Analytical Solution of Two-Dimensional Sine-Gordon Equation

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.


Introduction
Nonlinear phenomena, which appear in many areas of scientific fields such as solid-state physics, plasma physics, fluid dynamics, mathematical biology, and chemical kinetics, can be modeled by partial differential equations. A broad class of analytical and numerical solution methods were used to handle these problems. Recently, several research on the physical phenomena of the diverse fields of engineering and science was carried out, see for example [1][2][3][4][5][6][7][8][9] and the references therein.
The nonlinear sine-Gordon equation (SGE), a type of hyperbolic partial differential equation, is often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxions, and dislocation of metals, for details see [10] and the references therein. Because the sine-Gordon equation has many kinds of soliton solutions, it has attracted wide spread attention [11]. The sine-Gordon equation was first discovered in the nineteenth century in the course of study of various problems of differential geometry [12]. In the early 1970s, it was first realized that the sine-Gordon equation led to kink and antikink (so-called solitons) [13]. As one of the crucial equations in nonlinear science, the sine-Gordon equation has been constantly investigated and solved numerically and analytically in recent years [10,[14][15][16][17][18]. Different scholars employed different methods to solve the one-dimensional sine-Gordon equation, for example, the Adomian decomposition method (ADM) [19][20][21][22][23], the EXP function method [24], the homotopy perturbation method (HPM) [25][26][27], the homotopy analysis method (HAM) [28], the variable separated ODE method [29,30], and the variational iteration method (VIM) [31,32]. Further, Shukla et al. [33] obtained numerical solution of the twodimensional nonlinear sine-Gordon equation using a localized method of approximate particular solutions. Baccouch [34] developed and analyzed an energy-conserving local discontinuous Galerkin method for the two-dimensional SGE on Cartesian grids. Duan et al. [35] proposed a numerical model based on the lattice Boltzmann method to obtain the numerical solutions of the two-dimensional generalized sine-Gordon equation, and the method was extended to solve the nonlinear hyperbolic telegraph equation as indicated in [36].
The main aim of this study is to obtain the approximate analytical solutions for the two-dimensional nonlinear sine-Gordon equation (TDNLSGE), since most of the research focused on the numerical solutions for this problem. The reduced differential transform method is used for this purpose for several reasons. The first reason is that the method has not previously been studied to solve this problem. Secondly, the present method is easy to apply for multidimensional problems and the corresponding algebraic equation is simple and easy to implement. Thirdly, this method can reduce the size of the calculations and can provide an analytic approximation, in many cases exact solutions, in rapidly convergent power series form with elegantly computed terms ( [37] and see the references therein). Moreover, the reduced differential transform method (RDTM) has an alternative approach of solving problems to overcome the demerit of discretization, linearization, or perturbations of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration [37][38][39].
In this paper, we investigate the solution of the twodimensional nonlinear sine-Gordon equation [40]: subject to the initial conditions: by using RDTM, where Ω = fðx, yÞ: The function ϕðx, yÞ can be interpreted as a Josephson current density, and φ 1 ðx, yÞ and φ 2 ðx, yÞare wave modes or kinks and velocity, respectively. The parameter β is the socalled dissipative term, which is assumed to be a real number with β ≥ 0. When β = 0, Equation (1) reduces to the undamped SGE equation in two space variables, while when β > 0, to the damped one, and α is a nonnegative real number.
The paper is organized as follows. In Section 2, we begin with some basic definitions and operations of the proposed method, and we introduce some lemmas that will be used later in this paper. The implementation of the method is presented in Section 3. The convergence analysis of the method is presented in Section 4. In Section 5, we apply RDTM to solve four test problems to show the applicability, efficiency, and accuracy of the method. Section 6 presents graphical representation and physical interpretations of the solutions behavior of the considered examples. Conclusions are given in Section 7.

Preliminaries and Notations
In this section, we give the basic definitions and operations of the two-dimensional reduced differential transform method [37,[41][42][43].
Definition 1. If a function uðx, y, tÞ is analytic and differentiated continuously with respect to space variables x, y and time variable t in the domain of interest, then where U k ðx, yÞ is the t-dimensional spectrum function or the transformed function.
Furthermore, the inverse reduced differential transform of the set of fU k ðx, yÞg n k=0 gives an approximate solution as where n is the order of the approximate solution. Therefore, by Definition 2, the exact solution of the problem is given by From Equation (8), it can be found that the concept of the reduced differential transform method is derived from the power series expansion.
The fundamental mathematical operations performed by RDTM are listed in Table 1.
In addition to the properties of RDTM given in Table 1, we introduce the lemmas which provide us with a simple way to apply the RDTM to the two-dimensional nonlinear sine-Gordon Equations (1)-(3).

Implementation of the Method
To illustrate the basic concepts of the RDTM, we consider the NLSGE (1) with initial conditions (2) and (3).
According to the RDTM given in Table 1 and Lemma 4, we can construct the following iteration formula: where F k ðx, yÞ is the reduced differential transform of the nonlinear term sin uðx, y, tÞ and H k ðx, yÞ is the reduced differential transform of the inhomogeneous term hðx, y, tÞ. Thus, and so on.

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Using Lemma 5 on initial conditions (2) and (3), we get Substituting (24) and (23) into (22) and by straightforward iterative calculations, we get the following successive values of U k ðx, yÞ, i.e., U 2 ðx, yÞ, U 3 ðx, yÞ, U 4 ðx, yÞ, ⋯. Then, the inverse reduced differential transform of the set of values fU k ðx, yÞg n k=0 gives the n -term approximate solution: Therefore, the exact solution of problem (1) is given by

Convergence Analysis
In this section, we present the convergence analysis of the approximate analytical solutions which are computed from the application of RDTM [41]. Consider the SGE (1) in the following functional equation form: where F is a general nonlinear operator involving both linear and nonlinear terms. According to RDTM, the two-dimensional NLSGE given in Equation (1) has a solution of the form: It is noted that the solutions by RDTM is equivalent to determining the sequences by using the iterative scheme associated with the functional equation Hence, the solution obtained by RDTM, uðx, y, tÞ = The sufficient condition for convergence of the series solution fS n g ∞ n=0 is given in the following theorems.
Theorem 6. Let F be an operator from a from Hilbert space H in to H . Then, the series solution fS n g ∞ n=0 converges whenever there is α such that 0 < α < 1, and kβ k+1 k ≤ αkβ k k.
See [41] for the proof. Theorem 7. Let F be a nonlinear operator that satisfies the Lipschitz condition from Hilbert space H in to H and uðx, y , tÞ be the exact solution of the given SGE. If the series solution fS n g ∞ n=0 converges, then it converges to uðx, y, tÞ.

Numerical Results
In this section, we apply the reduced differential transform method (RDTM) for finding the approximate analytic solutions of four test examples associated with the nonlinear sine-Gordon equations (NLSGEs) in a two-dimensional space. To demonstrate the applicability of the method and accuracy of the solutions, the results obtained by the proposed method is compared with the exact solution existing in the literature, and the numerical results and the absolute errors are given using tables and figures.
Numerical results corresponding to the two-dimensional nonlinear sine-Gordon equation given in Example 1 are depicted in Table 2 and Figure 1.
Consider the two dimensional sine-Gordon equation [36,45] with initial conditions

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By taking the reduced differential transform of Equation (43), we obtain where F k ðx, yÞ and H k ðx, yÞ are the reduced differential transform of the nonlinear term sin ðuðx, y, tÞÞ and the inhomogeneous term respectively. Using RDTM to the initial conditions (44) and (45), we get Substituting Equations (48) and (49) into Equation (46), and applying Lemma 4, Definition 1, and properties of RDTM, we obtain the following successive iterated values for kðk = 0, 1, 2, ⋯Þ: and so on. Then by (8), we obtain the approximate analytic solution of Example 2 as follows: The exact solution of the problem is uðx, y, tÞ = e −t ð1 − cos ðπxÞÞð1 − cos ðπyÞÞ, as indicated in [36,45].
Numerical results corresponding to the two-dimensional nonlinear sine-Gordon equation given in Example 2 are depicted in Table 3 and Figure 2.
Example 3. Consider the two-dimensional inhomogeneous sine-Gordon equation [34], with initial conditions Applying the RDTM to Equation (52), we obtain the following recurrence relation where F k ðx, yÞ is the reduced differential transform of nonlinear term sin uðx, y, tÞ and H k ðx, yÞ is the reduced differential transform of inhomogeneous term ½sin ðsin ðx + y + tÞÞ + sin ðx + y + tÞ.
Numerical results corresponding to the two-dimensional nonlinear sine-Gordon equation given in Example 3 are depicted in Table 4 and Figure 3.
Example 4. Consider the following two dimensional sine-Gordon equation [33] subject to the initial conditions u x, y, 0 ð Þ= 4 tan −1 e x+y ð Þ, ð60Þ Applying RDTM technique to Equation (59), we obtain the following iterative formula: where F k ðx, yÞ is the reduced differential transform of nonlinear term sin uðx, y, tÞ.
Using RDTM to initial condition (60) and (61)  and so on.
Numerical results corresponding to the two-dimensional nonlinear sine-Gordon equation given in Example 4 are depicted in Table 5 and Figure 4.  Periodic traveling waves play an important role in numerous physical phenomena, including reaction-diffusion-advection systems, self-reinforcing systems, and impulsive systems. Mathematical modeling of many intricate physical events, for instance, physics, mathematical physics, engineering, and many more phenomena resemble periodic traveling wave solutions [46].

Conclusions
The reduced differential transform method (RDTM) is successfully implemented to find approximate analytical solutions or exact solutions of the two-dimensional nonlinear sine-Gordon equations subject to the appropriate initial conditions. The convergence analysis of the proposed method is also studied, and the results we obtained in Examples 1, 2, 3, and 4 are in excellent agreement with the exact solutions obtained by different methods available in the literature, see Refs. [33,34,36,40,45]. Furthermore, RDTM is much easier, more convenient, and efficient and this work illustrates the validity and great potential of the reduced differential transform method for solving nonlinear partial differential equations. As a result, the basic ideas of this approach are expected to be further employed to solve other nonlinear problems arising in sciences and engineering.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.