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

Saifullah, Sayed; Ali, Amir; Irfan, Muhammad; Shah, Kamal

doi:https://doi.org/10.1155/2021/6858592

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6858592 | https://doi.org/10.1155/2021/6858592

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

Sayed Saifullah,¹Amir Ali,¹Muhammad Irfan,²and Kamal Shah¹

Academic Editor: Libor Pekař

Received01 Jun 2021

Accepted22 Sept 2021

Published13 Oct 2021

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 coefficient 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.

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 in , 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]:with initial conditionswhere is a function of spatial and temporal variables and . The function contains the nonlinearity present in the model, and are real numbers, and 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], Bcklund 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 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.

2. Preliminaries

Here, we provide some basic definitions, which will be used throughout the article.

Definition 1. Let and , then the Caputo–Fabrizio operator of fractional order can be written as [31]such that . It should be noted that is the normalized function. When , then the above definition can be formulated for for any as

Definition 2. Let , with and , then the fractional-order operator in Atangana–Baleanu–Caputo’s sense is defined as [32]where is a fractional operator with Mittag–Leffler kernel in Caputo’s sense. is called normalization function having properties .

Remark 1. For Definitions 1 and 2, is the greatest integer not greater than and is the well-known gamma function which is defined as

Definition 3. Consider a function for in x, t-plane; the double Laplace transform of the function as given by [63] is defined bywhere and are the complex numbers.

Definition 4. Application of the double Laplace on fractional-order operator in Caputo–Fabrizio sense is given bywhere and .

Definition 5. Application of double Laplace transform on fractional-order operator in Atangana–Baleanu–Caputo’s sense is given bywhere .

From the above definitions, we conclude that

The inverse double Laplace transform is represented by a complex double integral formulawhere is an analytic function and is defined in the region by the inequalities and , where 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 formwhere in the above system, is a linear operator, R is an operator containing the linear terms, is a nonlinear operator, and 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.together with the subsidiary conditions

Comparing equations (13) with (12), we observe that , contain nonlinear term, and 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

Now, applying the single Laplace transform on initial conditions given in equation (14), we obtain

Consider the series solution of the formand the nonlinear terms are decomposed aswhere is a well-known polynomial called Adomian polynomial [64] of the functions , described by the formula

Solving equation (12) with the help of equations (16) and (20), we obtain the following series solution:

The other terms can be calculated in a similar way. The final solution can be written as

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,with subsidiary conditions

Solving equation (23) with the techniques used in Section 3, we obtain the following series solution:

The final solution can be written as

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:

Let , where [65], such thatwith and where then

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

Hypothesis 1. : and . : for , there exists a constant , and , with , we obtain .

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 hypothesis 1 for operator in equation (1), such that

On taking the inner product, we obtainwhen , then the above equation can be written asand we can put conditions on the operator in , such that for , we can define

Taking , we can write

Hence, hypothesis 1 is satisfied.

Next, we verify hypothesis 1 for operator .

For every , there exists a constant such that for with , we have

Now, for the proof, considering , we have

By applying Cauchy–Schwartz inequality and since and are bounded, we have

Therefore, we can write

Thus, hypothesis 1 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 equationwith . The subsidiary conditions are given aswhere

The parameters , , , and are real numbers to be chosen accordingly. The exact solution of equation (39) can be attained [66]. Case I: consider the TFKG equation (39) in Caputo–Fabrizio’s sense as The approximate solution of equation (43) by the techniques discussed in Section 3 is obtained as The final solution in the series form up to is given by Case II: similar to the previous section, the TFKG equation (39) in Atangana–Baleanu–Caputo’s sense,

The solution of equation (46) is obtained as

The final solution in series form up to is

(a)

(b)

5.2. Discussion

For numerical illustrations, we have considered the parameters as . 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 . One can see that TFKG equation (39) may admit the excitation of monotonic shocks in an inviscid dynamical system. This degree enhancement in 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 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 , while Figure 2(b) represents equation (48) for . It reveals the evolution of localized shock excitations. We have depicted the solution equation (45) versus x when , and in Figure 3(a), with , respectively. Obviously, the wave amplitude enhances with variations in t. By choosing , we have illustrated equation (48) in Figure 3(b). It infers that coefficient significantly increases the wave amplitudes.

(a)

(b)

(a)

(b)

5.3. Example

Consider a nonlinear TFKG equationwith , , and

The exact solution for of the above equation is [30] Case I: consider the TFKG equation (49) in Caputo–Fabrizio’s sense as The approximate solution of equation (51) together with initial conditions (50) by the techniques discussed in Section 3 is obtained as The final solution in series form up to is given by Case II: consider the TFKG equation (49) in Atangana–Baleanu–Caputo’s sense

We obtain the approximate solution in equation (52) together with initial condition (50) and using the techniques discussed in Section 3:

The final solution in series form up to is given by

(a)

(b)

5.4. Discussion

Figure 4(a) displays the absolute of wave solution equations (54) and (57), having variations in 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 , while Figure 5(b) l represents equation (57) for . It reveals the evolution of localized shock excitations. We have depicted the solution equation (54) versus x with in Figure 6(a), when , respectively. Obviously, the wave amplitude enhances with variations in t. By choosing , we have illustrated equation (57) in Figure 6(b). It infers that coefficient significantly increases the wave amplitudes.

(a)

(b)

(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 considering , 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].

Data Availability

The 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.

Authors’ Contributions

All the authors have equal contributions in the manuscript. Furthermore, the authors have checked and approved the final version of the manuscript.

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Copyright © 2021 Sayed Saifullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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