Research Article | Open Access
Alemayehu Tamirie Deresse, Yesuf Obsie Mussa, Ademe Kebede Gizaw, "Analytical Solution of Two-Dimensional Sine-Gordon Equation", Advances in Mathematical Physics, vol. 2021, Article ID 6610021, 15 pages, 2021. https://doi.org/10.1155/2021/6610021
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.
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 [1–9] 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  and the references therein. Because the sine-Gordon equation has many kinds of soliton solutions, it has attracted wide spread attention . The sine-Gordon equation was first discovered in the nineteenth century in the course of study of various problems of differential geometry . In the early 1970s, it was first realized that the sine-Gordon equation led to kink and antikink (so-called solitons) . 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, 14–18]. Different scholars employed different methods to solve the one-dimensional sine-Gordon equation, for example, the Adomian decomposition method (ADM) [19–23], the EXP function method , the homotopy perturbation method (HPM) [25–27], the homotopy analysis method (HAM) , the variable separated ODE method [29, 30], and the variational iteration method (VIM) [31, 32]. Further, Shukla et al.  obtained numerical solution of the two-dimensional nonlinear sine-Gordon equation using a localized method of approximate particular solutions. Baccouch  developed and analyzed an energy-conserving local discontinuous Galerkin method for the two-dimensional SGE on Cartesian grids. Duan et al.  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 .
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 ( 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 [37–39].
In this paper, we investigate the solution of the two-dimensional nonlinear sine-Gordon equation : subject to the initial conditions: by using RDTM, where .
The function can be interpreted as a Josephson current density, and and are wave modes or kinks and velocity, respectively. The parameter is the so-called dissipative term, which is assumed to be a real number with . When , Equation (1) reduces to the undamped SGE equation in two space variables, while when , to the damped one, and 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.
2. Preliminaries and Notations
Definition 1. If a function is analytic and differentiated continuously with respect to space variables and time variable in the domain of interest, then where is the -dimensional spectrum function or the transformed function.
Definition 2. The inverse reduced differential transform of a sequence is given by
Remark 3. The function is represented by a finite series (5) around and can be written as where the tail function is negligibly small.
Furthermore, the inverse reduced differential transform of the set of gives an approximate solution as where 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.
Lemma 4. Assume that and are the reduced differential transform of the functions and , respectively, then, we have the following RDTM results: (i)If then(ii)If , then
Proof. (i)Applying properties of RDTM on both sides of , we obtainThen, by Leibnitz rule for higher order derivatives of the products and properties of RDTM on , we obtain,
Hence, by using Definition 1, for , we get (ii)Applying properties of RDTM on both sides of , we get
Using Leibnitz rule of higher order derivatives of the products on , we get
Therefore, and then using Definition 1, for , we get
Lemma 5. If , then
Proof. By Definition 1, , and so, when is replaced by , we have,
and from the initial condition we get .
Furthermore, by convention, if , then and if , then .
3. Implementation of the Method
where is the reduced differential transform of the nonlinear term and is the reduced differential transform of the inhomogeneous term .
Thus, and so on.
Substituting (24) and (23) into (22) and by straightforward iterative calculations, we get the following successive values of , i.e., . Then, the inverse reduced differential transform of the set of values gives the -term approximate solution:
Therefore, the exact solution of problem (1) is given by
4. Convergence Analysis
In this section, we present the convergence analysis of the approximate analytical solutions which are computed from the application of RDTM .
Consider the SGE (1) in the following functional equation form: where 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, is equivalent to
The sufficient condition for convergence of the series solution is given in the following theorems.
Theorem 6. Let be an operator from a from Hilbert space in to . Then, the series solution converges whenever there is such that , and .
See  for the proof.
Theorem 7. Let be a nonlinear operator that satisfies the Lipschitz condition from Hilbert space in to and be the exact solution of the given SGE. If the series solution converges, then it converges to .
For proof see Ref. .
Definition 8. For , we define Then, we can say that the series approximate solution converges to the exact solution when for .
5. 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.
Example 1. Consider the sine-Gordon equation  with initial conditions Applying properties of RDTM to Equation (34), we construct the following recursive formula: where and are the reduced differential transform of the nonlinear term and the inhomogeneous term , respectively.
Then, by (8), we get
For the convergence of the approximate analytic solution given in Equation (41), we calculate using
Example 2. Consider the two dimensional sine-Gordon equation [36, 45] with initial conditions By taking the reduced differential transform of Equation (43), we obtain where and are the reduced differential transform of the nonlinear term and the inhomogeneous term respectively.
To test the convergence of the approximate solution, we calculate . Let us take and in the domain of interest, then using definition 8, we obtain, , ,, …., and by induction for all . Therefore, the solution of Equation (43) converges to the exact solution.
Example 3. Consider the two-dimensional inhomogeneous sine-Gordon equation , with initial conditions Applying the RDTM to Equation (52), we obtain the following recurrence relation where is the reduced differential transform of nonlinear term and is the reduced differential transform of inhomogeneous term .