# Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

Authors: Sayed Saifullah; Amir Ali; Muhammad Irfan; Kamal Shah
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficientα significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.


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## Body

## 1. Introduction

In recent years, the fractional-order calculus is extensively used as a promising tool in numerous areas of physical sciences [1–3] due to its extensive applications to study a diversity of real world phenomena in bioengineering [4–6], electronics [7, 8], visco-elasticity [9], robotic technology [10], signal processing [11], control theory [12], diffusion model, and relaxation processes [13–15]. In fractional-order calculus, the order of derivatives and integrals is arbitrary [16]. Therefore, the fractional-order nonlinear partial differential equations (FNPDEs) have established a fundamental interest to generalize an integer-order nonlinear partial differential equations (NPDEs) to represent complex problems in engineering, thermodynamics, optical physics, and fluid dynamics [17, 18].The most significant advantage of using fractional differential equations is their nonlocal property with memory preserving [2, 19]. It is assumed that the integer-order differential operators are local but fractional are nonlocal because fractional differential operators are global as they converge to ordinary differential operator when fractional-order becomes one [20]. This implies that not only the following condition of the system depends on its present condition but also on all its past states appropriate to memory and hereditary property [21].The nonlinear Klein–Gordon equation considered herein was first proposed to describe relativistic electrons by the well-known physicists O. Klein and W. Gordon in1926, while Klein–Gordon model was originally studied for quantum waves by Schrödinger [15, 22]. The Klein–Gordon equation has many applications in quantum mechanics, quantum field theory, relativistic physics, solid-state physics, plasma physics, nonlinear optics, dispersive wave-phenomena, and condensed type matter physics and also has soliton type solution [23–28].Here, we consider the TFKG equation [29, 30]:(1)∂αϕx,t∂tα−∂2ϕx,t∂x2+cϕ+bgϕx,t=hx,t,1<α≤2,with initial conditions(2)ϕx,0=fx,ϕtx,t=gx,where ϕ is a function of spatial and temporal variables x and t. The function gϕx,t contains the nonlinearity present in the model, b and c are real numbers, and h is an analytic/source function.There are several fractional operators that have been studied in fractional calculus. For example, Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo’s sense [31, 32]. These operators are very useful because of the complexity of fractional nonlinear differential equations (FNDEs), where classical operators cannot solve such equations to obtain explicit solutions. Due to this disadvantage of the classical operators, one needs a legitimate numerical method to obtain the coefficients of the series solutions of FNDEs [33–35]. The Caputo fractional operator is used widely in applied sciences, but this operator has some disadvantages about the singularity. To overcome this problem, Caputo and Fabrizio introduced a nonsingular fractional operator, using an exponential decay kernel. Similarly, another form of nonsingular and nonlocal fractional operator known as Atangana–Baleanu fractional operator was introduced, which produces efficient results due to the nonlocal and nonsingular kernel [36].There are numerous methods that have been used to find numerical as well as analytical solutions to FNDEs, for example, Laplace transforms method [37], Adomian decomposition method (ADM) [38, 39], multistep approach [40], double Laplace transform [41, 42], integral transform [43], iterative reproducing kernel method [44], homotopy perturbation method (HPM) [45, 46], Sumudu transform method (STM) [47], and variational iteration method (VIM) [48, 49].Similarly, the integer-order Klein–Gordon equation has been widely investigated by applying the inverse scattering method, variation iteration method [50, 51], Ba¨cklund transformation method, modified decomposition method and modified Adomain decomposition method [52], auxiliary equation method [53], homotopy perturbation transform method [54], radial basis function [55], homotopy analysis technique [56], sine-cosine and tanh-sech methods [57], pseudospectral method [58], and Hirota bilinear forms [59] together with the Jacobian elliptical function [60]. We will use a modified double Laplace transform to find the approximate solutions of the proposed model in CF and ABC sense. It should be noted that numerous numerical and analytical approaches have also been applied to study TFKG equations with Caputo and Riemann–Liouville (RL) operators [61, 62]. The advantage of the proposed technique is that it converges to an exact solution of a problem after some iterations and does not involve any perturbation or discretization.The remainder of this article is organized as follows: In Section2, some basic definitions are given associated with the fractional calculus. In Section 3, we present the solution to the nonlinear Klein–Gordon equation in CF and ABC by using the proposed method (MDLDM). In Section 4, convergence of the proposed method for the considered model is discussed. In Section 5, we present two numerical examples related to equation (1). In Section 6, we accomplish the article.


## 2. Preliminaries

Here, we provide some basic definitions, which will be used throughout the article.Definition 1.
Letϕ∈H1a,b and α∈0,1, then the Caputo–Fabrizio operator of fractional order can be written as [31](3)DCFαϕt=Mα1−α∫atϕ′texp−αt−s1−sds,such that  M0=M1=1. It should be noted that Mα is the normalized function. When ϕt∉H1a,b, then the above definition can be formulated for ϕϵL−1−∞,b for any αϵ0,1 as(4)DCFαϕt=αMα1−α∫−∞tϕ′t−ϕs−exp−αt−s1−sds.Definition 2.
Letϕ∈H1a,b, with b>a and α∈0,1, then the fractional-order operator in Atangana–Baleanu–Caputo’s sense is defined as [32](5)DABCαϕt=Bα1−α∫atϕ′sEα−αt−sα1−αds,where DABCαϕt is a fractional operator with Mittag–Leffler kernel in Caputo’s sense. Bα is called normalization function having properties B0=B1=1.Remark 1.
For Definitions1 and 2, n=α+1,α is the greatest integer not greater than α and Γ is the well-known gamma function which is defined as(6)Γα=∫0∞e−ssα−1ds.Definition 3.
Consider a functionϕx,t for x,t>0 in x, t-plane; the double Laplace transform of the function ϕx,t as given by [63] is defined by(7)ℒxℒtϕx,t=∫0∞e−px∫0∞e−stϕx,tdtdx,where p and s are the complex numbers.Definition 4.
Application of the double Laplace on fractional-order operator in Caputo–Fabrizio sense is given by(8)ℒxℒtDCFxα+nϕx,t=Mαp+1−pαpn+1ϕ¯p,s−∑k=0npn−kℒt∂kϕ0,t∂xk,ℒxℒtDCFtβ+mϕx,t=Mβs+1−sβsm+1ϕ¯p,s−∑k=0msm−kℒx∂kϕx,0∂tk,where m&n=0,1,2,… and ϕ¯p,s=ℒxℒtϕx,t.Definition 5.
Application of double Laplace transform on fractional-order operator in Atangana–Baleanu–Caputo’s sense is given by(9)ℒxℒtDABCxαϕx,t=Bα1−αpα+α/1−αpαϕ¯p,s−∑k=0n−1pα−1−kℒt∂kϕ0,t∂xk,ℒxℒtDABCtβϕx,t=Bβ1−βsβ+β/1−βsβϕ¯p,s−∑k=0m−1sβ−1−kℒx∂kϕx,0∂tk,where n=α+1,m=β+1.From the above definitions, we conclude that(10)ℒxℒtϕxϕt=u¯pϕ¯s=ℒxϕxℒtϕt.The inverse double Laplace transformℒx−1ℒt−1ϕ¯x,t=ϕx,t is represented by a complex double integral formula(11)ℒx−1ℒt−1ϕ¯x,t=12πi∫c−i∞c+i∞est∫d−i∞d+i∞epxϕ¯p,sdpds,where ϕ¯p,s is an analytic function ∀p and s is defined in the region by the inequalities Rep≥c and Res≥d, where c,d∈ℝ is to be considered accordingly.


## 3. Modified Double Laplace Transform Decomposition Method

In this section, we briefly present the proposed method MDLDM. It is the composition of double Laplace and the Adomian decomposition method used to obtain the solution in the series form of nonlinear partial differential equations (NPDEs) and nonlinear ordinary differential equations (NODEs). It is the most effective scheme to find the approximate solution of dynamic problems. Here, first, we briefly discuss the proposed approach and then apply to equation (1). Let us suppose the general nonlinear problem of the form(12)Lϕ+Rϕ+Nϕ=hx,t,∀t∈ℝ,where ϕ=ϕx,t in the above system, L is a linear operator, R is an operator containing the linear terms, N is a nonlinear operator, and hx,t is an external function.


### 3.1. The Proposed Model with Exponential Decay Kernel

In this subsection, we consider equation (1) in CF sense and use the proposed method to obtain series solution to equation (1), by using the technique defined in Section 3.(13)DCFtα+mϕ−∂2ϕ∂x2+cϕ+bgϕ=hx,t,0<α≤1,m=1,together with the subsidiary conditions(14)ϕx,0=fx,ϕtx,0=gx.Comparing equations (13) with (12), we observe that L=∂2/∂x2, N=bgϕ contain nonlinear term, and DCFtα+m is the fractional-order operator in Caputo–Fabrizio’s sense. Applying the double Laplace transform and using the definitions given in Section 2, we obtain(15)ℒxℒtDCFtα+mϕ−ℒxℒt∂2∂x2ϕ+cℒxℒtϕ+bℒxℒtgϕ=ℒxℒthx,t.Using double Laplace on the fractional-order operator in Caputo–Fabrizio’s sense, we obtain(16)ℒxℒtϕx,t=1sℒxϕx,0+1s2ℒxϕtx,0+s+1−αss2ℒxℒt∂2∂x2ϕx,t+cs+1−αss2ℒxℒtϕx,t+bs+1−αss2ℒxℒtgϕx,t+s+1−αss2ℒxℒthx,t.Now, applying the single Laplace transform on initial conditions given in equation (14), we obtain(17)ℒxϕx,0=f¯p,ℒxϕtx,0=tg¯p.Consider the series solution of the form(18)ϕx,t=∑n=0∞ϕnx,t,and the nonlinear terms are decomposed as(19)gϕx,t=∑i=0∞An,where An is a well-known polynomial called Adomian polynomial [64] of the functions ϕ0,ϕ1,ϕ2,…, described by the formula(20)An=1n!dndλn∑k=0nλkϕkx,tλ=0.Solving equation (12) with the help of equations (16) and (20), we obtain the following series solution:(21)ϕ0=ℒx−1ℒt−11sϕ¯p,0+tℒx−1ℒt−11s2ϕ¯p,0=ϕx,0,ϕ1=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ0xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ0−bℒx−1ℒt−1s+1−sαs2ℒxℒtA0+ℒx−1ℒt−1s+1−sαs2ℒxℒthx,t,ϕ2=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ1xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ1−bℒx−1ℒt−1s+1−sαs2ℒxℒtA1,ϕ3=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ2xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ2−bℒx−1ℒt−1s+1−sαs2ℒxℒtA2.The other terms can be calculated in a similar way. The final solution can be written as(22)ϕx,t=∑n=0∞ϕnx,t.


### 3.2. The Proposed Model with Mittag–Leffler Kernel

Here, we consider equation (1) in ABC sense and applying the proposed method with definitions discussed in Section 2,(23)DABCtαϕx,t−∂2ϕ∂x2+cϕ+bgϕ=hx,t,∀t∈ℝ,1<α≤2,with subsidiary conditions(24)ϕx,0=fx,ϕtx,0=gx.Solving equation (23) with the techniques used in Section 3, we obtain the following series solution:(25)ϕ0=ℒx−1ℒt−11sϕ¯p,0+tℒx−1ℒt−11s2ϕ¯p,0=ϕx,0,ϕ1=ℒx−1ℒt−11−α+αsαℒxℒtϕ0xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ0−bℒx−1ℒt−11−α+αsαℒxℒtA0+ℒx−1ℒt−11−α+αsαℒxℒthx,t,ϕ2=ℒx−1ℒt−11−α+αsαℒxℒtϕ1xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ1−bℒx−1ℒt−11−α+αsαℒxℒtA1,ϕ3=ℒx−1ℒt−11−α+αsαℒxℒtϕ2xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ2−bℒx−1ℒt−11−α+αsαℒxℒtA2.The final solution can be written as(26)ϕx,t=∑n=0∞ϕnx,t.Equations (22) and (26) are the general series solutions of equation (1) in both CF and ABC sense.


## 3.1. The Proposed Model with Exponential Decay Kernel

In this subsection, we consider equation (1) in CF sense and use the proposed method to obtain series solution to equation (1), by using the technique defined in Section 3.(13)DCFtα+mϕ−∂2ϕ∂x2+cϕ+bgϕ=hx,t,0<α≤1,m=1,together with the subsidiary conditions(14)ϕx,0=fx,ϕtx,0=gx.Comparing equations (13) with (12), we observe that L=∂2/∂x2, N=bgϕ contain nonlinear term, and DCFtα+m is the fractional-order operator in Caputo–Fabrizio’s sense. Applying the double Laplace transform and using the definitions given in Section 2, we obtain(15)ℒxℒtDCFtα+mϕ−ℒxℒt∂2∂x2ϕ+cℒxℒtϕ+bℒxℒtgϕ=ℒxℒthx,t.Using double Laplace on the fractional-order operator in Caputo–Fabrizio’s sense, we obtain(16)ℒxℒtϕx,t=1sℒxϕx,0+1s2ℒxϕtx,0+s+1−αss2ℒxℒt∂2∂x2ϕx,t+cs+1−αss2ℒxℒtϕx,t+bs+1−αss2ℒxℒtgϕx,t+s+1−αss2ℒxℒthx,t.Now, applying the single Laplace transform on initial conditions given in equation (14), we obtain(17)ℒxϕx,0=f¯p,ℒxϕtx,0=tg¯p.Consider the series solution of the form(18)ϕx,t=∑n=0∞ϕnx,t,and the nonlinear terms are decomposed as(19)gϕx,t=∑i=0∞An,where An is a well-known polynomial called Adomian polynomial [64] of the functions ϕ0,ϕ1,ϕ2,…, described by the formula(20)An=1n!dndλn∑k=0nλkϕkx,tλ=0.Solving equation (12) with the help of equations (16) and (20), we obtain the following series solution:(21)ϕ0=ℒx−1ℒt−11sϕ¯p,0+tℒx−1ℒt−11s2ϕ¯p,0=ϕx,0,ϕ1=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ0xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ0−bℒx−1ℒt−1s+1−sαs2ℒxℒtA0+ℒx−1ℒt−1s+1−sαs2ℒxℒthx,t,ϕ2=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ1xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ1−bℒx−1ℒt−1s+1−sαs2ℒxℒtA1,ϕ3=ℒx−1ℒt−1s+1−sαs2ℒxℒtϕ2xx−cℒx−1ℒt−1s+1−sαs2ℒxℒtϕ2−bℒx−1ℒt−1s+1−sαs2ℒxℒtA2.The other terms can be calculated in a similar way. The final solution can be written as(22)ϕx,t=∑n=0∞ϕnx,t.


## 3.2. The Proposed Model with Mittag–Leffler Kernel

Here, we consider equation (1) in ABC sense and applying the proposed method with definitions discussed in Section 2,(23)DABCtαϕx,t−∂2ϕ∂x2+cϕ+bgϕ=hx,t,∀t∈ℝ,1<α≤2,with subsidiary conditions(24)ϕx,0=fx,ϕtx,0=gx.Solving equation (23) with the techniques used in Section 3, we obtain the following series solution:(25)ϕ0=ℒx−1ℒt−11sϕ¯p,0+tℒx−1ℒt−11s2ϕ¯p,0=ϕx,0,ϕ1=ℒx−1ℒt−11−α+αsαℒxℒtϕ0xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ0−bℒx−1ℒt−11−α+αsαℒxℒtA0+ℒx−1ℒt−11−α+αsαℒxℒthx,t,ϕ2=ℒx−1ℒt−11−α+αsαℒxℒtϕ1xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ1−bℒx−1ℒt−11−α+αsαℒxℒtA1,ϕ3=ℒx−1ℒt−11−α+αsαℒxℒtϕ2xx−cℒx−1ℒt−11−α+αsαℒxℒtϕ2−bℒx−1ℒt−11−α+αsαℒxℒtA2.The final solution can be written as(26)ϕx,t=∑n=0∞ϕnx,t.Equations (22) and (26) are the general series solutions of equation (1) in both CF and ABC sense.


## 4. Convergence of MDLDM for the Proposed Model

Here, we discuss the convergence of the proposed method for the considered model equation (1). For this, we consider equation (1) in the operator form:(27)Tϕ=∂2ϕ∂t2=∂2ϕ∂x2−cϕ−bgϕ+hx,t.LetH∈L2T∀ϕ∈H, where H=ℒϕ2m,n×0,T [65], such that(28)ϕ≔m,n×0,T⟶ℝ2,with m≪0 and B=m,n×0,T where ϕH2=∫Bϕ2dxdt, then(29)ℒx−1ℒt−1ℒxℒtϕx,t<∞.For the operatorT to be hemicontinuous [65], we consider the hypothesis as follows.Hypothesis 1.
H1: Tϕ−Tψ,ϕ−ψ≥κ1ϕ−ψ2,ϕ,ψ∈H and κ1>0.H2: for q>0, there exists a constant Cq>0, and ∀ϕ,ψ∈H, with ϕ+ψ2≤q, we obtain Tϕ−Tψ≤cmϕ−ψ,∀ϕ,ψ∈H.Theorem 1 (sufficient conditions of convergence).
The proposed method is applied to equation (1) without initial and boundary conditions, converging to a particular solution.Here, we use hypothesis1 for operator Tϕ in equation (1), such that(30)Tϕ−Tψ=∂2ϕ∂x2−∂2ψ∂x2−cϕ+cψ−bgϕ+−bgψ+h−h,=∂2∂x2ϕ−ψ−cϕ−ψ−bgϕ−ψ.On taking the inner product, we obtain(31)Tϕ−Tψ,ϕ−ψ=∂2∂x2ϕ−ψ,ϕ−ψ−cϕ−ψ,ϕ−ψ−bgϕ−ψ,ϕ−ψ,when gϕ=ϕ2, then the above equation can be written as(32)Tϕ−Tψ,ϕ−ψ=∂2∂x2ϕ−ψ,ϕ−ψ−cϕ−ψ,ϕ−ψ−bϕ−ψ2,ϕ−ψ2,and we can put conditions on the operator ∂2/∂x2 in H, such that for η>0, we can define(33)∂2∂2xϕ−ψ≤ηϕ−ψ2,Tϕ−Tψ,ϕ−ψ≤ηϕ−ψ2−cϕ−ψ2−bϕ−ψ22=ηϕ−ψ2−cϕ−ψ2−bϕ+ψ2ϕ−ψ2Tϕ−Tψ,ϕ−ψ≤η−c−bσϕ−ψ2.Takingκ1=η−c−bσ>0, we can write(34)Tϕ−Tψ,ϕ−ψ≤κ1u−w2.Hence, hypothesis1 is satisfied.Next, we verify hypothesis1 for operator Tϕ.For everyq>0, there exists a constant Cq>0 such that for ϕ,ψ∈H with ϕ+ψ2≤q, we have(35)Tϕ−Tψ≤Cqϕ−ψ,∀ϕ,ψ∈H.Now, for the proof, consideringϖ1∈H, we have(36)Tϕ−Tψ,ϖ1=∂2∂x2ϕ−ψ,ϖ1−cϕ−ψ,ϖ1−bϕ−ψ2,ϖ12.By applying Cauchy–Schwartz inequality and sinceϕ and ψ are bounded, we have(37)∂2∂2xϕ−ψ,ϖ1≤η1ϕ−ψ≤η1ϕ−ψϖ1,c−ϕ−ψ,ϖ1≤cϕ−ψ≤cϕ−ψϖ1,b−ϕ−ψ2,ϖ12≤bϕ+ψϕ−ψ≤bqϕ−ψϖ1.Therefore, we can write(38)Tϕ−Tψ,ϖ1≤Cqϕ−ψϖ1,whereCq=η1+c+bq>0.Thus, hypothesis1 is satisfied. This completes the proof.


## 5. Numerical Examples

Here, we consider numerical examples of the nonlinear TFKG equation and discuss two cases.


### 5.1. Example

Here, we consider the time-fractional nonlinear KG equation(39)Dtαϕ−c∂2ϕ∂x2+bϕ3=0,1<α≤2,gϕ=ϕ3,hx,t=0,with b=c=1. The subsidiary conditions are given as(40)ϕx,0=BtanΩx,ϕtx,0=BɛΩsec2Ωx,0≤x≤2,where(41)B=ζγ,Ω=−ζ2σ+ϵ2.The parametersζ, γ, σ, and ϵ are real numbers to be chosen accordingly. The exact solution of equation (39) can be attained [66].(42)ϕx,t=BtanΩx+ϵt0.Case I: consider the TFKG equation (39) in Caputo–Fabrizio’s sense as(43)DCFtα+mϕx,t−∂2ϕ∂x2+ϕ3x,t=0,0<α≤1,m=1.The approximate solution of equation (43) by the techniques discussed in Section 3 is obtained as(44)ϕ0=BtanΩx+tBɛΩsec2Ωx,ϕ1=1−αt+αt22!2BΩ2sec2ΩxtanΩx−B2tan3Ωx+BϵΩ1−αt22!+αt33!4Ω2sec2Ωxtan2Ωx+2Ω2sec4Ωx−3B2tan2Ωxsec2Ωx−6B3Ω2ϵ21−αt33!+αt44!sec4ΩxtanΩx−6B3Ω3ϵ31−αt44!+αt55!sech6Ωx.The final solution in the series form up toO2 is given by(45)ϕx,t=ϕ0+ϕ1.Case II: similar to the previous section, the TFKG equation (39) in Atangana–Baleanu–Caputo’s sense,(46)DABCtαϕx,t−∂2ϕ∂x2+ϕ3x,t=0,1<α≤2.The solution of equation (46) is obtained as(47)ϕ0=BtanΩx+tBɛΩsec2Ωx,ϕ1=1−α+αtαΓα+12BΩ2sec2ΩxtanΩx−B2tan3Ωx+BϵΩ1−αt+αtα+1Γα+24Ω2sec2Ωxtan2Ωx+2Ω2sec4Ωx−3B2tan2Ωxsec2Ωx−6B3Ω2ϵ21−αt22!+αtα+2Γα+3sec4ΩxtanΩx−6B3Ω3c31−αt33!+αtα+3Γα+4sech6Ωx.The final solution in series form up toO2 is(48)ϕx,t=ϕ0+ϕ1.Figure 1
(a) The comparison between equations (42), (45), and (48) for different values of α; (b) solution profiles of ϕx,t against t (time) for a variety of α, as mentioned in the plots.
(a)(b)


### 5.2. Discussion

For numerical illustrations, we have considered the parameters asζ=−1,γ=1,σ=−8.5,andϵ=0.05. The numerical solutions, equations (45) and (48), and exact solution equation (42) associated with the Caputo–Fabrizio’s (CF) sense and Atangana–Baleanu–Caputo’s (ABC) sense are depicted in Figure 1(a), with variation in the time-fractional coefficient α. One can see that TFKG equation (39) may admit the excitation of monotonic shocks in an inviscid dynamical system. This degree enhancement in α suppresses the wave amplitude as it affects the nonlinearity/dispersion effects. To see the effect of a temporal variable (t) on the wave solutions, equations (45) and (48) are displayed in Figure 1(b); it reveals that ϕx,t rises with time. The three-dimensional profiles for equations (45) and (48) are shown versus x and t in Figure 2. Figure 2(a) represents the physical behavior of equation (45) for α=1, while Figure 2(b) represents equation (48) for α=2. It reveals the evolution of localized shock excitations. We have depicted the solution equation (45) versus x when t=0solid black curve,0.4circles,0.6solid green curve, and 0.8dotted curve in Figure 3(a), with α=1and0.8, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α=2and1.9, we have illustrated equation (48) in Figure 3(b). It infers that coefficient α significantly increases the wave amplitudes.Figure 2
The surface plots for CF equation (45) solution and ABC solution equation (48) for the parameters used in Figure 1(a).
(a)(b)Figure 3
The solution profiles ofϕx,t versus different values of t.
(a)(b)


### 5.3. Example

Consider a nonlinear TFKG equation(49)Dtαϕ−∂2ϕ∂x2+34ϕ−32ϕ3=0,1<α≤2,with c=3/4,b=3/2, gϕ=ϕ3,hx,t=0, and(50)ϕx,0=−sechx,ϕtx,0=12sechxtanhx.The exact solution forα=2 of the above equation is [30](51)ϕx,t=−sechx+t2.Case I: consider the TFKG equation (49) in Caputo–Fabrizio’s sense as(52)DCFtα+mϕ−∂2∂x2ϕ+34ϕ−32ϕ3=0,0<α≤1,m=1.The approximate solution of equation (51) together with initial conditions (50) by the techniques discussed in Section 3 is obtained as(53)ϕ0=t2sechxtanhx−sechx,ϕ1=1−α+αtsech3x−tanh2xsechx+34sechx−32sech3x+1−αt+αt22!tanh3xsechx−5sech3xtanhx+38sechxtanhx+94sech3xtanhx−1−αt22!+αt33!98sech3xtanh2x+1−αt33!+αt44!316sech3xtanh3x.The final solution in series form up toO2 is given by(54)ϕx,t=ϕ0+ϕ1.Case II: consider the TFKG equation (49) in Atangana–Baleanu–Caputo’s sense(55)DABCtαϕ−∂2∂x2ϕ+34ϕ−32ϕ3=0,1<α≤2.We obtain the approximate solution in equation (52) together with initial condition (50) and using the techniques discussed in Section 3:(56)ϕ0=t2sechxtanhx−sechx,ϕ1=1−α+αtsech3x−tanh2xsechx+34sechx−32sech3x+1−αt+αtα+1Γα+2tanh3xsechx−5sech3xtanhx+38sechxtanhx+94sech3xtanhx−1−αt22!+αtα+2Γα+398sech3xtanh2x+1−αt33!+αtα+3Γα+4316sech3xtanh3x,The final solution in series form up toO2 is given by(57)ϕx,t=ϕ0+ϕ1.Figure 4
(a) The comparison between equations (50), (52), and (54) for different values of alpha; (b) solution profiles of ϕx,t vs time t for different values of α.
(a)(b)


### 5.4. Discussion

Figure4(a) displays the absolute of wave solution equations (54) and (57), having variations in α with t = 0.3 and t = 1, respectively, with exact solution given in equation (51). Notice that the numerical solutions, CF equation (54), and ABC equation (57) exactly match to the exact solution equation (51). We observe that TFKG admits pulse-shaped solitons. We also know that the solution equations (54) and (57) in Figure 4(b) reveal that the amplitude of the solitary potentials goes up as t rises. The three-dimensional profiles for equations (54) and (57) are shown versus x and t in Figure 5. Figure 5(a) represents the physical behavior of equation (54) for α=1, while Figure 5(b) l represents equation (57) for α=2. It reveals the evolution of localized shock excitations. We have depicted the solution equation (54) versus x with t=0.6dashed line,0.4solid curve,and0.2dotted curve in Figure 6(a), when α=1and0.7, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α=2and1.7, we have illustrated equation (57) in Figure 6(b). It infers that coefficient α significantly increases the wave amplitudes.Figure 5
The surface plots of solution of TFKG with CF and ABC operators for the parameters used in Figure4(a).
(a)(b)Figure 6
Solution profiles ofϕx,t for different values of α with different values of time t.
(a)(b)


## 5.1. Example

Here, we consider the time-fractional nonlinear KG equation(39)Dtαϕ−c∂2ϕ∂x2+bϕ3=0,1<α≤2,gϕ=ϕ3,hx,t=0,with b=c=1. The subsidiary conditions are given as(40)ϕx,0=BtanΩx,ϕtx,0=BɛΩsec2Ωx,0≤x≤2,where(41)B=ζγ,Ω=−ζ2σ+ϵ2.The parametersζ, γ, σ, and ϵ are real numbers to be chosen accordingly. The exact solution of equation (39) can be attained [66].(42)ϕx,t=BtanΩx+ϵt0.Case I: consider the TFKG equation (39) in Caputo–Fabrizio’s sense as(43)DCFtα+mϕx,t−∂2ϕ∂x2+ϕ3x,t=0,0<α≤1,m=1.The approximate solution of equation (43) by the techniques discussed in Section 3 is obtained as(44)ϕ0=BtanΩx+tBɛΩsec2Ωx,ϕ1=1−αt+αt22!2BΩ2sec2ΩxtanΩx−B2tan3Ωx+BϵΩ1−αt22!+αt33!4Ω2sec2Ωxtan2Ωx+2Ω2sec4Ωx−3B2tan2Ωxsec2Ωx−6B3Ω2ϵ21−αt33!+αt44!sec4ΩxtanΩx−6B3Ω3ϵ31−αt44!+αt55!sech6Ωx.The final solution in the series form up toO2 is given by(45)ϕx,t=ϕ0+ϕ1.Case II: similar to the previous section, the TFKG equation (39) in Atangana–Baleanu–Caputo’s sense,(46)DABCtαϕx,t−∂2ϕ∂x2+ϕ3x,t=0,1<α≤2.The solution of equation (46) is obtained as(47)ϕ0=BtanΩx+tBɛΩsec2Ωx,ϕ1=1−α+αtαΓα+12BΩ2sec2ΩxtanΩx−B2tan3Ωx+BϵΩ1−αt+αtα+1Γα+24Ω2sec2Ωxtan2Ωx+2Ω2sec4Ωx−3B2tan2Ωxsec2Ωx−6B3Ω2ϵ21−αt22!+αtα+2Γα+3sec4ΩxtanΩx−6B3Ω3c31−αt33!+αtα+3Γα+4sech6Ωx.The final solution in series form up toO2 is(48)ϕx,t=ϕ0+ϕ1.Figure 1
(a) The comparison between equations (42), (45), and (48) for different values of α; (b) solution profiles of ϕx,t against t (time) for a variety of α, as mentioned in the plots.
(a)(b)


## 5.2. Discussion

For numerical illustrations, we have considered the parameters asζ=−1,γ=1,σ=−8.5,andϵ=0.05. The numerical solutions, equations (45) and (48), and exact solution equation (42) associated with the Caputo–Fabrizio’s (CF) sense and Atangana–Baleanu–Caputo’s (ABC) sense are depicted in Figure 1(a), with variation in the time-fractional coefficient α. One can see that TFKG equation (39) may admit the excitation of monotonic shocks in an inviscid dynamical system. This degree enhancement in α suppresses the wave amplitude as it affects the nonlinearity/dispersion effects. To see the effect of a temporal variable (t) on the wave solutions, equations (45) and (48) are displayed in Figure 1(b); it reveals that ϕx,t rises with time. The three-dimensional profiles for equations (45) and (48) are shown versus x and t in Figure 2. Figure 2(a) represents the physical behavior of equation (45) for α=1, while Figure 2(b) represents equation (48) for α=2. It reveals the evolution of localized shock excitations. We have depicted the solution equation (45) versus x when t=0solid black curve,0.4circles,0.6solid green curve, and 0.8dotted curve in Figure 3(a), with α=1and0.8, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α=2and1.9, we have illustrated equation (48) in Figure 3(b). It infers that coefficient α significantly increases the wave amplitudes.Figure 2
The surface plots for CF equation (45) solution and ABC solution equation (48) for the parameters used in Figure 1(a).
(a)(b)Figure 3
The solution profiles ofϕx,t versus different values of t.
(a)(b)


## 5.3. Example

Consider a nonlinear TFKG equation(49)Dtαϕ−∂2ϕ∂x2+34ϕ−32ϕ3=0,1<α≤2,with c=3/4,b=3/2, gϕ=ϕ3,hx,t=0, and(50)ϕx,0=−sechx,ϕtx,0=12sechxtanhx.The exact solution forα=2 of the above equation is [30](51)ϕx,t=−sechx+t2.Case I: consider the TFKG equation (49) in Caputo–Fabrizio’s sense as(52)DCFtα+mϕ−∂2∂x2ϕ+34ϕ−32ϕ3=0,0<α≤1,m=1.The approximate solution of equation (51) together with initial conditions (50) by the techniques discussed in Section 3 is obtained as(53)ϕ0=t2sechxtanhx−sechx,ϕ1=1−α+αtsech3x−tanh2xsechx+34sechx−32sech3x+1−αt+αt22!tanh3xsechx−5sech3xtanhx+38sechxtanhx+94sech3xtanhx−1−αt22!+αt33!98sech3xtanh2x+1−αt33!+αt44!316sech3xtanh3x.The final solution in series form up toO2 is given by(54)ϕx,t=ϕ0+ϕ1.Case II: consider the TFKG equation (49) in Atangana–Baleanu–Caputo’s sense(55)DABCtαϕ−∂2∂x2ϕ+34ϕ−32ϕ3=0,1<α≤2.We obtain the approximate solution in equation (52) together with initial condition (50) and using the techniques discussed in Section 3:(56)ϕ0=t2sechxtanhx−sechx,ϕ1=1−α+αtsech3x−tanh2xsechx+34sechx−32sech3x+1−αt+αtα+1Γα+2tanh3xsechx−5sech3xtanhx+38sechxtanhx+94sech3xtanhx−1−αt22!+αtα+2Γα+398sech3xtanh2x+1−αt33!+αtα+3Γα+4316sech3xtanh3x,The final solution in series form up toO2 is given by(57)ϕx,t=ϕ0+ϕ1.Figure 4
(a) The comparison between equations (50), (52), and (54) for different values of alpha; (b) solution profiles of ϕx,t vs time t for different values of α.
(a)(b)


## 5.4. Discussion

Figure4(a) displays the absolute of wave solution equations (54) and (57), having variations in α with t = 0.3 and t = 1, respectively, with exact solution given in equation (51). Notice that the numerical solutions, CF equation (54), and ABC equation (57) exactly match to the exact solution equation (51). We observe that TFKG admits pulse-shaped solitons. We also know that the solution equations (54) and (57) in Figure 4(b) reveal that the amplitude of the solitary potentials goes up as t rises. The three-dimensional profiles for equations (54) and (57) are shown versus x and t in Figure 5. Figure 5(a) represents the physical behavior of equation (54) for α=1, while Figure 5(b) l represents equation (57) for α=2. It reveals the evolution of localized shock excitations. We have depicted the solution equation (54) versus x with t=0.6dashed line,0.4solid curve,and0.2dotted curve in Figure 6(a), when α=1and0.7, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α=2and1.7, we have illustrated equation (57) in Figure 6(b). It infers that coefficient α significantly increases the wave amplitudes.Figure 5
The surface plots of solution of TFKG with CF and ABC operators for the parameters used in Figure4(a).
(a)(b)Figure 6
Solution profiles ofϕx,t for different values of α with different values of time t.
(a)(b)


## 6. Conclusion

We have studied the time-fractional Klein–Gordon equation using the MDLD method. The approximate solutions of nonlinear Klein–Gordon equation are obtained in the form of the series. It is very important to notice that even after some iterations, more accurate results are obtained. It is perceived that our proposed method provides accurate numerical results without perturbation and discretization for nonlinear differential equations with fractional operators. The numerical results obtained for particular examples are compared with the exact solutions at the classical order. The effect of the altered fractional orders of the considered numerical illustrations is shown explicitly, where good agreements are obtained. The Klein–Gordon equation is evidently a nonlinear PDE and thus a perfect model for understanding the nonlinearity/dispersion effects and the evolution of localized shock excitations. It is inferred in this manuscript that fractional order significantly increases the wave amplitudes. Similar to the Klein–Gordon equation, the Sine-Gordon equation bears kink-anti-kink phenomena. The Sine-Gordon potential has unbounded minimum points; however, two of them are assumed in the kink solutions of the system. By consideringu≃sinu, the considered system becomes the Sine-Gordon equation. In future work, it will be interesting to investigate the Sine-Gordon model with nonlinear AC/DC drives with different fractional operators to study the solitonic behavior, localized modes in single and in stacked long Josephson junctions with a variety of potentials, parity time symmetry, the nonlinearity/dispersion effects, and evolution of the localized monotonic shocks [67–73].



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