Nonlocal Telegraph Equation in Frame of the Conformable Time-Fractional Derivative

Abstract

We use the cosine family of linear operators to prove the existence, uniqueness, and stability of the integral solution of a nonlocal telegraph equation in frame of the conformable time-fractional derivative. Moreover, we give its implicit fundamental solution in terms of the classical trigonometric functions.

1. Introduction and Statement of the Problem

The telegraph equation is better than the heat equation in modeling of physical phenomena, which have a parabolic behavior [1]. The one-dimensional telegraph equation can be written as follows:where and are, respectively, the resistance and the conductance of resistor, is the capacitance of capacitor, and is the inductance of coil. Many concrete applications amount to replacing the time derivative in the telegraph equation with a fractional derivative. For example, in the works [2–8], the authors have extensively studied the time-fractional telegraph equation with Caputo fractional derivative. For more details about the good effect of the fractional derivative, we refer to monographs [9–13].

Recently, a new definition of fractional derivative, named “fractional conformable derivative,” is introduce by Khalil et al. [14]. This novel fractional derivative is compatible with the classical derivative and it is excellent for study nonregular solutions. Since the subject of the fractional conformable derivative has attracted the attention of many authors in domains such as mechanics [15], electronic [16], and anomalous diffusion [17]. We are interested in studying in this paper the telegraph model (1) in framework of the time-fractional conformable derivative. Precisely, we will propose the following transformations:where and are the time-fractional conformable derivative operators [14]. Then, we get the fractional conformable telegraph model associated with the transformations (2) and (3) as follows:where the time-parameter belongs to an interval , with is a fixed positive real number. The spatial parameter belongs to the interval .

We associate to (4) the boundary and the nonlocal initial conditions:where is a nonnull fixed integer and . The quantities , , and are physical measures. The condition appearing in (8) means the nonlocal condition [18]. For physical interpretations of this condition, we refer to works [19, 20]. For example, in [19] the author used a nonlocal condition of the form (8) to describe the diffusion phenomenon of a small amount of gas in a transparent tube.

We note that it is not easy to find the fundamental solution of (4) by using the Laplace transform method if . For this reason, we will propose the integral solution concept based on an operator theory approach. When , we investigate an implicit fundamental solution.

The content of this paper is organized as follows. In Section 2, we recall some needed results of the conformable fractional derivative and cosine family of linear operators. In Section 3, we prove the existence, uniqueness, and stability of the integral solution of (4) by using of an operator theory approach. Section 4 is devoted to an implicit fundamental solution of (4) in terms of the classical trigonometric functions.

2. Preliminaries

We start recalling some concepts on the conformable fractional calculus [14].

The conformable fractional derivative of a function of order at is defined by the following limit:When this limit exists, we say that is -differentiable at .

If is -differentiable at some and the limit exists. Then, we define the conformable fractional derivative of at byThe -fractional integral of a function is defined byIf is a continuous function in the domain of , then we haveAccording to [21], if is differentiable, then we haveWe remark that the classical Laplace transform is not compatible with the conformable fractional derivative. For this reason, the adapted transform is defined as [21]The above transformation is called fractional Laplace transform of order of the function .

If is differentiable, then the action of the fractional Laplace transform on the conformable fractional derivative is given as follows:Now, we introduce some results concerning the cosine family theory [22].

A one-parameter family of bounded linear operators on the Banach space is called a strongly continuous cosine family if and only if(1), (I is the identity operator),(2), for all ,(3) is continuous for each fixed .

We define also the sine family byThe infinitesimal generator of a strongly continuous cosine family on is defined byIf is the infinitesimal generator of a strongly cosine family on , then there exists a constant such that, for all with , we havewhere is the real part of the complex number .

Before presenting the main results, we introduce the following notations:

3. Integral Solution by Using an Operator Theory Approach

Define the operator byAccording to [23], the operator generates a cosine family on . Moreover, and , for all .

Next, we consider the following transformations:Then, we get the following nonlocal fractional ordinary differential equation:We denote the Banach space of continuously -differentiable functions from into with the norm . Here, is the classical norm in the space

3.1. Existence and Uniqueness of the Integral Solution

To explain integral Duhamel’s formula, we apply the fractional Laplace transform to (22), obtainingWe remark that, for , we have classical Duhamel’s formula [22]. Hence, we can introduce the following definition.

Definition 1. We say that is an integral solution of the equation (22) if the following assertion is true:

Theorem 2. The Cauchy problem (22) has a unique integral solution, provided that

Proof. Define the operator byLet ; then we haveAccordingly, we obtainThen, we getFinally, has an unique fixed point in , which is the integral solution of the equation (22).

Now, we give a result that is better than the previous one.

Theorem 3. The Cauchy problem (22) has a unique integral solution, provided that

Proof. Define the operator byNext, we define a new norm in bywhereFor and , we haveTherefore, we obtainAccordingly, we getHence, we conclude thatThe fact proves that has an unique fixed point in , which is the integral solution of equation (22).

3.2. Stability of the Integral Solution

Here, we give a result concerning the nonlocal-condition effect on the stability of the integral solution.

Theorem 4. Let and be solutions associated with and , respectively. Then, we have the following estimate:provided that

Proof. We haveConsequently, we getFinally, we obtain the following estimation:

4. Implicit Fundamental Solution in the Case When

Here, we give the implicit fundamental solution of (4) by using the separating variables method. To do so, let . Then, (4) becomes as follows:Then, there exists a constant such asAccording to (5) and (6), the solution of (47) is given byMoreover, the solution of (4) can written aswhere and is the solution of the following ordinary differential equation:with the nonlocal initial conditionsBy using the fractional Laplace transform in (50), we getFinally, replacing in (49), we get

Proposition 5. For and the formula (53) coincides with the fundamental solution of the classical telegraph equation (1).

Proof. Based on the separating variables method in (1), we obtain where is the solution of the following ordinary differential equation:with the initial conditionsBy using the classical Laplace transform in (55), we getReplacing in (54), we find the fundamental solution of the classical telegraph equation (1) as follows:

Remark 6. If we consider the fractional derivative in Caputo’s sense, the implicit fundamental solution of (4) can be written in terms of the Mittag-Leffler function. However, in our case we have found the implicit fundamental solution in terms of the classical trigonometric functions.

Remark 7. The integral solution does not impose any constraint concerning the choice of the derivation parameters. This provides more freedom concerning the choice of sensitive parameters and in practice situations for modeling naturel phenomena.

5. Conclusion

We have studied a time-conformable fractional telegraph equation with nonlocal condition. In the case when , we have given the implicit fundamental solution in terms of the classical trigonometric functions. In the general case, we have established the existence, uniqueness, and stability of the integral solution.

As for future work, we intend to give the fundamental solution for all values of and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.