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

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 coeﬃcient α 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.


Introduction
In recent years, the fractional-order calculus is extensively used as a promising tool in numerous areas of physical sciences [1][2][3] due to its extensive applications to study a diversity of real world phenomena in bioengineering [4][5][6], electronics [7,8], visco-elasticity [9], robotic technology [10], signal processing [11], control theory [12], diffusion model, and relaxation processes [13][14][15]. In fractional-order calculus, the order of derivatives and integrals is arbitrary [16]. erefore, 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]. e 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 fractionalorder becomes one [20].
is 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]. e nonlinear Klein-Gordon equation considered herein was first proposed to describe relativistic electrons by the well-known physicists O. Klein and W. Gordon in 1926, while Klein-Gordon model was originally studied for quantum waves by Schrödinger [15,22]. e 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][24][25][26][27][28].
Here, we consider the TFKG equation [29,30]: with initial conditions where ϕ is a function of spatial and temporal variables x and t. e function g(ϕ(x, t)) contains the nonlinearity present in the model, b and c are real numbers, and h is an analytic/ source function. ere are several fractional operators that have been studied in fractional calculus. For example, Caputo, Caputo-Fabrizio, and Atangana-Baleanu-Caputo's sense [31,32]. ese 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][34][35]. e 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].
Similarly, the integer-order Klein-Gordon equation has been widely investigated by applying the inverse scattering method, variation iteration method [50,51], B € acklund 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]. e 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. e remainder of this article is organized as follows: In Section 2, 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.

Remark 1.
For Definitions 1 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 Definition 3. Consider a function ϕ(x, t) for x, t > 0 in x, tplane; the double Laplace transform of the function ϕ(x, t) as given by [63] is defined by where p and s are the complex numbers.
Definition 5. Application of double Laplace transform on fractional-order operator in Atangana-Baleanu-Caputo's sense is given by From the above definitions, we conclude that where ϕ(p, s) is an analytic function ∀ p and s is defined in the region by the inequalities Re(p) ≥ c and Re(s) ≥ d, where c, d ∈ R is to be considered accordingly.

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 where ϕ � ϕ(x, t) in the above system, L is a linear operator, R is an operator containing the linear terms, N is a nonlinear operator, and h(x, t) is an external function.

e 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.
together with the subsidiary conditions Comparing equations (13) with (12), we observe that L � z 2 /zx 2 , N � bg(ϕ) contain nonlinear term, and CF D α+m t 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 Using double Laplace on the fractional-order operator in Caputo-Fabrizio's sense, we obtain Mathematical Problems in Engineering Now, applying the single Laplace transform on initial conditions given in equation (14), we obtain Consider the series solution of the form and the nonlinear terms are decomposed as where A n is a well-known polynomial called Adomian polynomial [64] of the functions ϕ 0 , ϕ 1 , ϕ 2 , . . ., described by the formula Solving equation (12) with the help of equations (16) and (20), we obtain the following series solution: e other terms can be calculated in a similar way. e final solution can be written as

e 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, with subsidiary conditions Solving equation (23) with the techniques used in Section 3, we obtain the following series solution: 4 Mathematical Problems in Engineering e final solution can be written as Equations (22) and (26) are the general series solutions of equation (1) in both CF and ABC sense.

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: with For the operator T to be hemicontinuous [65], we consider the hypothesis as follows.

Theorem 1 (sufficient conditions of convergence). e proposed method is applied to equation (1) without initial and boundary conditions, converging to a particular solution.
Here, we use hypothesis 1 for operator T(ϕ) in equation (1), such that On taking the inner product, we obtain when g(ϕ) � ϕ 2 , then the above equation can be written as

Mathematical Problems in Engineering
and we can put conditions on the operator z 2 /zx 2 in H, such that for η > 0, we can define Taking κ 1 � (η − c − bσ) > 0, we can write Hence, hypothesis 1 is satisfied. Next, we verify hypothesis 1 for operator T(ϕ).
For every q > 0, there exists a constant C(q) > 0 such that for ϕ, ψ ∈ H with ‖ϕ + ψ‖ 2 ≤ q, we have Now, for the proof, considering ϖ 1 ∈ H, we have By applying Cauchy-Schwartz inequality and since ϕ and ψ are bounded, we have erefore, we can write us, hypothesis 1 is satisfied. is completes the proof.

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

Example.
Here, we consider the time-fractional nonlinear KG equation with b � c � 1. e subsidiary conditions are given as where e parameters ζ, c, σ, and ϵ are real numbers to be chosen accordingly. e exact solution of equation (39) can be attained [66].

Mathematical Problems in Engineering
Case I: consider the TFKG equation (39) in Caputo-Fabrizio's sense as e approximate solution of equation (43) by the techniques discussed in Section 3 is obtained as e final solution in the series form up to O(2) is given by Case II: similar to the previous section, the TFKG equation (39) in Atangana-Baleanu-Caputo's sense, e solution of equation (46) is obtained as ϕ 0 � B tan(Ωx) + tBεΩsec 2 (Ωx), e final solution in series form up to O(2) is

Discussion.
For numerical illustrations, we have considered the parameters as ζ � − 1, c � 1, σ � − 8.5, and ϵ � 0.05. e 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. is degree enhancement in α suppresses the wave amplitude as it affects the nonlinearity/dispersion effects. To Mathematical Problems in Engineering 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 ϕ(x, t) rises with time. e 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 � 0(solid black curve), 0.4 (circles), 0.6(solid green curve), and 0.8(dotted curve) in Figure 3(a), with α � 1 and 0.8, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α � 2 and 1.9, we have illustrated equation (48) in Figure 3(b). It infers that coefficient (α) significantly increases the wave amplitudes.

Example. Consider a nonlinear TFKG equation
e exact solution for α � 2 of the above equation is [30] ϕ( Mathematical Problems in Engineering e final solution in series form up to O(2) is given by 5.4. Discussion. Figure 4(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. e 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 � 0.6(dashed line), 0.4(solid curve), and 0.2(dotted curve) in Figure 6(a), when α � 1 and 0.7, respectively. Obviously, the wave amplitude enhances with variations in t. By choosing α � 2 and 1.7, we have illustrated equation (57) in Figure 6(b). It infers that coefficient (α) significantly increases the wave amplitudes.

Conclusion
We have studied the time-fractional Klein-Gordon equation using the MDLD method. e 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. e numerical results obtained for particular examples are compared with the exact solutions at the classical order. e effect of the altered fractional orders of the considered numerical illustrations is shown explicitly, where good agreements are obtained. e 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. e Sine-Gordon potential has unbounded minimum points; however, two of them are assumed in the kink solutions of the system. By considering u≃ sin u, 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][68][69][70][71][72][73].
Data Availability e data that support the findings of this study are available on request to the corresponding author.

Conflicts of Interest
All the authors have no conflicts of interest regarding this article.