The Improved Moving Least-Square Ritz Method for the One-Dimensional Sine-Gordon Equation

Analysis of the one-dimensional sine-Gordon equation is performed using the improved moving least-square Ritz method (IMLSRitzmethod).The improvedmoving least-square approximation is employed to approximate the 1D displacement field. A system of discrete equations is obtained by application of the Ritz minimization procedure. The effectiveness and accuracy of the IMLS-Ritz method for the sine-Gordon equation are investigated by numerical examples in this paper.


Introduction
It is well known that many physical phenomena in one or higher-dimensional space can be described by a soliton model.Many of these models are based on simple integrable models such as Korteweg-de Vries equation and the nonlinear Schr ö dinger equation.Solitons have found to model among others shallow-water waves, optical fibres, Josephsonjunction oscillators, and so forth.Equations which also lead to solitary waves are the sine-Gordon.The sine-Gordon equation arises in extended rectangular Josephson junctions, which consist of two layers of super conducting materials separated by an isolating barrier.A typical arrangement is a layer of lead and a layer of niobium separated by a layer of niobium oxide.A quantum particle has a nonzero significant probability of being able to penetrate to the other side of a potential barrier that would be impenetrable to the corresponding classical particle.This phenomenon is usually referred to as quantum tunneling [1][2][3].
The moving least-square (MLS) technique was originally used for data fitting.Nowadays, the MLS technique has been employed as the shape functions of the meshless method or element-free Galerkin (EFG) method [47].Though EFG method is now a very popular numerical computational method, a disadvantage of the method is that the final algebraic equations system is sometimes illconditioned.Sometimes a poor solution will be obtained due to the ill-conditioned system.The improved moving leastsquare (IMLS) approximation has been proposed [48][49][50][51][52] to overcome this disadvantage.In the IMLS, the orthogonal function system with a weight function is chosen to be the basis functions.The algebraic equations system in the IMLS approximation will be no more ill-conditioned.
The Ritz [53] approximation technique is a generalization of the Rayleigh [54] method and it has been widely used in computational mechanics.The element-free kp-Ritz method is firstly developed and implemented for the free vibration analysis of rotating cylindrical panels by Liew et al. [55].The kp-Ritz method was widely applied and used in many kinds of problems, such as free vibration of two-side simply-supported laminated cylindrical panels [56], nonlinear analysis of laminated composite plates [57], Sine-Gordon equation [16], 3D wave equation [58], and biological population problem [59].
A new numerical method which is named the IMLS-Ritz method for the sine-Gordon equation is presented in this paper.In this paper, the unknown function is approximated by these IMLS approximation; a system of nonlinear discrete equations is obtained by the Ritz minimization procedure, and the boundary conditions are enforced by the penalty method.Numerical examples are presented to validate the accuracy and efficiency of the proposed method.

IMLS-Ritz Formulation for the Sine-Gordon Equation
Consider the following one-dimensional sine-Gordon equation: with initial conditions and boundary conditions where Γ denotes the domain of , Γ  denotes the boundaries, and  1 () and  2 () are wave modes or kinks and velocity, respectively.Parameter  is the so-called dissipative term, assumed to be a real number with  ≥ 0.
The weighted integral form of (1a) is obtained as follows: The weak form of ( 2) is The energy functional Π() can be written as In the improved moving least-square approximation [19], define a local approximation by This defines the quadratic form Equation ( 6) can be rewritten in the vector form To find the coefficients a(x), we obtain the extremum of  by which results in the equation system If the functions  1 (x),  2 (x), . . .,   (x) satisfy the conditions then  1 (x),  2 (x), . . .,   (x) is called a weighted orthogonal function set with a weight function {  } about points {x  }.The weighted orthogonal basis function set p = (  ) can be formed with the Schmidt method [39][40][41], Equation ( 9) can be rewritten as The coefficients   (x) can be directly obtained as follows: that is, where From ( 12), the approximation function  ℎ (x) can be rewritten as where Φ(x) is the shape function and Taking derivatives of ( 17), we can obtain the first derivatives of shape function Imposing boundary conditions by penalty method, the total energy functional for this problem will be obtained: By (16), we can derive the approximation function Substituting ( 20) into (19) and applying the Ritz minimization procedure to the energy function Π * (), we obtain In the matrix form, the results can be expressed as where Making time discretization of ( 22) by the center difference method, we get where The numerical solution of the one-dimensional sine-Gordon equation will be obtained by solving the above iteration equation.

Numerical Examples and Analysis
To verify the efficiency and accuracy of the proposed IMLS-Ritz method for the sine-Gordon equation, two examples are studied and the numerical results are presented.The weight function is chosen to be cubic spline and the bases are chosen to be linear in all examples.
Example 1.Consider the sine-Gordon Equation (1a)-(1d) without nonlinear term sin() over the region −1 ≤  ≤ 1 with initial conditions with boundary conditions The exact solution is  (, ) = 1 2 (sin  ( + ) + sin  ( − )) .The IMLS-Ritz method is applied to solve the above equation with penalty factor  = 10 4 and time step length Δ = 0.001,  max = 3.0.In Figure 1, the numerical solution and exact solution are plotted at times  = 0.1, 0.2, 0.3, and 0.4, respectively.In Figures 2, 3, and 4, the graphs of error function  ℎ (, ) − (, ) are plotted at times  = 0.1, 0.2, and 0.3, respectively, where (, ) is the exact solution and numerical solution  ℎ (, ) is obtained by using the IMLS-Ritz method.Table 1 shows the comparison of exact solutions and numerical solutions by IMLS-Ritz method and EFG method.From the results of Table 1, we can draw the conclusion that IMLS-Ritz method has higher accuracy than the EFG method.The surfaces of the numerical solution with the IMLS-Ritz method and exact solutions are plotted in Figures 5 and 6.
which derives the analytic solution and the boundary conditions can be obtained from (30).
The IMLS-Ritz method is applied to solve the above equation with penalty factor  = 10 5 and time step length Δ = 0.01,  max = 2.2. Figure 7 depicts the numerical and exact solution when  = 1, 5, 10, 20, and 30, respectively.In Figures 8, 9, and 10, the graphs of error function  ℎ (, ) − (, ) are plotted at times  = 1, 5, and 10, respectively, where   (, ) is the exact solution and numerical solution  ℎ (, ) is obtained by using the IMLS-Ritz method.Table 2 shows the comparison of exact solutions and numerical solutions by IMLS-Ritz method and EFG method.From the results of Table 2, it is shown that IMLS-Ritz method has higher accuracy than the EFG method.The surfaces of the numerical solution with the IMLS-Ritz method and exact solution are plotted in Figures 11 and 12   and the boundary conditions can be derived from the following exact solitary wave solution: where  = / √ 1 −  2 and  is the velocity of solitary wave.
The IMLS-Ritz method is applied to solve the above equation with penalty factor  = 10 7 and time step length Δ = 0.01,  max = 2.7,  = 0.5.Table 3 shows the comparison of exact solutions and numerical solutions by IMLS-Ritz method and EFG method.From the results of Table 3, it is shown that IMLS-Ritz method has higher accuracy than the EFG method.The surfaces of the numerical solution with the IMLS-Ritz method and exact solution are plotted in Figures     solution and the numerical solution  ℎ (, ) is obtained by using the IMLS-Ritz method.

Table 1 :
The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 41 nodes at  = 0.1 with  = 0.001 and  max = 3.0 (Example 1).

Table 2 :
The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 40 nodes at  = 1 with  = 0.01 and  max = 2.2 (Example 2).

Table 3 :
The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 81 nodes at  = 1 with  = 0.001 and  max = 2.7 (Example 3).