Using Corollary 2 we get

  $$\begin{array}{cc}\hfill \left(D^{\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )& =D^{n_r,\gamma ,v}{D}^{\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x(\ell )\hfill \\ & =D^{n_r,\gamma ,v}\left[{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}{D}^{\gamma ,v}-{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{n_r-r-1}{e}^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }(n_r-r)\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r-1}}}x\left({\ell }_i\right)\right]\hfill \\ & ={}_{{\ell }_i}\mathit{I}^{-r,\gamma ,v}{D}^{\gamma ,v}x(\ell )-{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{-r-1}{e}^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r-1}\mathrm{\Gamma }(-r)}}x\left({\ell }_i\right).\hfill \end{array}$$
• Corollary 2 is valid for any real r.

The subsequent theorems illustrate the influence of the fractional integral on the fractional derivative of the same order.

Theorem 6.

Let $\mathrm{Re}\left(r\right)>0,n_r=-[-\mathrm{Re}\left(r\right)],x\in L_1]{\ell }_i,{\ell }_f[$ and $\left({}_{{\ell }_i}\mathit{I}^{r,\gamma }x\right)(\ell ),\left(I_{{\ell }_f}^{r,\gamma }x\right)(\ell )\in A{C}^{n_r}[{\ell }_i,{\ell }_f]$. Then

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )& =x(\ell )-e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}\sum _{m=1}^{n_r}\left({}_{{\ell }_i}\mathit{I}^{m-r,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right)\hfill \\ \hfill & \times {\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-m}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-m}\mathrm{\Gamma }(r+1-m)}},\hfill \end{array}$$

(49)

and

$$\begin{array}{cc}\hfill \left(I_{{\ell }_f}^{r,\gamma ,v}{D}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )& =x(\ell )-e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}\sum _{m=1}^{n_r}\left(I_{{\ell }_f}^{m-r,\gamma ,v}x\right)\left({{\ell }_f}^{-}\right)\hfill \\ & \times {\displaystyle \frac{{(v\left({\ell }_f\right)-v(\ell ))}^{r-m}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-m}\mathrm{\Gamma }(r+1-m)}}.\hfill \end{array}$$

Proof. 

By Definition 5, we have

$$\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{D}^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)(\ell ),$$

and by applying (48) in Corollary 2, we get

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )=& \left(D^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)(\ell )\hfill \\ & -\sum _{k=0}^{n_r-1}{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-n_r+k}{e}^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }(r+k-n_r+1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-n_r+k}}}\hfill \\ & \times \left(D^{k,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)\left({\ell }_i\right),\hfill \end{array}$$

Furthermore, by employing the first observation stated in Remark 4, we obtain

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )=& \left(D^{n_r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)(\ell )\hfill \\ & -\sum _{k=0}^{n_r-1}{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-n_r+k}{e}^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }(r+k-n_r+1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-n_r+k}}}\hfill \\ & \times \left({}_{{\ell }_i}\mathit{I}^{-k,\gamma }{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)\left({\ell }_i\right),\hfill \end{array}$$

and with Theorem 1, we have

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )=& x(\ell )-\sum _{k=0}^{n_r-1}{\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-n_r+k}{e}^{-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}(v(\ell )-v\left({\ell }_i\right))}}{\mathrm{\Gamma }(r+k-n_r+1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-n_r+k}}}\left({}_{{\ell }_i}\mathit{I}^{n_r-r-k,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right)\hfill \\ \hfill =& x(\ell )-e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{m=1}^{n_r}\left({}_{{\ell }_i}\mathit{I}^{m-r,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right)\hfill \\ & \times {\displaystyle \frac{{(v(\ell )-v\left({\ell }_i\right))}^{r-m}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-m}\mathrm{\Gamma }(r+1-m)}}.\hfill \end{array}$$

The substitution $m=n_r-k$ has been employed to facilitate the change in variable. □

Theorem 7.

Consider x to be of exponential order, $\gamma \in ]0,1]$ and $r\in \mathbb{C}$, where $\mathrm{Re}\left(r\right)>0$. We get

$${\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)={\displaystyle \frac{1}{{(\cos \left(\frac{\pi }{2}\gamma \right)+\sin \left(\frac{\pi }{2}\gamma \right)p)}^r}}{\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right),p>d.$$

(50)

Proof. 

We have

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)& ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{\ast }_{v}x(\ell )\right\}\left(p\right)\hfill \\ & ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\displaystyle \frac{\mathrm{\Gamma }\left(r\right)}{{\left(p-\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}\right)}^r}}{\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)\hfill \\ & ={\displaystyle \frac{1}{{(\cos \left(\frac{\pi }{2}\gamma \right)+\sin \left(\frac{\pi }{2}\gamma \right)p)}^r}}{\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right).\hfill \end{array}$$

□

Theorem 8.

Let r be such that $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$ and $x\in C^{n_r-1}[{\ell }_i,+\infty ]$ be such that $x^{\left(k\right)},k=1,2,\dots ,n_r-1$ are of exponential order on each subinterval $[{\ell }_i,{\ell }_f]$. We get

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{\left(D^{n_r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)& ={(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r}{\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)\hfill \\ \hfill & -\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r-1-k}\hfill \\ \hfill & \times \left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right).\hfill \end{array}$$

(51)

Proof. 

By using the fact that ${\mathcal{L}}_v\left\{x^{\prime }(\ell )\right\}\left(p\right)=p{\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)-x\left({\ell }_i\right)$, we have for $n_r=1$

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{\left(D^{1,\gamma ,v}x\right)(\ell )\right\}\left(p\right)& ={\mathcal{L}}_v\left\{\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)x^{\prime }(\ell )\right\}\left(p\right)\hfill \\ \hfill & =(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p){\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)\hfill \\ \hfill & -\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)x\left({\ell }_i\right).\hfill \end{array}$$

(52)

By applying (52) we get relation (51). □

Theorem 9.

Consider $r\in \mathbb{C}$, where $\mathrm{Re}\left(r\right)>0$ and $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$ and $\gamma \in [0,1]$; then

$$\begin{array}{cc}& {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)={(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^r{X}_{{\ell }_i}\left(p\right)-\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\sum _{k=0}^{n_r-1}(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)\hfill \\ & +\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right){p)}^{n_r-k-1}\left(I^{n_r-r-k,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right),\hfill \end{array}$$

with $X_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)$. In particular, if x is continuous at ${\ell }_i$, then

$${\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)={(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)s)}^r{X}_{{\ell }_i}\left(p\right).$$

Proof. 

Utilizing Theorems 8 and 14, we obtain the following result:

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)& =\left.{\mathcal{L}}_v\left\{{}_{{\ell }_i}\mathit{D}_{{\ell }_i}^{n_r,\gamma ,v}{I}^{n_r-r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)\hfill \\ & =\left.{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r}{\mathcal{L}}_v\left\{{}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)\hfill \\ & -\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r-1-k}\hfill \\ & \times \left(D_{{\ell }_i}^{k,\gamma ,v}{I}^{n_r-r,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right)\hfill \\ & ={(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-n_r}\hfill \\ & \times X_{{\ell }_i}\left(p\right)-\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r-1-k}\hfill \\ & \times \left({}_{{\ell }_i}\mathit{I}^{n_r-r-k,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right).\hfill \end{array}$$

□

Theorem 10.

Consider the linear v-cotangent differential equation

$$\left\{\begin{array}{cc}{}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x(\ell )-\lambda x(\ell )=y(\ell ),\hfill & 0<r,\gamma \le 1,\hfill \\ \left({}_{{\ell }_i}\mathit{I}^{1-r,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right)=x_{{\ell }_i}.\hfill \end{array}\right.$$

(53)

Then, the solution of Equation (53) is

$$\begin{array}{cc}\hfill x(\ell )=& {\cot }_{r,r}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r)\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{1-r}{x}_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\int }_{{\ell }_i}^{\ell }{\cot }_{r,r}^{\gamma }(v(\ell )-v\left(\rho \right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left(\rho \right))}^r)\hfill \\ \hfill & {(v(\ell )-v\left(\rho \right))}^{r-1}y\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(54)

Proof. 

We apply ${\mathcal{L}}_v$ to (53) and make use of Theorem 9 with $n_r=1$; then we get

$$\left({(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^r-\lambda \right)X_{{\ell }_i}\left(p\right)=\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)x_{{\ell }_i}+Y_{{\ell }_i}\left(p\right),$$

where $X_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)$ and $Y_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{y(\ell )\right\}\left(p\right)$. Hence,

$$X_{{\ell }_i}\left(p\right)={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{1-r}{x}_{{\ell }_i}}{{\left(p+\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}\right)}^r-\lambda \sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}}}+{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}{Y}_{{\ell }_i}\left(p\right)}{{\left(p+\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}\right)}^r-\lambda \sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}}}.$$

From Equation (39), we have

$$\begin{array}{cc}\hfill & {\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\mathcal{L}}_v\left\{{\mathrm{e}}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right).\hfill \end{array}$$

(55)

Using Equation (40), we get

$$\begin{array}{c}\hfill {\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\\ \hfill ={\left[{(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}})}^r+\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda \right]}^{-1}.\end{array}$$

(56)

Hence, by using Equations (55) and (56), we get

$$\begin{array}{cc}\hfill & {\left[{\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^r+\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda \right]}^{-1}\hfill \\ \hfill & ={\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right).\hfill \end{array}$$

(57)

We can get the following result by combining Equation (57) with Equation (53). We have

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)& ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{1-r}{x}_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)\hfill \\ \hfill & {\mathcal{L}}_v[\Lambda ]\left(p\right).\hfill \end{array}$$

(58)

Hence, by using the v-convolution formula in Equation (58), we get

$$\begin{array}{cc}\hfill & {\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{1-r}{x}_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right){\ast }_v\Lambda \right\}\left(p\right).\hfill \end{array}$$

(59)

Applying the Laplace inverse of Equation (59), we get

$$\begin{array}{cc}& x(\ell )=e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{1-r}{x}_{{\ell }_i}\hfill \\ & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right){\ast }_v\Lambda \right\}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & x(\ell )={\cot }_{r,r}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r)\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{1-r}{x}_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\int }_{{\ell }_i}^{\ell }{\cot }_{r,r}^{\gamma }(v(\ell )-v\left(\rho \right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left(\rho \right))}^r){(v(\ell )-v\left(\rho \right))}^{r-1}y\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(60)

□

5. The Caputo Cotangent Derivative with RAF

Now, we present the Caputo cotangent derivatives with RAF and the solution of linear v-cotangent equations.

Definition 8.

Consider $\gamma \in ]0,1]$ and $Re\left(r\right)\ge 0$. The left Caputo cotangent derivative with respect to v is

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}^C{D}^{r,\gamma ,v}x)(\ell )=& {(}_{{\ell }_i}{I}^{n_r-r,\gamma ,v}{D}^{n_r,\gamma ,v}x)(\ell ))\hfill \\ \hfill =& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{n_r-r-1}\hfill \\ \hfill & \times \left(D^{n_r,\gamma ,v}x\right)\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(61)

and the right Caputo cotangent derivative with respect to v is

$$\begin{array}{cc}\hfill {(}^C{D}_{{\ell }_f}^{r,\gamma ,v}x)(\ell )& =\left(I_{{\ell }_f}^{n_r-r,\gamma ,v}{D}^{n_r,\gamma ,v}x\right)(\ell ))\hfill \\ \hfill =& {\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left(\rho \right)-v(\ell ))}{(v\left(\rho \right)-v(\ell ))}^{n_r-r-1}\hfill \\ \hfill & \times \left(D^{n_r,\gamma ,v}x\right)\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(62)

where $n_r=\left[Re\left(r\right)\right]+1$.

Remark 5.

If $v(\ell )=\ell$, the fractional derivatives in (61) and (62) coincide with the cotangent derivatives (23) and (24).

Proposition 2.

Consider $\gamma \in ]0,1]$, $\mathrm{Re}\left(r_1\right)>0$ and $\mathrm{Re}\left(r_2\right)>0$. Let

$$f(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1},$$

and

$$g(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^{r_2-1}.$$

We have

1.

${}_{{\ell }_i}{}^C\mathit{D}^{r_1,\gamma ,v}(f(\ell ))={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1-r_1}$.

2.

${}^C\mathit{D}_{{\ell }_f}^{r_1,\gamma ,v}(g(\ell ))={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^{r_2-1-r_1}$.

Let $n_{r_1}=\left[Re\left(r_1\right)\right]+1$, for $k=0,1,\dots ,n_{r_1}-1$. We have

$${}_{{\ell }_i}{}^C\mathit{D}^{r_1,\gamma ,v}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^k=0,$$

and

$${}^C\mathit{D}_{{\ell }_f}^{r_1,\gamma ,v}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}{(v\left({\ell }_f\right)-v(\ell ))}^k=0.$$

In particular, ${}_{{\ell }_i}{}^C\mathit{D}^{r_1,\gamma ,v}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}=0$ and ${}^C\mathit{D}_{{\ell }_f}^{r_1,\gamma ,v}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}=0$.

Proof. 

Let $n_{r_1}=\left[Re\left(r_1\right)\right]+1$. Using Proposition 1, we have

$$\begin{array}{cc}& {}_{{\ell }_i}{}^C\mathit{D}^{r_1,\gamma ,v}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}\hfill \\ & ={}_{{\ell }_i}\mathit{I}^{n_{r_1}-r_1,\gamma ,v}{D}^{n_{r_1},\gamma ,v}\left[e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1}\right]\hfill \\ \hfill =& {}_{{\ell }_i}\mathit{I}^{n_{r_1}-r_1,\gamma ,v}\hfill \\ & \times \left[\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{n_{r_1}}(r_2-1)(r_2-2)\dots (r_2-1-n_{r_1}){(v(\ell )-v\left({\ell }_i\right))}^{r_2-n_{r_1}-1}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}t}\right]\hfill \\ \hfill =& {\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_{r_1}}(r_2-1)(r_2-2)\dots (r_2-1-n_{r_1})\mathrm{\Gamma }(r_2-n)}{\mathrm{\Gamma }(r_2-r_1)\sin {\left(\frac{\pi }{2}\gamma \right)}^{n-r_1}}}\hfill \\ & \times {(v(\ell )-v\left({\ell }_i\right))}^{r_2-r_1-1}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\hfill \\ \hfill =& {\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r_1}\mathrm{\Gamma }\left(r_2\right)}{\mathrm{\Gamma }(r_2-r_1)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r_2-1-r_1}.\hfill \end{array}$$

The same reasoning can be applied to relation 2. □

Lemma 6.

Consider $\gamma \in ]0,1]$; then

$${}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}{\cot }_{r,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^{\alpha })$$

$$=\lambda {\cot }_{r,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r).$$

Proof. 

$$\begin{array}{ccc}& & {}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}{\cot }_{r,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^{\alpha })\hfill \\ & & ={}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^{r_1}\right)\right)\hfill \\ & & ={}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}\left(\sum _{k=0}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-kr}{\lambda }^k}{\mathrm{\Gamma }(rk+1)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{kr}\right)\hfill \\ & & =\sum _{k=0}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-kr}{\lambda }^k}{\mathrm{\Gamma }(rk+1)}}{}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}\left(e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{kr}\right)\hfill \\ & & =\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-kr}{\lambda }^k}{\mathrm{\Gamma }(rk+1)}}{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }(rk+1)}{\mathrm{\Gamma }(rk+1-r)}}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{rk-r}\hfill \\ & & =e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r(k-1)}{\lambda }^k}{\mathrm{\Gamma }\left(r\right(k-1)+1)}}{(v(\ell )-v\left({\ell }_i\right))}^{r(k-1)}\hfill \\ & & =\lambda e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{k=1}^{+\infty }{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r(k-1)}{\lambda }^{k-1}}{\mathrm{\Gamma }\left(r\right(k-1)+1)}}{(v(\ell )-v\left({\ell }_i\right))}^{r(k-1)}\hfill \\ & & =\lambda e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r\right).\hfill \end{array}$$

□

Theorem 11.

Let $\gamma \in ]0,1]$ and $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$; then

$${}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x\right)(\ell )=x(\ell )-\sum _{k=0}^{n-1}{\displaystyle \frac{\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{k}k!}}{(v(\ell )-v\left({\ell }_i\right))}^k{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}.$$

(63)

Proof. 

From Theorem 6, where $r=n_r$, we get

$$\begin{array}{cc}\hfill {}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x\right)(\ell )& ={}_{{\ell }_i}\mathit{I}^{r,\gamma ,v}\left({}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}{D}^{n_r,\gamma ,v}x\right)(\ell )=\left({}_{{\ell }_i}\mathit{I}^{n_r,\gamma ,v}{D}^{n_r,\gamma ,v}x\right)(\ell )\hfill \\ & =x(\ell )-e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}\sum _{j=1}^{n_r}{\displaystyle \frac{\left({}_{{\ell }_i}\mathit{I}^{j-n_r,\gamma ,v}x\right)\left({{\ell }_i}^{+}\right){(v(\ell )-v\left({\ell }_i\right))}^{n_r-j}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-j}\mathrm{\Gamma }(n_r-j+1)}}\hfill \\ & =x(\ell )-\sum _{k=0}^{n_r-1}{\displaystyle \frac{\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right)}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{k}k!}}{(v(\ell )-v\left({\ell }_i\right))}^k{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}.\hfill \end{array}$$

□

Theorem 12.

Consider $\gamma \in ]0,1]$ and $\mathrm{Re}\left(r\right)>0$, $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$. Let $X_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)$; then

$$\begin{array}{cc}\hfill & {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)={(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^r{X}_{{\ell }_i}\left(p\right)\hfill \\ \hfill & -\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-1-k}\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right).\hfill \end{array}$$

(64)

Proof. 

By using Theorems 8 and 14, we get

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)=& {\mathcal{L}}_v\left\{\left({}_{{\ell }_i}\mathit{I}^{n_r-r,\gamma ,v}{D}^{n_r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)\hfill \\ \hfill =& {(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-n_r}{\mathcal{L}}_v\left\{\left(D^{r,\gamma ,v}x\right)(\ell )\right\}\left(p\right)\hfill \\ \hfill =& {(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-n_r}\hfill \\ & \times [{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r}{X}_{{\ell }_i}\left(p\right)-\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\hfill \\ & \sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{n_r-1-k}\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right)]\hfill \\ \hfill =& {(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^r{X}_{{\ell }_i}\left(p\right)-\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)\hfill \\ & \sum _{k=0}^{n_r-1}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-1-k}\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right).\hfill \end{array}$$

□

Based on the application of Theorems 9 and 12, the following proposition can be derived.

Proposition 3.

Consider $\gamma \in ]0,1]$ and $r\in \mathbb{C}$, where $\mathrm{Re}\left(r\right)>0$ $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$; then

$$\begin{array}{cc}\hfill \left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x\right)(\ell )& =\left({}_{{\ell }_i}\mathit{D}^{r,\gamma ,v}x\right)(\ell )-\sum _{k=0}^{n_r-1}{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-k}}{\mathrm{\Gamma }(k+1-r)}}{(v(\ell )-v\left({\ell }_i\right))}^{k-r}\hfill \\ \hfill & \times e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left({\ell }_i\right))}\left(D^{k,\gamma ,v}x\right)\left({\ell }_i\right),\hfill \end{array}$$

(65)

and

$$\begin{array}{cc}\hfill \left({}^C\mathit{D}_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )& =\left(D_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )-\sum _{k=0}^{n_r-1}{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{r-k}}{\mathrm{\Gamma }(k+1-r)}}{(v\left({\ell }_f\right)-v(\ell ))}^{k-r}\hfill \\ \hfill & \times e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v\left({\ell }_f\right)-v(\ell ))}\left(D^{k,\gamma ,v}x\right)\left({\ell }_f\right).\hfill \end{array}$$

(66)

Remark 6.

We have

• $\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}c\right)(\ell )\ne 0,\forall \gamma \in ]0,1].$
• Let $n_r=\left[\mathrm{Re}\left(r\right)\right]+1$; then ${\mathcal{L}}_v\left\{\left({}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}1\right)(\ell )\right\}\left(p\right)={\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}{\left(\cos \left(\frac{\pi }{2}\gamma \right)\right)}^{n_r}}{{(\cos \left(\frac{\pi }{2}\gamma \right)+\sin \left(\frac{\pi }{2}\gamma \right)p)}^{n_r}}}{\displaystyle \frac{1}{p}},\forall \gamma \in ]0,1[$.
• $D^{\gamma ,v}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}=0$ implies that ${}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}{e}^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}=0$.

Theorem 13.

Consider the linear v-Caputo cotangent differential equation

$$\left\{\begin{array}{cc}{}_{{\ell }_i}{}^C\mathit{D}^{r,\gamma ,v}x(\ell )-\lambda x(\ell )=y(\ell ),\hfill & 0<r,\gamma \le 1,\hfill \\ x\left({\ell }_i\right)=x_{{\ell }_i},\hfill \end{array}\right.$$

(67)

Then the solution of (67) is

$$\begin{array}{cc}\hfill & x(\ell )={\cot }_{r,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r)x_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\int }_{{\ell }_i}^{\ell }{\cot }_{r,r}^{\gamma }(v(\ell )-v\left(\rho \right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left(\rho \right))}^r){(v(\ell )-v\left(\rho \right))}^{r-1}y\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(68)

Proof. 

Applying ${\mathcal{L}}_v$ to (67) and using Theorem 12, where $n_r=1$, we get

$$\begin{array}{cc}\hfill \left({(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^r-\lambda \right)X_{{\ell }_i}\left(p\right)& =\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)x_{{\ell }_i}{(\cos \left({\displaystyle \frac{\pi }{2}}\gamma \right)+\sin \left({\displaystyle \frac{\pi }{2}}\gamma \right)p)}^{r-1}+Y_{{\ell }_i}\left(p\right).\hfill \end{array}$$

where $X_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)$ and $Y_{{\ell }_i}\left(p\right)={\mathcal{L}}_v\left\{y(\ell )\right\}\left(p\right)$. Hence,

$$X_{{\ell }_i}\left(p\right)={\displaystyle \frac{{\left(p+\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}\right)}^{r-1}{x}_{{\ell }_i}}{{\left(p+\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}\right)}^r-\lambda \sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}}}+{\displaystyle \frac{\sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}{Y}_{{\ell }_i}\left(p\right)}{{\left(\frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}+p\right)}^r-\lambda \sin {\left(\frac{\pi }{2}\gamma \right)}^{-r}}}.$$

From Equation (39), we have

$$\begin{array}{cc}\hfill & {\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}-\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right),\hfill \end{array}$$

(69)

and

$$\begin{array}{cc}\hfill & {\mathcal{L}}_v\left\{E_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right).\hfill \end{array}$$

(70)

From Equations (40) and (41), we get

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\{& \left.{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill =& {\left[{(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}})}^r+\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda \right]}^{-1},\hfill \end{array}$$

(71)

and

$$\begin{array}{cc}\hfill {\mathcal{L}}_v& \left\{E_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^{r-1}{\left[{\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^r+\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda \right]}^{-1}.\hfill \end{array}$$

(72)

Hence, by using Equations (70) and (72), we have

$$\begin{array}{c}\hfill {\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^{r-1}{\left[{\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^r-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\Lambda \right]}^{-1}\\ \hfill ={\mathcal{L}}_v\left\{E_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\\ \hfill ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right).\end{array}$$

(73)

Further, by using Equations (69) and (71), we get

$$\begin{array}{cc}\hfill & {\left[{\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)}^r+\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda \right]}^{-1}\hfill \\ \hfill & ={\mathcal{L}}_v\left\{{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p+{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}\right)\hfill \\ \hfill & ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right).\hfill \end{array}$$

(74)

We can get the following result by combining Equations (73) and (74) in Equation (67). We get

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)& ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)x_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\mathcal{L}}_v\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}\hfill \\ \hfill & \times E_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\}\left(p\right){\mathcal{L}}_v\{\Lambda \}\left(p\right).\hfill \end{array}$$

(75)

Hence, by using the v-convolution formula in Equation (75), we get

$$\begin{array}{cc}\hfill {\mathcal{L}}_v\left\{x(\ell )\right\}\left(p\right)& ={\mathcal{L}}_v\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)\right\}\left(p\right)x_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\mathcal{L}}_v\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}\hfill \\ \hfill & \times E_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right){\ast }_v\Lambda \}\left(p\right).\hfill \end{array}$$

(76)

Applying the Laplace inverse of Equation (76), we get

$$\begin{array}{cc}& x(\ell )=e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{E}_{r,1}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right)x_{{\ell }_i}\hfill \\ & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\left\{e^{-{\displaystyle \frac{\cos \left(\frac{\pi }{2}\gamma \right)}{\sin \left(\frac{\pi }{2}\gamma \right)}}(v(\ell )-v\left({\ell }_i\right))}{(v(\ell )-v\left({\ell }_i\right))}^{r-1}{E}_{r,r}\left(-\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}\lambda {(v(\ell )-v\left({\ell }_i\right))}^r\right){\ast }_v\Lambda \right\}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & x(\ell )={\cot }_{r,1}^{\gamma }(v(\ell )-v\left({\ell }_i\right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left({\ell }_i\right))}^r)x_{{\ell }_i}\hfill \\ \hfill & +\sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{\int }_{{\ell }_i}^{\ell }{\cot }_{r,r}^{\gamma }(v(\ell )-v\left(\rho \right),\lambda \sin {\left({\displaystyle \frac{\pi }{2}}\gamma \right)}^{-r}{(v(\ell )-v\left(\rho \right))}^r){(v(\ell )-v\left(\rho \right))}^{r-1}y\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(77)

□

Remark 7.

Consider $\gamma \in [0,1]$ and $K:[0,1]\to [0,+\infty ]$ to be continuous such that

$$\underset{\gamma \to 1^{-}}{\lim }K\left(\gamma \right)={\displaystyle \frac{\pi }{2}}+2k\pi ,\underset{\gamma \to 0^{+}}{\lim }K\left(\gamma \right)=2k\pi ,k\in \mathbb{Z},$$

Then, the left Riemann–Liouville cotangent fractional integral of x with respect to v is

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{I}^{r,\gamma ,v}x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(K\left(\gamma \right)\right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\ell )-v(\rho \left)\right)}{(v(\ell )-v\left(\rho \right))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(78)

and the right Riemann–Liouville cotangent integral of x with respect to v is

$$\begin{array}{cc}\hfill \left(I_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )=& {\displaystyle \frac{1}{\sin {\left(K\left(\gamma \right)\right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\rho )-v(\ell \left)\right)}{(v\left(\rho \right)-v(\ell ))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(79)

The left Riemann–Liouville cotangent derivative of x with respect to v is

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}{D}^{r,\gamma ,v}x)(\ell )& ={\displaystyle \frac{D^{n_r,\gamma }}{\sin {\left(K\left(\gamma \right)\right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\ell )-v(\rho \left)\right)}\hfill \\ \hfill & \times {(v(\ell )-v\left(\rho \right))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(80)

and the right Riemann–Liouville cotangent derivative of x with respect to v is

$$\begin{array}{cc}\hfill \left(D_{{\ell }_f}^{r,\gamma ,v}x\right)(\ell )& ={\displaystyle \frac{D^{n_r,\gamma }}{\sin {\left(K\left(\gamma \right)\right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\rho )-v(\ell \left)\right)}\hfill \\ \hfill & \times {(v\left(\rho \right)-v(\ell ))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(81)

The left Caputo cotangent derivative of x with respect to v is

$$\begin{array}{cc}\hfill {(}_{{\ell }_i}^C{D}^{r,\gamma ,v}x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(K\left(\gamma \right)\right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\ell )-v(\rho \left)\right)}\hfill \\ \hfill & \times {(v(\ell )-v\left(\rho \right))}^{n_r-r-1}\left(D^{n_r,\gamma ,v}x\right)\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(82)

and the right Caputo cotangent derivative of x with respect to v is

$$\begin{array}{cc}\hfill {(}^C{D}_{{\ell }_f}^{r,\gamma ,v}x)(\ell )=& {\displaystyle \frac{1}{\sin {\left(K\left(\gamma \right)\right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{\ell }^{{\ell }_f}{e}^{-\cot \left(K\right(\gamma \left)\right)\left(v\right(\rho )-v(\ell \left)\right)}\hfill \\ \hfill & \times {(v\left(\rho \right)-v(\ell ))}^{n_r-r-1}\left(D^{n_r,\gamma }x\right)\left(\rho \right)v^{\prime }\left(\rho \right)d\rho ,\hfill \end{array}$$

(83)

where $n_r=\left[Re\left(r\right)\right]+1$,

$$\left(D^{n_r,\gamma ,v}x\right)(\ell )=\left(\underset{n_r\mathit{times}}{\underbrace{D^{\gamma ,v}{D}^{\gamma ,v}\dots D^{\gamma ,v}}}x\right)(\ell ).$$

and $D^{\gamma ,v}x(\ell )=\cos \left(K\left(\gamma \right)\right)x(\ell )+\sin \left(K\left(\gamma \right)\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}$.

The introduction of this novel form of fractional calculus can provide valuable insights for researchers engaged in the ongoing exploration of this relevant topic, as exemplified by recent works such as [25,26,27,28].

6. Numerical Approach

In this section, we present the numerical approach and the application for solving the Caputo cotangent fractional Cauchy problem.

Theorem 14.

Let $m\ge 1$ be an integer and $x:[a,L]\to \mathbb{R}$ be a function of class $A{C}^2$. Consider

$$\begin{array}{cc}\hfill {\mathcal{A}}_m& ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}\sum _{i=0}^m{\displaystyle \frac{\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)i!}},\hfill \\ \hfill {\mathcal{B}}_{m,i}& ={\displaystyle \frac{\mathrm{\Gamma }(i+\gamma -1)}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )\mathrm{\Gamma }(\gamma -1)(i-1)!}},i=1,2,\dots ,m,\hfill \end{array}$$

(84)

and functions ${\mathcal{C}}_i:[a,L]\to \mathbb{R}$ by

$$\begin{array}{c}\hfill {\mathcal{C}}_i(\ell )={\int }_a^{\ell }{(v\left(s\right)-v\left(a\right))}^{i-1}{v}^{\prime }\left(\tau \right)e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(s\right)}\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(s\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(s\right)}{v^{\prime }\left(s\right)}}\right)ds.\end{array}$$

(85)

Then

$$\begin{array}{cc}\hfill {}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )=& {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}\right)\hfill \\ & -e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)v(\ell )}\sum _{i=1}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{\mathcal{C}}_i(\ell )+{\mathcal{E}}_m(\ell ),\hfill \end{array}$$

with

$$\underset{m\to \infty }{\lim }{\mathcal{E}}_m(\ell )=0,\forall \ell \in (a,L].$$

Proof. 

We have

$${}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )={\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(1-\gamma )}}{\int }_a^{\ell }{z}^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,$$

where $z^{\prime }\left(\tau \right)={(v(\ell )-v\left(\tau \right))}^{-\gamma }{v}^{\prime }\left(\tau \right)$ and $w\left(\tau \right)=e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})$. Then, integrating by parts, we get

$$\begin{array}{cc}& {}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )={\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{e}^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(a\right)}{(v(\ell )-v\left(a\right))}^{1-\gamma }\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(a\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(a\right)}{v^{\prime }\left(a\right)}}\right)\hfill \\ & +{\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{\int }_a^{\ell }{(v(\ell )-v\left(\tau \right))}^{1-\gamma }{\displaystyle \frac{d}{ds}}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}\left[\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}}\right]\right)d\tau .\hfill \end{array}$$

Using the generalized binomial theorem, we obtain

$$\begin{array}{cc}\hfill {(v(\ell )-v\left(\tau \right))}^{1-\gamma }& ={(v(\ell )-v\left(a\right))}^{1-\gamma }{\left(1-{\displaystyle \frac{v\left(\tau \right)-v\left(a\right)}{v(\ell )-v\left(a\right)}}\right)}^{1-\gamma }\hfill \\ & ={(v(\ell )-v\left(a\right))}^{1-\gamma }\sum _{k=0}^{\infty }{\displaystyle \frac{\mathrm{\Gamma }(k+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)k!}}{\left({\displaystyle \frac{v\left(s\right)-v\left(a\right)}{v(\ell )-v\left(a\right)}}\right)}^k.\hfill \end{array}$$

By substituting the previously mentioned equality into the fractional derivative expression, we arrive at

$$\begin{array}{cc}\hfill {}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )& ={\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{e}^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(a\right)}{(v(\ell )-v\left(a\right))}^{1-\gamma }(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(a\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(a\right)}{v^{\prime }\left(a\right)}})\hfill \\ & +{\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{\int }_a^{\ell }{(v(\ell )-v\left(a\right))}^{1-\gamma }\sum _{k=0}^m{\displaystyle \frac{\mathrm{\Gamma }(k+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)k!}}{\left({\displaystyle \frac{v\left(\tau \right)-v\left(a\right)}{v(\ell )-v\left(a\right)}}\right)}^k\hfill \\ & \times {\displaystyle \frac{d}{d\tau }}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})\right)d\tau +{\mathcal{E}}_m(\ell ),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{E}}_m(\ell )=& {\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{\int }_a^{\ell }{(v(\ell )-v\left(a\right))}^{1-\gamma }\sum _{i=m+1}^{\infty }{\displaystyle \frac{\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)i!}}{\left({\displaystyle \frac{v\left(\tau \right)-v\left(a\right)}{v(\ell )-v\left(a\right)}}\right)}^i\hfill \\ & \times {\displaystyle \frac{d}{d\tau }}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})\right)d\tau .\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill {}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )& ={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{(v(\ell )-v\left(a\right))}^{1-\gamma }(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(a\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(a\right)}{v^{\prime }\left(a\right)}})\hfill \\ & +{\displaystyle \frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{(v(\ell )-v\left(a\right))}^{1-\gamma }\sum _{i=1}^m{\displaystyle \frac{\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)i!{(v(\ell )-v\left(a\right))}^i}}\hfill \\ & \times {\int }_a^{\ell }{\left(v\left(\tau \right)-v\left(a\right)\right)}^i{\displaystyle \frac{d}{d\tau }}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})\right)d\tau +{\mathcal{E}}_m(\ell ).\hfill \end{array}$$

Setting $u\left(\tau \right)={(v\left(\tau \right)-v\left(a\right))}^i$ and $g^{\prime }\left(\tau \right)={\displaystyle \frac{d}{d\tau }}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})\right)$ and integrating by parts, we get

$$\begin{array}{cc}\hfill {}^C\mathit{D}_a^{\gamma ,r,v}x(\ell )& {\displaystyle =\frac{1}{\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}{(v(\ell )-v\left(a\right))}^{1-\gamma }(\cos \left(\frac{\pi }{2}r\right)x(\ell )+\sin \left(\frac{\pi }{2}r\right)\frac{x^{\prime }(\ell )}{v^{\prime }(\ell )})}\hfill \\ & {\displaystyle \times \left(1+\sum _{i=1}^m\frac{\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)i!}\right)}\hfill \\ & {\displaystyle -\frac{e^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{\mathrm{\Gamma }(2-\gamma )\sin {\left(\frac{\pi }{2}r\right)}^{\gamma -1}}\sum _{i=1}^m\frac{{(v(\ell )-v\left(a\right))}^{1-\gamma -i}\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)(i-1)!}{\int }_a^{\ell }{(v\left(\tau \right)-v\left(a\right))}^{i-1}{v}^{\prime }\left(\tau \right)}\hfill \\ & {\displaystyle \times e^{\cot \left(\frac{\pi }{2}r\right)v\left(\tau \right)}(\cos \left(\frac{\pi }{2}r\right)x\left(\tau \right)+\sin \left(\frac{\pi }{2}r\right)\frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)})d\tau +{\mathcal{E}}_m(\ell ).}\hfill \end{array}$$

On the other side, we show that

$$\underset{m\to +\infty }{\lim }{\mathcal{E}}_m\left(\tau \right)=0,\forall \tau \in [a,L].$$

We have

$$\begin{array}{ccc}\hfill \sum _{i=m+1}^{\infty }\left|{\displaystyle \frac{\mathrm{\Gamma }(i+\gamma -1)}{\mathrm{\Gamma }(\gamma -1)i!}}{\left({\displaystyle \frac{v\left(s\right)-v\left(a\right)}{v(\ell )-v\left(a\right)}}\right)}^i\right|& \le \hfill & \hfill \sum _{i=m+1}^{\infty }{\displaystyle \frac{e^{{(1-\gamma )}^2+(1-\gamma )}}{i^{2-\gamma }}}\\ & \le \hfill & \hfill {\int }_m^{\infty }{\displaystyle \frac{e^{{(1-\gamma )}^2+(1-\gamma )}}{s^{2-\gamma }}}ds\\ & \le \hfill & \hfill {\displaystyle \frac{e^{{(1-\gamma )}^2+(1-\gamma )}}{m^{1-\gamma }(1-\gamma )}},\end{array}$$

so

$$\begin{array}{ccc}\hfill |{\mathcal{E}}_m(\ell )|& \le & {\displaystyle \frac{e^{{(1-\gamma )}^2+(1-\gamma )}{e}^{-\cot \left(\frac{\pi }{2}r\right)v(\ell )}}{m^{1-\gamma }(1-\gamma )\sin {\left(\frac{\pi }{2}r\right)}^{1-\gamma }\mathrm{\Gamma }(2-\gamma )}}{(v(\ell )-v\left(a\right))}^{1-\gamma }\hfill \\ & & \times {\int }_a^{\ell }\left|{\displaystyle \frac{d}{d\tau }}\left(e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v\left(\tau \right)}(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(\tau \right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(\tau \right)}{v^{\prime }\left(\tau \right)}})\right)\right|d\tau .\hfill \end{array}$$

(86)

As $m\to \infty$, the right-hand side of the above inequality tends to zero for all $\ell \in [a,L]$. The proof is finished. □

Now consider the following Caputo cotangent fractional Cauchy problem:

$$\left\{\begin{array}{cc}{}_a{}^c\mathit{D}^{\gamma ,r,v}x(\ell )=f(\ell ,x(\ell )),& 0<\gamma <1,r\in (0,1],\\ x\left(a\right)=x_a,& a\in \mathbb{R},\end{array}\right.$$

(87)

where ${}^c\mathit{D}_a^{\gamma ,r,v}$ is the Caputo cotangent derivative and $f\in C(J\times \mathbb{R},\mathbb{R})$, with $J=[a,L]$.

Theorem 15.

Let $m\in \mathbb{N}$, ${\mathcal{A}}_m$ and ${\mathcal{B}}_{m,i}$, $i=1,2,\dots ,m$ be defined as in Equation (84). Let

$$\begin{array}{cc}\hfill f_1(\ell ,Y)& ={\displaystyle \frac{v^{\prime }(\ell )}{{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\sin \left(\frac{\pi }{2}r\right)}}[f(\ell ,Y_1)+Y_1[{\mathcal{B}}_{m,1}\sin \left({\displaystyle \frac{\pi }{2}}r\right){(v(\ell )-v\left(a\right))}^{-\gamma }\hfill \\ & -{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\cos \left({\displaystyle \frac{\pi }{2}}r\right)]+e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)v(\ell )}\sum _{i=2}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{Y}_i\hfill \\ & +\sin \left({\displaystyle \frac{\pi }{2}}r\right)e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(a\right))}{\mathcal{B}}_{m,1}{(v(\ell )-v\left(a\right))}^{-\gamma }{x}_a],\hfill \\ \hfill f_k(\ell ,Y)& ={\displaystyle \frac{{(v(\ell )-v\left(a\right))}^{k-1}{v}^{\prime }(\ell )}{{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }}}[f(\ell ,Y_1)e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v(\ell )}+{\mathcal{B}}_{m,1}{(v(\ell )-v\left(a\right))}^{-\gamma }\sin \left({\displaystyle \frac{\pi }{2}}r\right)[Y_1{e}^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)v(\ell )}\hfill \\ & -x_a{e}_r\left(v\left(a\right)\right)]+\sum _{i=2}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{Y}_i],\forall k\in \{2,\dots ,m\}.\hfill \end{array}$$

The numerical solution of problem (87) is equivalent to solving the following ordinary differential equation:

$$\left\{\begin{array}{cc}\dot{Y}(\ell )=F(\ell ,Y(\ell )),\hfill & \\ Y\left(a\right)=\left(\begin{array}{c}{x}_a\\ 0\\ 0\\ ⋮\\ 0\end{array}\right)\in {\mathbb{R}}^{m\times 1},\hfill \end{array}\right.$$

(88)

where

$$F\left(\ell ,Y(\ell )\right)=\left(\begin{array}{c}{f}_1(\ell ,Y(\ell ))\\ f_2(\ell ,Y(\ell ))\\ ⋮\\ f_m(\ell ,Y(\ell ))\end{array}\right),$$

and the numerical solution is $x=Y_1$.

Proof. 

Let $e_r$ be function defined by

$$\begin{array}{ccc}\hfill e_r(\ell )& =& e^{\cot \left({\displaystyle \frac{\pi }{2}}r\right)\ell }.\hfill \end{array}$$

According to Theorem 14, we have

$$\begin{array}{cc}\hfill {}^c\mathit{D}_c^{\gamma ,r,v}x(\ell )=& {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}\right)\hfill \\ \hfill & -e_r(-v(\ell ))\sum _{i=1}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{\mathcal{C}}_i(\ell )+{\mathcal{E}}_m(\ell )\hfill \\ \hfill & \approx {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}\right)\hfill \\ \hfill & -e_r(-v(\ell ))\sum _{i=1}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{\mathcal{C}}_i(\ell ),\hfill \end{array}$$

(89)

and from Equation (85), ${\mathcal{V}}_i$ is defined as being the solution of the equation

$${\mathcal{C}}_i^{\prime }\left(s\right)={(v\left(s\right)-v\left(a\right))}^{i-1}{v}^{\prime }\left(s\right)e_r\left(v\left(s\right)\right)\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(s\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(s\right)}{v^{\prime }\left(s\right)}}\right),$$

(90)

$${\mathcal{C}}_i\left(a\right)=0,i=1,\dots ,m.$$

(91)

According to problem (87) and approximation Equation (89), we obtain

$$\begin{array}{cc}\hfill & {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}\right)-e_r(-v(\ell ))\sum _{i=1}^m{\mathcal{B}}_{m,i}{(v(\ell )-v\left(a\right))}^{1-\gamma -i}{\mathcal{C}}_i(\ell )\hfill \\ \hfill & =f(\ell ,x(\ell \left)\right),\hfill \end{array}$$

(92)

and from Equation (90), we have

$$\begin{array}{cc}\hfill {\mathcal{C}}_1^{\prime }\left(s\right)& =v^{\prime }\left(s\right)e_r\left(v\left(s\right)\right)\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x\left(s\right)+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }\left(s\right)}{v^{\prime }\left(s\right)}}\right)\hfill \\ \hfill & =\sin \left({\displaystyle \frac{\pi }{2}}r\right){\left[x\left(s\right)e_r\left(v\left(s\right)\right)\right]}^{\prime },\hfill \end{array}$$

(93)

and then

$$\begin{array}{cc}\hfill x^{\prime }(\ell )& ={\displaystyle \frac{v^{\prime }(\ell )}{{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\sin \left(\frac{\pi }{2}r\right)}}[f(\ell ,x(\ell ))+x(\ell )[{\mathcal{B}}_{m,1}\sin \left({\displaystyle \frac{\pi }{2}}r\right){(v(\ell )-v\left(a\right))}^{-\gamma }\hfill \\ \hfill & -{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\cos \left({\displaystyle \frac{\pi }{2}}r\right)]+e_r(-v(\ell ))\sum _{k=2}^m{\mathcal{B}}_{m,k}{(v(\ell )-v\left(a\right))}^{1-\gamma -k}{\mathcal{C}}_k(\ell )\hfill \\ \hfill & +\sin \left({\displaystyle \frac{\pi }{2}}r\right)e_r(v\left(a\right)-v(\ell )){\mathcal{B}}_{m,1}{(v(\ell )-v\left(a\right))}^{-\gamma }{x}_a].\hfill \end{array}$$

(94)

On the other hand, from Equation (90), we have

$$\left(\cos \left({\displaystyle \frac{\pi }{2}}r\right)x(\ell )+\sin \left({\displaystyle \frac{\pi }{2}}r\right){\displaystyle \frac{x^{\prime }(\ell )}{v^{\prime }(\ell )}}\right)={\displaystyle \frac{{\mathcal{C}}_i^{\prime }(\ell )}{{(v(\ell )-v\left(a\right))}^{i-1}{v}^{\prime }(\ell )e_r\left(v(\ell )\right)}},i\in \{1,\dots ,m\},$$

(95)

and from Equations (92) and (95), we establish

$$\begin{array}{cc}& {\displaystyle {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }\frac{{\mathcal{C}}_i^{\prime }(\ell )}{{(v(\ell )-v\left(a\right))}^{i-1}{v}^{\prime }(\ell )e_r\left(v(\ell )\right)}-e_r(-v(\ell ))\sum _{k=1}^m{\mathcal{B}}_{m,k}{(v(\ell )-v\left(a\right))}^{1-\gamma -k}{\mathcal{C}}_k(\ell )}\hfill \\ & =f(\ell ,x(\ell \left)\right).\hfill \end{array}$$

From Equation (93), we have

$$\begin{array}{cc}& {\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }{\displaystyle \frac{{\mathcal{V}}_i^{\prime }(\ell )}{{(v(\ell )-v\left(a\right))}^{i-1}{v}^{\prime }(\ell )e_r\left(v(\ell )\right)}}-e_r(-v(\ell )){\mathcal{B}}_{m,1}{(v(\ell )-v\left(a\right))}^{-\gamma }\sin \left({\displaystyle \frac{\pi }{2}}r\right)\hfill \\ & {\displaystyle \times \left[x(\ell )e_r\left(v(\ell )\right)-x_a{e}_r\left(v\left(a\right)\right)\right]-e_r(-v(\ell ))\sum _{k=2}^m{\mathcal{B}}_{m,k}{(v(\ell )-v\left(a\right))}^{1-\gamma -k}{\mathcal{C}}_k(\ell )=f(\ell ,x(\ell )),}\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill {\mathcal{C}}_i^{\prime }(\ell )& ={\displaystyle \frac{{(v(\ell )-v\left(a\right))}^{i-1}{v}^{\prime }(\ell )}{{\mathcal{A}}_m{(v(\ell )-v\left(a\right))}^{1-\gamma }}}[f(\ell ,x(\ell ))e_r\left(v(\ell )\right)+{\mathcal{B}}_{m,1}{(v(\ell )-v\left(a\right))}^{-\gamma }\sin \left({\displaystyle \frac{\pi }{2}}r\right)\hfill \\ \hfill & \left[x(\ell )e_r\left(v(\ell )\right)-x_a{e}_r\left(v\left(a\right)\right)\right]\hfill \\ \hfill & +\sum _{k=2}^m{\mathcal{B}}_{m,k}{(v(\ell )-v\left(a\right))}^{1-\gamma -k}{\mathcal{C}}_k(\ell )],\forall i\in \{1,\dots ,m\}.\hfill \end{array}$$

(96)

From Equations (94) and (96), we obtain the system (88). □

7. Numerical Examples

In this section, we discuss two numerical examples to verify the theoretical results.

Example 1.

Let $0<\gamma <1$ and $r\in (0,1]$. Consider the following Cauchy-type initial value problem:

$$\left\{\begin{array}{c}{}^C\mathit{D}_1^{\gamma ,r,v}x(\ell )={\displaystyle \frac{\sin {\left(\frac{\pi }{2}r\right)}^{\gamma }\mathrm{\Gamma }\left(3\right)}{\mathrm{\Gamma }(3-\gamma )}}{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(1\right))}{(v(\ell )-v\left(1\right))}^{2-\gamma },\ell \in [1,2],\hfill \\ x\left(1\right)=0.\hfill \end{array}\right.$$

(97)

The exact solution of this system is given by $x(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(1\right))}{(v(\ell )-v\left(1\right))}^2$. The numerical solution is carried out in Matlab using the Explicit Euler method. Presented below are two graphs, Figure 1 and Figure 2, illustrating the exact solution and the approximate solution for various values of $m=\{8,16,32\}$, within the interval $[1,2]$. Assessing the maximum absolute error $E_m$ involves the following formula:

$$E_m={\parallel x-x_m\parallel }_{\infty }.$$

To test the stability of our model (97) against variations in initial conditions, we applied a low-amplitude but representative perturbation to the initial condition of the Cauchy problem (97). To assess the model’s stability, we modified the initial condition to $x^{*}\left(1\right)=1e^{-4}$, creating a perturbed system. For the values in

Table 1. The maximum absolute error for $v(\ell )=\sqrt{\ell }$ of $\ell \in [1,2]$ for Example 1.

	$\mathit{r}=0.9$	$\mathit{\gamma }=0.8$			$\mathit{r}=0.6$	$\mathit{\gamma }=0.9$			$\mathit{r}=1$	$\mathit{\gamma }=0.5$
$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$
8	0.0020	0.0020	0.0001	8	0.0011	0.0010	0.0001	8	0.0019	0.0019	0.0001
16	0.0009	0.0009	0.0001	16	0.0005	0.0005	0.0001	16	0.0007	0.0007	0.0001
32	0.0004	0.0004	0.0001	32	0.0002	0.0002	0.0001	32	0.0003	0.0003	0.0001

and

Table 2. The maximum absolute error for $v(\ell )=\ln (\ell )$ of $\ell \in [1,2]$ for Example 1.

	$\mathit{r}=0.9$	$\mathit{\gamma }=0.8$			$\mathit{r}=0.6$	$\mathit{\gamma }=0.9$			$\mathit{r}=1$	$\mathit{\gamma }=0.5$
$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$
8	0.0053	0.0053	0.0001	8	0.0024	0.0024	0.0001	8	0.0052	0.0052	0.0001
16	0.0025	0.0025	0.0001	16	0.0011	0.0012	0.0001	16	0.0020	0.0020	0.0001
32	0.0011	0.0011	0.0001	32	0.0005	0.0005	0.0001	32	0.0007	0.0007	0.0001

, we determined the absolute error $E_m$ by comparing the exact solution x with the approximate solution $x_m$ for the original (unperturbed) system. Additionally, we calculated the absolute error $E_m^p$ for the perturbed system by comparing the exact solution with its approximate counterpart $x_m^p$. We also computed the absolute difference $AD$ between the approximate solutions of the unperturbed and perturbed systems, $x_m$ and $x_m^p(\ell )$, respectively. The data in Table 1 and Table 2 indicates that the solutions from the perturbed system closely align with the exact solution, thereby verifying the model’s stability. Figure 3 and Figure 4 present a graphical illustration of the absolute error value between the exact solution and the approximate solution for ℓ belonging to $[1,2]$ associated with Equation (97). In Figure 5, we compare solutions for the Caputo cotangent with RAF with $v(\ell )=\sqrt{\ell }$, $r=0.9$, and $\gamma =0.5$ and the Caputo classical $r=1$, $\gamma =0.5$, within the interval $[1,2]$ for Example 1. In Figure 6, we show the effect of function v, $r=0.2$, within the interval $[1,2]$ for Example 1.

Example 2.

Let $0<\gamma <1$ and $r\in (0,1]$. Consider the following Cauchy-type initial value problem:

$$\begin{array}{cc}\hfill {}^C\mathit{D}_0^{\gamma ,r,v}x(\ell )& =x(\ell )+{\displaystyle \frac{\sin {\left(\frac{\pi }{2}r\right)}^{\gamma }\mathrm{\Gamma }\left(3\right)}{\mathrm{\Gamma }(3-\gamma )}}\hfill \\ \hfill & e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(0\right))}{(v(\ell )-v\left(0\right))}^{2-\gamma }\hfill \\ \hfill & -e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(0\right))}{(v(\ell )-v\left(0\right))}^2,\ell \in [0,1],\hfill \\ \hfill x\left(0\right)& =0.\hfill \end{array}$$

(98)

The exact solution of this system is given by $x(\ell )=e^{-\cot \left({\displaystyle \frac{\pi }{2}}r\right)(v(\ell )-v\left(0\right))}{(v(\ell )-v\left(0\right))}^2$. Presented below are graphs in Figure 7 illustrating the exact solution and the approximate solution for various values of $m=\{8,16,20\}$ within the interval $[0,1]$. To test the stability of our model Equation (98) against variations in initial conditions, we applied a low-amplitude but representative perturbation to the initial condition of the Cauchy problem (98). To assess the model’s stability, we modified the initial condition to $x^{*}\left(1\right)=1e^{-4}$, creating a perturbed system. For the values in

Table 3. The maximum absolute error for $v(\ell )={\ell }^2$ of $\ell \in [0,1]$ for Example 2.

	$\mathit{r}=0.9$	$\mathit{\gamma }=0.8$			$\mathit{r}=0.6$	$\mathit{\gamma }=0.9$			$\mathit{r}=1$	$\mathit{\gamma }=0.5$
$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$	$\mathbf{m}$	${\mathbf{E}}_{\mathbf{m}}$	${\mathbf{E}}_{\mathbf{m}}^{\mathbf{p}}$	$\mathbf{AD}$
8	0.0170	0.0170	0.0001	8	0.0067	0.0067	0.0001	8	0.0237	0.0237	0.0001
16	0.0080	0.0080	0.0001	16	0.0032	0.0032	0.0001	16	0.0092	0.0092	0.0001
20	0.0062	0.0062	0.0001	20	0.0023	0.0023	0.0001	20	0.0067	0.0067	0.0001

, we determined the absolute error $E_m$ by comparing the exact solution x with the approximate solution $x_m$ for the original (unperturbed) system. Additionally, we calculated the absolute error $E_m^p$ for the perturbed system by comparing the exact solution with its approximate counterpart $x_m^p$. We also computed the absolute difference $AD$ between the approximate solutions of the unperturbed and perturbed systems, $x_m$ and $x_m^p(\ell )$, respectively. The data in Table 3 indicates that the solutions from the perturbed system closely align with the exact solution, thereby verifying the model’s stability. Figure 8 presents a graphical illustration of the absolute error value between the exact solution and the approximate solution for ℓ belonging to $[0,1]$ associated with Equation (98).

8. Physical Interpretation and Real-World Applications

The proposed cotangent fractional derivatives with respect to another function (RAF) are not merely mathematical generalizations but possess significant physical interpretations and practical applicability across various scientific domains.

8.1. Physical Interpretation of the Cotangent Kernel

The kernel $e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}$ in our operators embodies several important physical characteristics:

• Memory Effects: The exponential decay term $e^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}$ represents fading memory, where recent events ($\rho$ close to ℓ) have a stronger influence than distant past events. This is particularly relevant in viscoelastic materials where stress depends on the entire deformation history.
• Damped Oscillatory Behavior: The cotangent function $\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)$ introduces oscillatory characteristics when $\gamma \in (0.5,1)$, modeling systems with both damping and oscillation, such as vibrating structures with energy dissipation.
• Time Scaling: The function $v(\ell )$ serves as an internal time or process variable, allowing the derivative to adapt to different temporal scales or material heterogeneities.

8.2. Applications in Viscoelastic Materials

The cotangent derivatives naturally model viscoelastic behavior, where materials exhibit both elastic (instantaneous response) and viscous (time-dependent) characteristics:

$$\sigma (\ell )=E·ϵ(\ell )+\eta ·{}^C\mathit{D}_0^{\gamma ,r,v}ϵ(\ell )$$

(99)

where $\sigma$ is stress, $ϵ$ is strain, E is the elastic modulus, and $\eta$ is the fractional viscosity coefficient. The function $v(\ell )$ can represent the following:

• Temperature-dependent time scaling: $v(\ell )={\int }_0^{\ell }\alpha \left(T\left(s\right)\right)ds$, where $\alpha \left(T\right)$ accounts for temperature effects on relaxation times.
• Non-uniform aging: $v(\ell )={\ell }^{\beta }$, modeling accelerated/decelerated material aging.

8.3. Anomalous Diffusion Processes

The cotangent operators effectively describe anomalous diffusion phenomena, where mean-squared displacement follows $\langle x^2\left(t\right)\rangle \sim t^{\alpha }$ with $\alpha \ne 1$:

$${}^C\mathit{D}_0^{\gamma ,r,v}C(x,\ell )=D{\displaystyle \frac{{\partial }^{2}C}{\partial x^2}}$$

(100)

Here, $v(\ell )$ enables modeling of the following:

• Heterogeneous media: $v(\ell )={\ell }^{1/d_w}$, where $d_w$ is the walk dimension in fractal media.
• Time-dependent diffusivity: $v(\ell )={\int }_0^{\ell }D\left(s\right)ds$ for spatially/temporally varying diffusion coefficients.

8.4. Biological Systems and Growth Models

In biological applications, the cotangent derivatives capture memory effects in growth processes and physiological systems:

$${}^C\mathit{D}_0^{\gamma ,r,v}N(\ell )=rN(\ell )\left(1-{\displaystyle \frac{N(\ell )}{K}}\right)$$

(101)

where the function $v(\ell )$ can represent the following’:

• Metabolic time: $v(\ell )=\ln (1+\ell )$ for biological processes operating on logarithmic time scales.
• Environmental factors: $v(\ell )={\int }_0^{\ell }f(T\left(s\right),pH\left(s\right),\dots )ds$ incorporating multiple environmental variables.
• Developmental stages: Piecewise $v(\ell )$ functions modeling different growth phases (embryonic, juvenile, adult).

8.5. Memory-Dependent Processes

The non-local nature of cotangent derivatives makes them ideal for systems with long-range temporal dependencies:

• Economics: Market memory effects in financial time series.
• Psychology: Learning processes where past experiences influence current behavior.
• Climate science: Long-memory processes in temperature and precipitation records.

8.6. Engineering Applications

• Control Systems: Fractional-order controllers with $v(\ell )$ adapting to operating conditions.
• Signal Processing: Analysis of signals with time-varying frequency content using appropriate $v(\ell )$.
• Material Science: Modeling complex material responses under varying environmental conditions.

8.7. Advantages over Classical Fractional Operators

The cotangent derivatives with RAF offer several advantages:

• Flexibility: The function $v(\ell )$ provides adaptability to various physical scenarios.
• Physical interpretability: The exponential-cotangent kernel has clear physical meaning.
• Mathematical consistency: Preservation of semigroup properties ensures physical predictability.
• Computational efficiency: The Laplace transform approach enables analytical solutions.

The numerical examples presented in Section 7 demonstrate the practical utility of these operators in solving real-world problems with complex memory effects and heterogeneous characteristics.

9. Comprehensive Comparison with Modern Fractional Operators

To position our work within the broader context of modern fractional calculus, we provide a detailed comparison between the proposed cotangent derivatives with RAF and other prominent nonsingular fractional operators, including the Prabhakar and Atangana–Baleanu derivatives.

Left Riemann-Liouville integral:

$$\begin{array}{cc}\hfill & \left({\ell }_i{I}^{r,\gamma ,v}x\right)(\ell )={\displaystyle \frac{1}{\sin {\left(\frac{\pi }{2}\gamma \right)}^r\mathrm{\Gamma }\left(r\right)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(102)

Left Riemann-Liouville derivative:

$$\begin{array}{cc}\hfill & \left({\ell }_i{D}^{r,\gamma ,v}x\right)(\ell )={\displaystyle \frac{D^{n_r,\gamma ,v}}{\sin {\left(\frac{\pi }{2}\gamma \right)}^{n_r-r}\mathrm{\Gamma }(n_r-r)}}{\int }_{{\ell }_i}^{\ell }{e}^{-\cot \left({\displaystyle \frac{\pi }{2}}\gamma \right)(v(\ell )-v\left(\rho \right))}{(v(\ell )-v\left(\rho \right))}^{n_r-r-1}x\left(\rho \right)v^{\prime }\left(\rho \right)d\rho .\hfill \end{array}$$

(103)

The Atangana–Baleanu derivative: [16]:

$$\begin{array}{cc}\hfill & {}^{ABC}\mathit{D}_{a+}^{\alpha }f(\ell )={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_a^{\ell }{f}^{\prime }\left(\xi \right)E_{\alpha }\left(-{\displaystyle \frac{\alpha }{1-\alpha }}{(\ell -\xi )}^{\alpha }\right)d\xi .\hfill \end{array}$$

(104)

The Prabhakar Prabhakar integral [29]:

$$\begin{array}{cc}\hfill & \left({\mathcal{E}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)(\ell )={\int }_a^{\ell }{(\ell -\xi )}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {(\ell -\xi )}^{\rho }\right]f\left(\xi \right)d\xi .\hfill \end{array}$$

(105)

The Caputo–Fabrizio Operator [12]:

$$\begin{array}{cc}\hfill & {}^{CF}\mathit{D}_{a+}^{\alpha }f(\ell )={\displaystyle \frac{1}{1-\alpha }}{\int }_a^{\ell }{f}^{\prime }\left(\xi \right)\exp \left(-{\displaystyle \frac{\alpha }{1-\alpha }}(\ell -\xi )\right)d\xi .\hfill \end{array}$$

(106)

In

Table 4. Parameter sensitivity and physical meaning.

Operator	Parameters	Physical Interpretation
Cotangent RAF	$\gamma ,r,v(\ell )$	$\gamma$: Memory strength and oscillation frequency r: Derivative order and power-law exponent $v(\ell )$: Internal time scaling and process adaptation
Atangana–Baleanu	$\alpha$	$\alpha$: Memory length and fractional order
Prabhakar	$\alpha ,\rho ,\gamma ,\omega$	$\alpha$: Basic fractional order $\rho$: Time scaling parameter $\gamma$: Memory complexity $\omega$: Coupling strength
Caputo–Fabrizio	$\alpha$	$\alpha$: Memory decay rate

, the Parameter Sensitivity Analysis.

The kernel functions for each fractional operator are defined as follows: The Cotangent RAF is

$$K(\ell ,\tau )=e^{-\cot \left({\displaystyle \frac{\pi \gamma }{2}}\right)(v(\ell )-v\left(\tau \right))}{(v(\ell )-v\left(\tau \right))}^{r-1},$$

(107)

where $v(\ell )$ is an increasing function defining the time scaling. The Atangana–Baleanu is

$$K(\ell ,\tau )=E_{\alpha }\left(-{\displaystyle \frac{\alpha }{1-\alpha }}{(\ell -\tau )}^{\alpha }\right),$$

(108)

where $E_{\alpha }(·)$ is the Mittag–Leffler function. The Prabhakar is

$$K(\ell ,\tau )={(\ell -\tau )}^{\alpha -1}{E}_{\rho ,\alpha }^{\gamma }\left(\omega {(\ell -\tau )}^{\rho }\right),$$

(109)

where $E_{\rho ,\alpha }^{\gamma }(·)$ is the Prabhakar (three-parameter Mittag–Leffler) function. The Caputo–Fabrizio is

$$K(\ell ,\tau )=e^{-{\displaystyle \frac{\alpha }{1-\alpha }}(\ell -\tau )}.$$

(110)

• Interpretation of Results

The comparative analysis reveals that each fractional operator possesses unique characteristics that make it suitable for specific application domains (Figure 9):

• The Cotangent RAF operator demonstrates the highest memory strength (integral = 1.7940) and unique oscillatory capabilities, making it ideal for systems combining memory effects with periodic behavior.
• The Atangana–Baleanu operator, while having the lowest kernel integral (0.4560), excels in modeling systems with strong long-range memory dependencies, particularly those exhibiting power-law decay.
• The Prabhakar operator offers intermediate memory strength (1.1011) with exceptional flexibility through its multiple parameters, suitable for complex multi-scale phenomena.
• The Caputo–Fabrizio operator provides a balanced approach with good memory strength (1.0769) and computational efficiency, ideal for standard applications where complex memory patterns are not required.

The choice of operator should be guided by the specific physical characteristics of the system under study, computational constraints, and the required level of modeling sophistication.

10. Conclusions

We have introduced the cotangent derivatives with RAF, denoted as $D^{r,\gamma ,v}$ (Riemann–Liouville type) and ${}^C\mathit{D}^{r,\gamma ,v}$ (Caputo type), which utilize an exponential cotangent function as their kernel. The notable advantage of these new fractional derivatives is their ability to exhibit a semigroup property. Furthermore, we have identified special cases of these derivatives: when $\gamma =1$, we obtain the RL-D, C-D, and RL-I with RAF, and when $\gamma =1$ and $v(\ell )=\ell$, we obtain the RL-D, C-D, and RL-I. By utilizing the v-Laplace transform of cotangent derivatives and integrals, we provide exact solutions to linear cotangent differential equations. Moreover, we present the numerical approach, the application for solving the Caputo cotangent fractional Cauchy problem, and two examples for testing this approach. The introduction of this novel fractional calculus type can prove beneficial to researchers working on this ever-relevant subject matter.