MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/6460480 6460480 Research Article Meshless Singular Boundary Method for Nonlinear Sine-Gordon Equation http://orcid.org/0000-0001-9580-5634 Ji Yun 1 Torrisi Mariano Institute of Applied Electronics Chongqing College of Electronic Engineering Chongqing 401331 China cqcet.com 2018 572018 2018 08 05 2018 12 06 2018 572018 2018 Copyright © 2018 Yun Ji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  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 , 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  have been developed to alleviate the meshing-related drawbacks. Up to now, a lot of meshless methods have been developed. Recently, some meshless methods  have been used to obtain the numerical solutions of the sine-Gordon equation.

The method of fundamental solutions (MFS)  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)  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 . 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 Ω be a two-dimensional bounded domain with boundary Γ. This paper is devoted to the numerical solution of the nonlinear sine-Gordon equation(1)2ut2+βut=Δu-ψxsinu,xΩ,  t>0with initial conditions(2)ux,0=fx,xΩ(3)ux,ttt=0=gx,xΩand boundary condition(4)ux,tn=q¯x,t,xΓ,  t>0where ux,t is the unknown wave displacement at position x and time t, ψx, fx, gx, and q¯x,t are known functions, and n=n1,n2 is the unit outward normal on Γ. The function ψx can be explained as the Josephson current density, the function fx is the wave modes or kinks, and the function gx is the wave velocity. In addition, β>0 denotes the dissipative term. Equation (1) is the damped equation for β>0, and it is the undamped one for β=0.

3. Numerical Computational Formulation 3.1. Time Differencing

For discretization of time variable, the time derivatives can be approximated by the time-stepping scheme as(5)2ux,tt2=uk+1x-2ukx+uk-1xτ2(6)ux,tt=uk+1x-uk-1x2τwhere τ>0 denotes the step size of t and(7)ukx=ux,kτ,k=0,1,2,In addition, Δu can be approximated by the Crank-Nicolson scheme as(8)Δux,t=13Δuk+1x+Δukx+Δuk-1xThen, substituting (5)-(8) into (1) yields(9)uk+1-2uk+uk-1τ2+βuk+1-uk-12τ=13Δuk+1+Δuk+Δuk-1-ψxsinukx

Clearly, initial values u0 and u-1 are required in the iterative computation of (9). Using the initial conditions in (2) and (3), we have(10)u0x=fx,xΩ(11)u-1x=u1x-2τgx,xΩIn (9), letting k=0 and using (10) and (11) lead to(12)Δu1x-3τ2u1x=b0xwhere(13)b0x=τΔgx-12Δfx-3τ2fx-3τ-3β2gx+32ψxsinfxis known from the given functions.

When k1, we can write (9) as(14)Δuk+1x-3τ2+3β2τuk+1x=bkx,k=1,2,where(15)bkx=-Δukx-Δuk-1x-6ukxτ2+3τ2-3β2τuk-1x+3ψxsinukxis known at the (k+1)th iteration.

Finally, using (12) and (14), the original time-dependent problem (1)-(4) is reduced to the following time-independent problem:(16)Δuk+1x-λuk+1x=bkx,xΩ(17)uk+1xn=q¯x,kτ,xΓwhere k=0,1,2, and λ=3/τ2 for k=0 and λ=3/τ2+3β/2τ for k=1,2,

3.2. Approximation of the Particular Solution

In (16), the constant λ 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 upk+1 and a homogeneous solution uhk+1; i.e.,(18)uk+1x=uhk+1x+upk+1xwhere upk+1 satisfies (16) as(19)Δupk+1x-λupk+1x=bkx,xΩbut not necessarily satisfies the boundary condition given in (17). Then, the homogeneous solution uhk+1 satisfies the following homogeneous equation:(20)Δuhk+1x-λuhk+1x=0,xΩwith modified boundary condition(21)uhk+1xn=q¯x,kτ-upk+1xn,xΓ

To obtain the particular solution upk+1, let xii=1NΓ be a set of N boundary nodes and xii=N+1N+LΩ be a set of L interior nodes. Then, we can approximate the right-hand side of (19) by a finite linear combination of radial basis functions (RBFs) ϕj  as(22)bkx=j=1N+Lajϕjxwhere aj is the undetermined coefficient.

Collocating (22) for all nodes provides(23)bkxi=j=1N+Lajϕjxi,i=1,2,,N+L;namely,(24)Φa=bkwhere(25)Φ=ϕ1x1ϕ2x1ϕN+Lx1ϕ1x2ϕ2x2ϕN+Lx2ϕ1xN+Lϕ2xN+LϕN+LxN+L(26)a=a1,a2,,aN+LT(27)bk=bkx1,bkx2,,bkxN+LT

Solving (24), we obtain(28)a=Φ-1bkFinally, from (19) and (22) we obtain(29)upk+1x=j=1N+Lajφjx=φTxa=φTxΦ-1bk,xΩwhere φj is related to the RBF ϕj according to(30)Δφjx-λφjx=ϕjx,j=1,2,,N+Land(31)φTx=φ1x,φ2x,,φN+Lx

Many RBFs , 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: (32)ϕjx=rj2xlnrjxAnd then, using the annihilator method , we have(33)φjx=-1λrj2xlnrjx-4λ2K0λrjx+lnrjx+1where rjx=x-xj and K0· 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) as(34)uhk+1x=j=1Ncjk+1K0λrjxwhere the zeroth-order modified Bessel function K0· is the fundamental solution of the modified Helmholtz operator Δ-λ, xii=1NΓ is the set of source points, and cjk+1 is the unknown coefficient.

From (34) we have(35)uhk+1xnx=j=1Ncjk+1nxK0λrjxwhere(36)nxK0λrjx=-λx-xj·nxrjxK1λrjxand K1· denotes the modified Bessel function of the second kind of order one.

To obtain the unknown coefficient cjk+1, using (34) and (35) and collocating the boundary condition given in (21) at xi  (i=1,2,,N) yield(37)j=1Ncjk+1nxiK0λrjxi=q¯xi,kτ-upk+1xinxi,for  all  xiΓ

Equation (37) contains N linear algebraic equations, from which we can gain N unknowns cjk+1. However, when source points xj and collocation points xi coincide, the singularity of the modified Bessel function K0· 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 as(38)j=1,jiNcjk+1nxiK0λrjxi+cik+1Qii=q¯xi,kτ-upk+1xinxi,for  all  xiΓwhere Qii is the source intensity factor corresponding to the singular term K0λrixi/nxi.

The computational formulae of the source intensity factor Qii are derived as follows. On the one hand, applying the subtracting and adding-back technique , from (35) we have(39)uhk+1xinxi=j=1Ncjk+1nxiK0λrjxi=j=1,jiNcjk+1-ljlicik+1K0λrjxinxi+cik+1lij=1NljK0λrjxinxiwhere xi,xjΓ and li is the half distance of the curve between xi-1 and xi+1. When i=j, the second term in the right-hand side of (39) has a singularity. After a lengthy deduction , we have(40)j=1NljK0λrjxinxi=j=1,jiNljK0λrjxinxi+ljlnrjxinxjSubstituting this equation into (39) yields(41)uhk+1xinxi=j=1,jiNcjk+1K0λrjxinxi+cik+1lij=1,jiNljlnrjxinxjBesides, collocating (35) at the boundary node xi yields(42)uhk+1xinxi=j=1,jiNcjk+1nxiK0λrjxi+cik+1QiiTherefore, from the above two equations we have(43)Qii=1lij=1,jiNljlnrjxinxj,i=1,2,,N

Finally, the unknown coefficient cjk+1 can be obtained from (38) by solving(44)j=1,jiNcjk+1nxiK0λrjxi+cik+1Qii=q¯xi,kτ-φTxinxiΦ-1bk,for  all  xiΓwhere we have used (29). Equation (44) can be expressed as(45)Ack+1=γk+Bbkwhere ck+1=c1k+1,c2k+1,,cNk+1T is an unknown vector, γk is a known vector containing the given boundary values q¯xi,kτ, A and B are N×N matrices, and bk is a vector given in (27) and is known at the (k+1)th iteration.

After solving (45) to obtain cjk+1, then, according to (18), (29), and (34), approximate solution of the original nonlinear problem (1)-(4) can be computed as(46)uk+1x=uhk+1x+upk+1x=j=1Ncjk+1K0λrjx+φTxΦ-1bk,xΩ

4. Numerical Results 4.1. Test Problem

The test problem is the following nonlinear sine-Gordon equation:(47)2ut2=2ux12+2ux22-sinu,x1,x2-7,7,  t>0subject to initial conditions(48)ux1,x2,0=4tan-1expx1+x2,ux1,x2,ttt=0=-4expx1+x21+exp2x1+2x2,x1,x2-7,7and boundary conditions(49)ux1,x2,tx1=4expx1+x2+texp2t+exp2x1+2x2,x1=±7,  x2-7,7,  t>0(50)ux1,x2,tx2=4expx1+x2+texp2t+exp2x1+2x2,x2=±7,  x1-7,7,  t>0The analytical solution is (51)ux1,x2,t=4tan-1expx1+x2-t

Figure 1 gives the space-time graph of the numerical solution and the graph of the associated error. The results are obtained by using τ=0.01 and h=0.5.

Graphs of (a) space-time and (b) error of the numerical solution up to t=7 with τ=0.01 and h=0.5 for the test problem.

For the error estimation and convergence analysis, the following L and root-mean-square (RMS) errors are used:(52)L=max1iNsui-uhi,RMS=1Nsi=1Nsui-uhi2where Ns is the number of test points in Ω and ui and uhi are the analytical and numerical values at the i-th test point, respectively.

L and RMS errors at t=1, 3, 5, and 7 are plotted in Figure 2 for the SBM using τ=0.01 and h=0.25. For comparison, the errors of the RBF method , the explicit method , and the moving Kriging-based meshless local Petrov-Galerkin (MK-MLPG) method  obtained by using τ=0.001 and h=0.25 are also given. Comparing the errors of these methods confirms the good accuracy of the SBM.

(a) L error and (b) RMS error of the SBM and the RBF method for the test problem.

Figure 3 gives the errors at times t=1, 3, 5, and 7 for the SBM against the nodal spacing h. In this analysis, τ=0.01. Clearly, as the nodal spacing decreases, the error becomes smaller. In addition, the SBM possesses high experimentally convergence rate.

(a) L error and (b) RMS error versus the nodal spacing h for the test problem.

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 β=0 and [711, 21](53)ψx1,x2=1+sech2x12+x22and initial conditions(54)fx1,x2=4tan-1expx1-3.50.954,gx1,x2=0.629sechx1-3.50.954in the domain -7x1,x27. The boundary condition is(55)ux,tn=0,xΓ,  t0

Figure 4 presents the results of this problem for τ=0.01 and h=0.5 at t=0, 6, 12, and 18. In this figure, both the contours and surface plots of the ring soliton are given in terms of sinu/2. We can see that the soliton is moving in x1-direction as a straight line soliton during the transmission through inhomogeneity. When t=6, we can observe a deformation in its straightness. Then, this movement seems to be impeded as t tends to 12. Finally, the soliton recovers its straightness when t=18. These graphical results match those obtained by the FEM , the FDM , the BEM [9, 10], the differential quadrature method (DQM) , and the RBF .

Initial condition and numerical solutions at t=6, 12, and 18 for the line soliton in an inhomogeneous medium: surface plots and contours of sin(u/2).

4.3. Circular Ring Soliton

The circular ring soliton can be simulated by solving the sine-Gordon equation for ψx1,x2=1 and β=0, with initial conditions [7, 911, 21, 23](56)fx1,x2=4tan-1exp3-x12+x22,gx1,x2=0,x1,x2-7,7and boundary condition(57)ux,tn=0,xΓ,  t0

Figure 5 depicts the results of this problem for τ=0.01 and h=0.5 at times t=0, 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 sinu/2. It can be observed from this figure that the ring soliton shrinks at the initial stage (t=0). Then, as t goes on, oscillations and radiations start to form and continue to form up to t=8.4. And an expanding phase is observed at t=5.6 and after it. The expansion continues until t=11.2, where the ring soliton is nearly formed again. When t=12.6, the ring soliton appears to be again in its shrinking phase. These graphical results match those obtained by the FEM , the BEM [9, 10], the DQM , the RBF , and the mesh-free reproducing kernel particle Ritz method .

Initial condition and numerical solutions at t=5.6, 8.4, 11.2, and 12.6 for the circular ring soliton: surface plots (left panel) and contours (right panel) of sin(u/2).

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 ψx1,x2=1 with initial conditions [6, 7, 10, 21](58)fx1,x2=4tan-1exp4-x1+32+x2+320.436,xΩ(59)gx1,x2=4.13sech4-x1+32+x2+320.436,xΩand boundary condition(60)ux,tn=0,xΓ,  t0

As in [6, 7, 10, 21], the solution is computed by the SBM over -10x110 and -7x27 and then is expanded at the lines x1=-10 and x2=-7 by symmetry relations.

When the dissipative parameter β=0.05, Figure 6 depicts the results for τ=0.01 and h=0.5 at times t=0, 4, 8, and 11 in terms of sinu/2. 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 , the FDM , the BEM , and the RBF .

Results of (a) initial condition (t=0) and numerical solutions at (b) t=4, (c) t=8, and (d) t=11 for the collision of two circular ring solitons when β=0.

The dissipative term β 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 t=8 for β=0.05, 0.35, 1 and 5. From Figures 6 and 7, we can observe that the ring soliton changes slowly as β increases. Additionally, it can be found that with β=0.35 the solitons at time t=8 are close to those given at t=4 for β=0.

Numerical solutions at t=8 for the collision of two circular ring solitons when (a) β=0.05, (b) β=0.35, (c) β=1, and (d) β=5.

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.

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