Abstract
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.
1. 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, (Klimek, 2005; Baleanu et al., 2009; Kilbas et al., 2010; Jumarie, 2009; Mainardi, 2010; Ortigueira, 2010). 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], examining 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 [7–11] 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 [12–14]. 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: with the boundary conditions
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.
2. Fractional Calculus
This section provides some fundamental concepts of fractional calculus.
Definition 1. The Liouville-Caputo operator (C) is given as [25] where is the derivative of integer th order of , and . If ; 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]:
is a normalization form, and . The exponential law is used as the nonsingular kernel in this fractional operator.
If , then we define the Caputo-Fabrizio of the Laplace transformation for the fractional derivative is given as
3. The Iterative Transform Method Basic Procedure
Consider a particular type of a FPDE. where the functions of linear and nonlinear are and , 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 is linear; therefore and the operator is nonlinear; we have the following
Putting Equations (12)–(14) in Equation (11), we obtain
The new iterative transform method is defined as
Finally, Equations (7) and (8) provide the -terms solution in a series form given as
4. Applications of the Proposed Method
4.1. Example
Consider the fractional-order Klein-Gordon equation [18] 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 :
In Figure 1, the exact and the approximate solutions of example 1 at are shown, and the second graph shows the 3D graph of different fractional-order , 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 for and . It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.
(a)
(b)
4.2. 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
The series form result is
The problem has the exact solution at :
In Figure 3, the exact and the approximate solutions of example 2 at 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 for and . It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.
4.3. Example
Consider the fractional-order nonlinear Klein-Gordon equation [18]: with the initial conditions
Using the Laplace transform to Equation (34), we get
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 :
Figure 5 compares the exact solution and approximate solution of example 3 for the nonlinear fractional-order Klein-Gordon equation at . The figure shows the close relationship between the exact and an approximate solution.
4.4. Example
Consider the fractional-order nonlinear Klein-Gordon equation [18]: with the initial conditions
Using the Laplace transform to Equation (42), we get
Applying the inverse Laplace transform of Equation (45), we have
Now, by using the suggested analytical method, we get
The series form result is
The problem has the exact solution at :
In Figure 6, the exact and the approximate solutions of example 4 at are shown, and the second graph shows the 3D graph of different fractional-order , respectively. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.
5. 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1B05030422).