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

Authors: Mohamed Bouaouid; Khalid Hilal; Said Melliani
Journal: Advances in Mathematical Physics
 (2019)

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## 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.


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## Body

## 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:(1)∂2ut,l∂t2=1LC∂2ut,l∂l2-RGLCut,l-RL+GC∂ut,l∂t,where R and G are, respectively, the resistance and the conductance of resistor, C is the capacitance of capacitor, and L 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:(2)∂2∂t2→∂α∂tα∂α∂tα,0<α<1,(3)∂∂t→∂γ∂tγ,0<γ≤α<1,where ∂α/∂tα and ∂γ/∂tγ 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:(4)∂α∂tα∂αut,l∂tα=1LC∂2ut,l∂l2-RGLCut,l-RL+GC∂γut,l∂tγ,where the time-parameter t belongs to an interval [0,τ], with τ is a fixed positive real number. The spatial parameter l belongs to the interval [0,π].We associate to (4) the boundary and the nonlocal initial conditions:(5)ut,0=0,t∈0,τ,(6)ut,π=0,t∈0,τ,(7)u0,l=x0,l∈0,π,(8)∂αu0,l∂tα=x1+∑i=1pεiuti,l,l∈0,π,where p is a nonnull fixed integer and 0<t1<⋯<tp<τ. The quantities x0, x1, and ε1,…,εp 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 α≠γ. For this reason, we will propose the integral solution concept based on an operator theory approach. When α=γ, we investigate an implicit fundamental solution.The content of this paper is organized as follows. In Section2, 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 functionx of order α at t>0 is defined by the following limit:(9)dαxtdtα=limε→0⁡xt+εt1-α-xtε.When this limit exists, we say that x is (α)-differentiable at t.Ifx is (α)-differentiable at some t~0 and the limit limt→0+dαx(t)/dtα exists. Then, we define the conformable fractional derivative of x at 0 by(10)dαx0dtα=limt→0+⁡dαxtdtα.The (α)-fractional integral of a function x is defined by(11)Iαxt=∫0tsα-1xsds.If x is a continuous function in the domain of Iα, then we have(12)dαIαxtdtα=xt.According to [21], if x is differentiable, then we have(13)Iαdαxdtαt=xt-x0.We remark that the classical Laplace transform is not compatible with the conformable fractional derivative. For this reason, the adapted transform is defined as [21](14)Lαxtλ≔∫0+∞tα-1e-λtα/αxtdt.The above transformation is called fractional Laplace transform of order α of the function x.Ifx is differentiable, then the action of the fractional Laplace transform on the conformable fractional derivative is given as follows:(15)Lαdαxtdtαλ=λLαxtλ-x0.Now, we introduce some results concerning the cosine family theory [22].A one-parameter family(C(t))t∈R of bounded linear operators on the Banach space X is called a strongly continuous cosine family if and only if(1)
C(0)=I, (I is the identity operator),(2)
C(s+t)+C(s-t)=2C(s)C(t), for all t,s∈R,(3)
t↦C(t)x is continuous for each fixed x∈X.We define also the sine family by(16)St≔∫0tCsds.The infinitesimal generator A of a strongly continuous cosine family ((C(t))t∈R,(S(t))t∈R) on X is defined by(17)DA=x∈X,t↦Ctxisatwicecontinuouslydifferentiablefunction,A=d2Ctdt2t=0.If A is the infinitesimal generator of a strongly cosine family ((C(t))t∈R,(S(t))t∈R) on X, then there exists a constant ω≥0 such that, for all λ with Re(λ)>ω, we have(18)λ2∈ρA,ρA:  is  the  resolvent  set  ofA,λλ2I-A-1=∫0+∞e-λtCtdt,λ2I-A-1=∫0+∞e-λtStdt,where Re(λ) is the real part of the complex number λ.Before presenting the main results, we introduce the following notations:(19)a=RGLC,b=RL+GC,c=1LCandε=∑i=1pεi.


## 3. Integral Solution by Using an Operator Theory Approach

Define the operatorA:L2([0,π],dl)→L2([0,π],dl) by(20)A=c∂2.∂l2andDA=u∈H20,π,dl,u0=uπ=0.According to [23], the operator A generates a cosine family ((C(t))t∈R,(S(t))t∈R) on L2([0,π],dl). Moreover, |C(t)|≤1 and |S(t)|≤1, for all t∈[0,τ].Next, we consider the following transformations:(21)xtl=ut,l,ft,xt,dγxtdtγ=-axt-bdγxtdtγ,gx=∑i=1pεixti.Then, we get the following nonlocal fractional ordinary differential equation:(22)dαdtαdαxtdtα=Axt+ft,xt,dγxtdtγ,(23)x0=x0,(24)dαx0dtα=x1+gx.We denote Cα the Banach space of continuously (α)-differentiable functions from [0,τ] into L2([0,π],dl) with the norm |x|=supt∈[0,τ]x(t)+supt∈[0,τ]dαx(t)/dtα. Here, . is the classical norm in the space L2([0,π],dl).


### 3.1. Existence and Uniqueness of the Integral Solution

To explain integral Duhamel’s formula, we apply the fractional Laplace transform to (22), obtaining(25)xt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds.We remark that, for α=γ=1, we have classical Duhamel’s formula [22]. Hence, we can introduce the following definition.Definition 1.
We say thatx∈Cα is an integral solution of the equation (22) if the following assertion is true:(26)xt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds,t∈0,τ.Theorem 2.
The Cauchy problem (22) has a unique integral solution, provided that (27)ε+max⁡aταα,bτ2α-γ2α-γ<12.Proof.
Define the operatorΓ:Cα→Cα by(28)Γxt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds.Let x,y∈Cα; then we have(29)Γyt-Γxt=Stααgy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds,dαdtαΓyt-Γxt=Ctααgy-gx+∫0tsα-1Ctα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Accordingly, we obtain(30)Γyt-Γxt+dαdtαΓyt-Γxt≤2ε+max⁡aταα,bτ2α-γ2α-γy-x.Then, we get(31)Γy-Γx≤2ε+max⁡aταα,bτ2α-γ2α-γy-x.Finally, Γ has an unique fixed point x in (Cα,|.|), 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 (32)ε<12.Proof.
Define the operatorΓ:Cα→Cα by(33)Γxt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds,t∈0,τ.Next, we define a new norm |.|α in Cα by(34)xα=exp⁡-θ.ααx,where(35)θ=3max⁡a,bτα-γ1-2ε.For x,y∈Cα and t∈[0,τ], we have(36)Γyt-Γxt=Stααgy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Therefore, we obtain(37)Γyt-Γxt≤εexp⁡θtαα+max⁡a,bτα-γ∫0tsα-1exp⁡θsααdsy-xα,dαdtαΓyt-Γxt≤εexp⁡θtαα+max⁡a,bτα-γ∫0tsα-1exp⁡θsααdsy-xα.Accordingly, we get(38)Γy-Γxα≤2ε+max⁡a,bτα-γθy-xα.Hence, we conclude that(39)Γy-Γxα≤2ε+13y-xα.The fact 2(ε+1)/3<1 proves that Γ has an unique fixed point x in (Cα,|.|α), 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.
Letx and y be solutions associated with (x0,x1) and (x0,y1), respectively. Then, we have the following estimate:(40)y-x≤21-2ε+max⁡aτα/α,bτ2α-γ/2α-γy1-x1,provided that (41)ε+max⁡aταα,bτ2α-γ2α-γ<12.Proof.
We have(42)yt-xt=Stααy1-x1+gy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds,dαdtαyt-xt=Ctααy1-x1+gy-gx+∫0tsα-1Ctα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Consequently, we get(43)y-x≤2ε+max⁡aταα,bτ2α-γ2α-γy-x+2y1-x1.
Finally, we obtain the following estimation:(44)y-x≤21-2ε+max⁡aτα/α,bτ2α-γ/2α-γy1-x1.


## 3.1. Existence and Uniqueness of the Integral Solution

To explain integral Duhamel’s formula, we apply the fractional Laplace transform to (22), obtaining(25)xt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds.We remark that, for α=γ=1, we have classical Duhamel’s formula [22]. Hence, we can introduce the following definition.Definition 1.
We say thatx∈Cα is an integral solution of the equation (22) if the following assertion is true:(26)xt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds,t∈0,τ.Theorem 2.
The Cauchy problem (22) has a unique integral solution, provided that (27)ε+max⁡aταα,bτ2α-γ2α-γ<12.Proof.
Define the operatorΓ:Cα→Cα by(28)Γxt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds.Let x,y∈Cα; then we have(29)Γyt-Γxt=Stααgy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds,dαdtαΓyt-Γxt=Ctααgy-gx+∫0tsα-1Ctα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Accordingly, we obtain(30)Γyt-Γxt+dαdtαΓyt-Γxt≤2ε+max⁡aταα,bτ2α-γ2α-γy-x.Then, we get(31)Γy-Γx≤2ε+max⁡aταα,bτ2α-γ2α-γy-x.Finally, Γ has an unique fixed point x in (Cα,|.|), 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 (32)ε<12.Proof.
Define the operatorΓ:Cα→Cα by(33)Γxt=Ctααx0+Stααx1+gx+∫0tsα-1Stα-sααfs,xs,dγxsdtγds,t∈0,τ.Next, we define a new norm |.|α in Cα by(34)xα=exp⁡-θ.ααx,where(35)θ=3max⁡a,bτα-γ1-2ε.For x,y∈Cα and t∈[0,τ], we have(36)Γyt-Γxt=Stααgy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Therefore, we obtain(37)Γyt-Γxt≤εexp⁡θtαα+max⁡a,bτα-γ∫0tsα-1exp⁡θsααdsy-xα,dαdtαΓyt-Γxt≤εexp⁡θtαα+max⁡a,bτα-γ∫0tsα-1exp⁡θsααdsy-xα.Accordingly, we get(38)Γy-Γxα≤2ε+max⁡a,bτα-γθy-xα.Hence, we conclude that(39)Γy-Γxα≤2ε+13y-xα.The fact 2(ε+1)/3<1 proves that Γ has an unique fixed point x in (Cα,|.|α), 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.
Letx and y be solutions associated with (x0,x1) and (x0,y1), respectively. Then, we have the following estimate:(40)y-x≤21-2ε+max⁡aτα/α,bτ2α-γ/2α-γy1-x1,provided that (41)ε+max⁡aταα,bτ2α-γ2α-γ<12.Proof.
We have(42)yt-xt=Stααy1-x1+gy-gx+∫0tsα-1Stα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds,dαdtαyt-xt=Ctααy1-x1+gy-gx+∫0tsα-1Ctα-sααfs,ys,dγysdtγ-fs,xs,dγxsdtγds.Consequently, we get(43)y-x≤2ε+max⁡aταα,bτ2α-γ2α-γy-x+2y1-x1.
Finally, we obtain the following estimation:(44)y-x≤21-2ε+max⁡aτα/α,bτ2α-γ/2α-γy1-x1.


## 4. Implicit Fundamental Solution in the Case Whenγ=α

Here, we give the implicit fundamental solution of (4) by using the separating variables method. To do so, let u(t,l)=x(t)y(l). Then, (4) becomes as follows:(45)axtyl+byldαxtdtα+yldαdtαdαxtdtα=cxtd2yldl2.Then, there exists a constant λ2 such as(46)a+λ2xt+bdαxtdtα+dαdtαdαxtdtα=0,(47)cd2yldl2=-λ2yl.According to (5) and (6), the solution of (47) is given by(48)ynl=sin⁡nl.Moreover, the solution of (4) can written as(49)ut,l=∑n=0+∞xntynl,where yn(l)=sin⁡(nl) and xn(t) is the solution of the following ordinary differential equation:(50)a+cn2xnt+bdαxntdtα+dαdtαdαxntdtα=0,with the nonlocal initial conditions(51)xn0=2π∫0πsin⁡nsu0,sds,dαxn0dtα=2π∫0πsin⁡ns∂αu0,s∂tαds.By using the fractional Laplace transform in (50), we get(52)xnt=2πexp⁡-btα2α∫0πcos⁡4cn2+4a-b22αtαx0sin⁡nsds+2bπexp⁡-btα2α∫0πsin⁡4cn2+4a-b2/2αtα4cn2+4a-b2x0sin⁡nsds+4πexp⁡-btα2α∑i=1p∫0πsin4cn2+4a-b2/2αtα4cn2+4a-b2x1p+εiuti,ssin⁡nsds.Finally, replacing xn(t) in (49), we get(53)ut,l=2πexp⁡-btα2α∑n=0+∞∫0πcos⁡4cn2+4a-b22αtαx0sin⁡nlsin⁡nsds+2bπexp⁡-btα2α∑n=0+∞∫0πsin⁡4cn2+4a-b2/2αtα4cn2+4a-b2x0sin⁡nlsin⁡nsds+4πexp⁡-btα2α∑n=0+∞∑i=1p∫0πsin⁡4cn2+4a-b2/2αtα4cn2+4a-b2x1p+εiuti,ssin⁡nlsin⁡nsds.Proposition 5.
Forα=1 and ε=0 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 (54)ut,l=∑n=0+∞xntsin⁡nl,where xn(t) is the solution of the following ordinary differential equation:(55)a+cn2xnt+bdxntdt+d2xntdt2=0,with the initial conditions(56)xn0=2π∫0πsin⁡nsu0,sds,dxn0dt=2π∫0πsin⁡ns∂u0,s∂tds.By using the classical Laplace transform in (55), we get(57)xnt=2πexp-b2t∫0πcos⁡4cn2+4a-b22tx0sin⁡nsds+2bπexp⁡-b2t∫0πsin⁡4cn2+4a-b2/2t4cn2+4a-b2x0sin⁡nsds+4πexp⁡-b2t∫0πsin⁡4cn2+4a-b2/2t4cn2+4a-b2x1sin⁡nsds.Replacing xn(t) in (54), we find the fundamental solution of the classical telegraph equation (1) as follows:(58)ut,l=2πexp⁡-b2t∑n=0+∞∫0πcos⁡4cn2+4a-b22tx0sin⁡nlsin⁡nsds+2bπexp-b2t∑n=0+∞∫0πsin⁡4cn2+4a-b2/2t4cn2+4a-b2x0sin⁡nlsin⁡nsds+4πexp-b2t∑n=0+∞∫0πsin⁡4cn2+4a-b2/2t4cn2+4a-b2x1sin⁡nlsin⁡nsds.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 parametersα and γ in practice situations for modeling naturel phenomena.


## 5. Conclusion

We have studied a time-conformable fractional telegraph equation with nonlocal condition. In the case whenα=γ, 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 ofα and γ.



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