Numerical Analysis of the Klein-Gordon Equations by Using the New Iteration Transform Method

This paper presents an analysis based on a mixture of the Laplace transform and the new iteration method to obtain new approximate results of the fractional-order Klein-Gordon equations in the Caputo-Fabrizio sense. So, a general system to investigate the approximate results of the fractional-order Klein-Gordon equations is obtained. This technique’s effectiveness is demonstrated by comparing the actual results of the fractional-order equations suggested with the results achieved.


Introduction
Fractional partial differential equations (FPDEs) are critical tools for analyzing and simulating numerous narrative models in physics and mathematical models, such as electrical circuits, fluid dynamics, damping, induction, mathematical biology, ad relaxation, ( . Fractional derivatives provide more precise representations of real-world problems than integer-order derivatives; they are regarded as an effective technique for describing such physical problems. The subject of fractional calculus is an important and valuable branch of mathematics that plays a critical and severe role in explaining complex dynamic behavior in a wide range of application areas, helps to understand the essence of the matter as well as simplify the control design without any lack of inherited behavior, and describes even more complex structures [1,2]. The Klein-Gordon equations (KGEs) play an important role in physics, nonlinear optics, quantum field theory and solid state physics, plasma physics, kinematics, mathematical biology, and the recurrence of the initial state. The modeling of many phenomena, including the behavior of elementary particles and dislocation of crystals propagation, is the important applications of KGEs. To study solitons [3], exam-ining nonlinear wave equations [4] and condensed matter physics equations gained the attention of scholars. In the previous few years, mathematicians have made many considerable efforts to find the solutions to these equations. There are many methods introduced to find the solution of these equations such as the radial basis functions [5], B-spline collocation method [5], auxiliary approach [6], and exponential-type potential, and there are some more methods mentioned in [7][8][9][10][11] for the solution of these equations. To solve the KGEs of a nonlinear type got tremendous attention of scholar, and a verity of methods were developed as mentioned in [12][13][14]. Some other methods are the stationary solution [15], the Homotopy perturbation technique [16], the tanh technique [17], the variation iteration technique [18], the traveling wave solutions, and so on.
In the recent paper, we are applying new iterative transform method to KGEs of both linear and nonlinear orders of the following form: Daftardar-Gejji and Jafari developed a new iterative approach for solving nonlinear equations in 2006 [19,20]. Jafari et al. first applied the Laplace transformation in the iterative technique. They proposed a new straightforward technique called the iterative Laplace transform method (ILTM) [21] to look for the numerical solution of the FPDE system. The iterative Laplace transform method was used to solve linear and nonlinear partial differential equations such as the time-fractional Fokker-Planck equation [22], Zakharov-Kuznetsov equation [23], and Fornberg-Whitham equation [24]. The Elzaki transform was used to modify the iterative technique, known as the new iterative transform method.
The new iterative transform method is implemented to investigate the fractional-order of the Klein-Gordon equations. The solution of the fractional-order problems and integral-order models is calculated applying the current techniques. The proposed approach is also helpful for dealing with other fractional-orders of linear and nonlinear PDEs.

Fractional Calculus
This section provides some fundamental concepts of fractional calculus. Definition 1. The Liouville-Caputo operator (C) is given as [25] where u n ðζ, θÞ is the derivative of integer nth order of uðζ, IÞ, n = 1, 2, ⋯ ∈ N and n − 1 < ϱ ≤ n. If 0 < ϱ ≤ 1; then, we defined the Laplace transformation for the Caputo fractional derivative as follows: Definition 2. The Caputo-Fabrizio operator (CF) is define as given [25]: MðϱÞ is a normalization form, and Mð0Þ = Mð1Þ = 1. The exponential law is used as the nonsingular kernel in this fractional operator.
If 0 < ρ ≤ 1, then we define the Caputo-Fabrizio of the Laplace transformation for the fractional derivative is given as

The Iterative Transform Method Basic Procedure
Consider a particular type of a FPDE.
where the functions of linear and nonlinear are M and N, respectively.
With the initial condition implementing the Laplace transformation of Equation (7), we have Applying the Laplace differentiation is given to using the inverse Laplace transformation of Equation (10) into As through the iterative technique, we have Further, the operator M is linear; therefore and the operator N is nonlinear; we have the following Journal of Function Spaces Putting Equations (12)- (14) in Equation (11), we obtain The new iterative transform method is defined as Finally, Equations (7) and (8) provide the m-terms solution in a series form given as with the initial conditions Applying the Laplace transform to Equation (18), we have Applying the inverse Laplace transform of Equation (21), we have Now, by using the suggested analytical method, we get The series form result is The problem has the exact solution at ϱ = 1: In Figure 1, the exact and the approximate solutions of 3 Journal of Function Spaces example 1 at ϱ = 1 are shown, and the second graph shows the 3D graph of different fractional-order ρ, respectively. From the given graphs, it can be shown that both the approximate and exact solutions are in close relation with each other. Also, in Figure 2, the 2D figure of the approximate solutions of problem 1 is analysis at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the timefractional evaluation to integer-order.

Example.
Consider the fractional-order Klein-Gordon equation [18]: with the initial conditions We apply the Laplace transformation to Equation (26), and we get Now, using the inverse Laplace transformation of Equation (29), we have Now, by using the suggested analytical method, we get 4200τϱ − 1680τ − 1680τϱ 2 − 336τ 3 ϱ 2 À + 126ϱτ 3 + 126τ 3 ϱ 3 + ϱ 3 τ 5 + 2520 − 2520ϱ + 1470τ 2 ϱ 2 − 1470τ 2 ϱ + 210τ 2 − 210τ 2 ϱ 3 + 21ϱ 2 τ 4 − 21τ 4 ϱ 3 Þ,  Journal of Function Spaces The series form result is The problem has the exact solution at ϱ = 1: In Figure 3, the exact and the approximate solutions of example 2 at ϱ = 1 are shown. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. Also, in Figure 4, the 2D graph of the approximate results of problem 2 is investigated at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.

Example. Consider the fractional-order nonlinear
Klein-Gordon equation [18]: with the initial conditions Using the Laplace transform to Equation (34), we get

Journal of Function Spaces
Applying the inverse Laplace transform of Equation (37), we have Now, by using the suggested an approximate method, we get The series form result is The problem has the exact solution at ϱ = 2:

Conclusion
In this paper, the iterative transformation method is implemented to achieve approximate analytical results of the fractional-order Klein-Gordon equations, which is widely applied in problems for spatial effects in applied sciences. In physical models, the technique yields series form results that converge very quickly. The obtained results in this article are expected to be important for further analysis of the sophisticated nonlinear models. The calculations of this method are very simple and straightforward. As a result, we conclude that this technique can be used to solve a variety of nonlinear fractional-order partial differential equation systems.

Data Availability
The numerical data used to support the findings of this study are included within the article.  Journal of Function Spaces