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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.

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 [

In the past thirty years, a variety of numerical methods [

The method of fundamental solutions (MFS) [

The singular boundary method (SBM) [

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

Let

For discretization of time variable, the time derivatives can be approximated by the time-stepping scheme as

Clearly, initial values

When

Finally, using (

In (

To obtain the particular solution

Collocating (

Solving (

Many RBFs [

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 (

From (

To obtain the unknown coefficient

Equation (

The computational formulae of the source intensity factor

Finally, the unknown coefficient

After solving (

The test problem is the following nonlinear sine-Gordon equation:

Figure

Graphs of (a) space-time and (b) error of the numerical solution up to

For the error estimation and convergence analysis, the following

(a)

Figure

(a)

A line soliton for an inhomogeneity on large-area Josephson junction can be simulated by solving the sine-Gordon equation for

Figure

Initial condition and numerical solutions at

The circular ring soliton can be simulated by solving the sine-Gordon equation for

Figure

Initial condition and numerical solutions at

The collision of two circular ring solitons can be simulated by solving the sine-Gordon equation for

As in [

When the dissipative parameter

Results of (a) initial condition (

The dissipative term

Numerical solutions at

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 [

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.

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

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