A meshless method based on the singular boundary method is developed for the numerical solution of the time-dependent nonlinear sine-Gordon equation with Neumann boundary condition. In this method, by using a time discrete scheme to approximate the time derivatives, the time-dependent nonlinear problem is transformed into a sequence of time-independent linear boundary value problems. Then, the singular boundary method is used to establish the system of discrete algebraic equations. The present method is meshless, integration-free, and easy to implement. Numerical examples involving line and ring solitons are given to show the performance and efficiency of the proposed method. The numerical results are found to be in good agreement with the analytical solutions and the numerical results that exist in literature.

1. Introduction

The sine-Gordon equation is nonlinear and has drawn considerable attention as it comes up in a broad class of modeling situations such as nonlinear optics and solid state physics [1, 2]. The solutions of nonlinear equations are important in mathematical physics and engineering. Many analytical methods [35] have been proposed to derive analytical solutions. It should be stressed that analytical solutions can only be gained in special cases. For instance, the study of analytical solutions of the sine-Gordon equation mostly focused on the undamped case.

In the past thirty years, a variety of numerical methods [615], such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM), have been used to numerical simulation of the sine-Gordon equation. The success of these methods relies on the ability to construct and retain meshes of sufficient quality in the process of computation. In the past twenty years, meshless methods [1620] have been developed to alleviate the meshing-related drawbacks. Up to now, a lot of meshless methods have been developed. Recently, some meshless methods [2128] have been used to obtain the numerical solutions of the sine-Gordon equation.

The method of fundamental solutions (MFS) [29] is an extensively used meshless method. The approximate solution in the MFS is expressed as a linear combination of fundamental solutions. Compared with the BEM, the MFS does not need to compute singular integrals. However, to avoid the singularity of the fundamental solution, source points in the MFS are required to locate on a fictitious boundary outside the computational domain. Besides, the final system of algebraic equations in the MFS is sometimes ill-conditioned or nearly singular.

The singular boundary method (SBM) [30] is a recently developed meshless method. As in the MFS, the approximate solution in the SBM is also expressed as a linear combination of fundamental solutions. However, to isolate the singularity of fundamental solutions, the source intensity factor is proposed in the SBM. Thus, compared with the MFS, source points in the SBM are located on the real boundary instead of the fictitious boundary. The SBM has been used successfully for solving many boundary value problems [3033]. Numerical results indicate that the SBM has much smaller condition number than the MFS.

In previous works, the SBM has only been used for problems governed by linear partial differential equations. The aim of this paper is to extend the SBM to the nonlinear sine-Gordon equation. The time derivatives in the nonlinear sine-Gordon equation are approximated by a time discrete scheme. Then, the SBM is used to establish the discrete algebraic equations. The present numerical method is truly meshless, free of integration, and easy to implement. Numerical examples are provided to illustrate the performance and capability of the method. Convergence studies are conducted to demonstrate the accuracy and efficiency of the present method.

The following discussions start with a description of the sine-Gordon equation. Then, detailed computational formulas of the SBM are presented for the sine-Gordon equation in Section 3. Finally, some numerical examples and conclusions are shown in Sections 4 and 5, respectively.

2. Problem Description

Let be a two-dimensional bounded domain with boundary . This paper is devoted to the numerical solution of the nonlinear sine-Gordon equationwith initial conditionsand boundary conditionwhere is the unknown wave displacement at position and time , , , , and are known functions, and is the unit outward normal on . The function can be explained as the Josephson current density, the function is the wave modes or kinks, and the function is the wave velocity. In addition, denotes the dissipative term. Equation (1) is the damped equation for , and it is the undamped one for .

3. Numerical Computational Formulation

3.1. Time Differencing

For discretization of time variable, the time derivatives can be approximated by the time-stepping scheme aswhere denotes the step size of andIn addition, can be approximated by the Crank-Nicolson scheme asThen, substituting (5)-(8) into (1) yields

Clearly, initial values and are required in the iterative computation of (9). Using the initial conditions in (2) and (3), we haveIn (9), letting and using (10) and (11) lead towhereis known from the given functions.

When , we can write (9) aswhereis known at the ()th iteration.

Finally, using (12) and (14), the original time-dependent problem (1)-(4) is reduced to the following time-independent problem:where and for and for

3.2. Approximation of the Particular Solution

In (16), the constant is positive. Thus, (16)-(17) compose a sequence of inhomogeneous problems for the modified Helmholtz equation. The solution can be represented as the sum of a particular solution and a homogeneous solution ; i.e.,where satisfies (16) asbut not necessarily satisfies the boundary condition given in (17). Then, the homogeneous solution satisfies the following homogeneous equation:with modified boundary condition

To obtain the particular solution , let be a set of boundary nodes and be a set of interior nodes. Then, we can approximate the right-hand side of (19) by a finite linear combination of radial basis functions (RBFs) [34] aswhere is the undetermined coefficient.

Collocating (22) for all nodes providesnamely,where

Solving (24), we obtainFinally, from (19) and (22) we obtainwhere is related to the RBF according toand

Many RBFs [34], such as multiquadric, Gaussians, linear distance functions, thin plate splines (TPS), and RBFs with compact supports, have been proposed and can be used for obtaining the particular solution. In this research, the following TPS is chosen: And then, using the annihilator method [35], we havewhere and is the modified Bessel function of the second kind of order zero.

3.3. SBM for the Homogeneous Problem

In the SBM, the set of source points coincides with the set of collocation points. Then, we can express the solution of the homogeneous modified Helmholtz equation (20) aswhere the zeroth-order modified Bessel function is the fundamental solution of the modified Helmholtz operator , is the set of source points, and is the unknown coefficient.

From (34) we havewhereand denotes the modified Bessel function of the second kind of order one.

To obtain the unknown coefficient , using (34) and (35) and collocating the boundary condition given in (21) at yield

Equation (37) contains linear algebraic equations, from which we can gain unknowns . However, when source points and collocation points coincide, the singularity of the modified Bessel function occurs, and then the value of this function and its normal derivative can not be obtained directly. To isolate the singularity, the source intensity factor is proposed. Then, the SBM interpolation formula for (20) can be expressed aswhere is the source intensity factor corresponding to the singular term .

The computational formulae of the source intensity factor are derived as follows. On the one hand, applying the subtracting and adding-back technique [36], from (35) we havewhere and is the half distance of the curve between and . When , the second term in the right-hand side of (39) has a singularity. After a lengthy deduction [31], we haveSubstituting this equation into (39) yieldsBesides, collocating (35) at the boundary node yieldsTherefore, from the above two equations we have

Finally, the unknown coefficient can be obtained from (38) by solvingwhere we have used (29). Equation (44) can be expressed aswhere is an unknown vector, is a known vector containing the given boundary values , and are matrices, and is a vector given in (27) and is known at the ()th iteration.

After solving (45) to obtain , then, according to (18), (29), and (34), approximate solution of the original nonlinear problem (1)-(4) can be computed as

4. Numerical Results

4.1. Test Problem

The test problem is the following nonlinear sine-Gordon equation:subject to initial conditionsand boundary conditionsThe analytical solution is [21]

Figure 1 gives the space-time graph of the numerical solution and the graph of the associated error. The results are obtained by using and .

For the error estimation and convergence analysis, the following and root-mean-square (RMS) errors are used:where is the number of test points in and and are the analytical and numerical values at the -th test point, respectively.

and RMS errors at , 3, 5, and 7 are plotted in Figure 2 for the SBM using and . For comparison, the errors of the RBF method [21], the explicit method [6], and the moving Kriging-based meshless local Petrov-Galerkin (MK-MLPG) method [25] obtained by using and are also given. Comparing the errors of these methods confirms the good accuracy of the SBM.

Figure 3 gives the errors at times , 3, 5, and 7 for the SBM against the nodal spacing . In this analysis, . Clearly, as the nodal spacing decreases, the error becomes smaller. In addition, the SBM possesses high experimentally convergence rate.

4.2. Line Soliton in an Inhomogeneous Medium

A line soliton for an inhomogeneity on large-area Josephson junction can be simulated by solving the sine-Gordon equation for and [711, 21]and initial conditionsin the domain . The boundary condition is

Figure 4 presents the results of this problem for and at , 6, 12, and 18. In this figure, both the contours and surface plots of the ring soliton are given in terms of . We can see that the soliton is moving in -direction as a straight line soliton during the transmission through inhomogeneity. When , we can observe a deformation in its straightness. Then, this movement seems to be impeded as tends to 12. Finally, the soliton recovers its straightness when . These graphical results match those obtained by the FEM [7], the FDM [8], the BEM [9, 10], the differential quadrature method (DQM) [11], and the RBF [21].

4.3. Circular Ring Soliton

The circular ring soliton can be simulated by solving the sine-Gordon equation for and , with initial conditions [7, 911, 21, 23]and boundary condition

Figure 5 depicts the results of this problem for and at times , 5.6, 8.4, 11.2 and 12.6. In this figure, both the contours and surface plots of the ring soliton are given in terms of . It can be observed from this figure that the ring soliton shrinks at the initial stage . Then, as goes on, oscillations and radiations start to form and continue to form up to . And an expanding phase is observed at and after it. The expansion continues until , where the ring soliton is nearly formed again. When , the ring soliton appears to be again in its shrinking phase. These graphical results match those obtained by the FEM [7], the BEM [9, 10], the DQM [11], the RBF [21], and the mesh-free reproducing kernel particle Ritz method [23].

4.4. Collision of Two Circular Ring Solitons

The collision of two circular ring solitons can be simulated by solving the sine-Gordon equation for with initial conditions [6, 7, 10, 21]and boundary condition

As in [6, 7, 10, 21], the solution is computed by the SBM over and and then is expanded at the lines and by symmetry relations.

When the dissipative parameter , Figure 6 depicts the results for and at times , 4, 8, and 11 in terms of . The temporal behavior of the soliton wave consists of a shrinking and an expanding phase. We can observe from Figure 6 the collision between two ring solitons in which, as a result of the collision, two ring solitons bounding an annular area emerge into a larger oval ring soliton. These graphical results match those obtained by the FEM [7], the FDM [6], the BEM [10], and the RBF [21].

The dissipative term is expected to slow down the two initial ring solitons to emerge into a larger oval ring soliton. In order to show this, Figure 7 gives the solution at for , 0.35, 1 and 5. From Figures 6 and 7, we can observe that the ring soliton changes slowly as increases. Additionally, it can be found that with the solitons at time are close to those given at for .

5. Conclusions

In this paper, a meshless method based on the meshless singular boundary method has been developed to obtain numerical solutions of the time-dependent nonlinear sine-Gordon equation. In this method, no mesh is required to discretize the problem domain, and the approximate solution is generated entirely based on scattered nodes. Several numerical examples have been studied to demonstrate the accuracy and efficiency of the method. The numerical results are compared with the results obtained by other methods. For the example with known analytical solution, it is found that the current method obtains more accurate numerical results. For all other examples, the current numerical method produces similar accurate results as those given in [611, 21, 23]. Therefore, the SBM is well-suited for solving the nonlinear sine-Gordon equation. However, although numerical results verify the accuracy and the convergence of the method, theoretical proof of stability and convergence is not presented and will be an important research topic for future research.

The current method can be extended to solve other nonlinear equations such as the Klein-Gordon equation, the double sine-Gordon and sinh-Gordon equations, and the hyperbolic telegraph equation. Besides, this meshless method with some modifications is extensible to solve problems in mathematical physics and engineering such as nonlinear optics, plasma physics, solid state physics, and relativistic quantum mechanics. Nevertheless, more research work is required.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.