Abstract

We have presented a numerical scheme for solving one-dimensional nonlinear sine-Gordon equation. We apply the spectral method with a basis of a new orthogonal polynomial which is orthogonal over the interval [0,1] with weighting function one. The results show the accuracy and efficiency of the proposed method.

1. Introduction

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation (PDE). It was originally considered in the nineteenth century in the course of surface of constant negative curvature. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions [1, 2]. The sine-Gordon equation appears in a number of physical applications [35], including applications in the chain of coupled pendulums and modelling the propagation of transverse electromagnetic (TEM) wave on a superconductor transmission system. Consider the one-dimensional nonlinear sine-Gordon equation𝜕2𝑢(𝑥,𝑡)𝜕𝑡2=𝜕2𝑢(𝑥,𝑡)𝜕𝑥2[]×𝑡sin(𝑢(𝑥,𝑡)),(𝑥,𝑡)0,10,,𝑇(1.1) with the initial conditions𝑢𝑥,𝑡0=𝑓0(𝑥),𝜕𝑢𝜕𝑡𝑥,𝑡0=𝑓1[](𝑥),𝑥0,1,(1.2) and the boundary conditions𝑢(0,𝑡)=𝑔0(𝑡),𝑢(1,𝑡)=𝑔1(𝑡),𝑡𝑡0.(1.3) In the last decade, several numerical schemes have been developed for solving (1.1), for instance, high-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods [3]. The authors of [6] proposed a numerical scheme for solving (1.1) using collocation and radial basis functions. Also, the boundary integral equation approach is used in [7]. Bratsos has proposed a numerical method for solving one-dimensional sine-Gordon equation and a third-order numerical scheme for the two-dimensional sine-Gordon equation in [8, 9], respectively. Also, in [10], a numerical scheme using radial basis function for the solution of two-dimensional sine-Gordon equation is used. In addition, several authors proposed spectral methods and Fourier pseudospectral method for solving nonlinear wave equation using a discrete Fourier series and Chebyshev orthogonal polynomials [1113].

In this paper, we have proposed a numerical scheme for solving (1.1) using spectral method with a basis of a new orthogonal polynomial. This polynomial is introduced by Chelyshkov in [14].

The outline of this paper is as follows. In Section 2, we introduce a new orthogonal polynomial. In Section 3, we apply spectral method for solving (1.1). In Section 4, we show the numerical results. Finally, a brief conclusion is given in Section 5.

2. The New Orthogonal Polynomial

In this section, we briefly introduce the new orthogonal polynomial, that is introduced in [14].

Let 𝑛 be a fixed whole number, and 𝑃𝑛 is a sequence of polynomial, that is,𝑃𝑛=𝑃𝑛𝑘0𝑘=𝑛,𝑃𝑛𝑘𝑃𝑛𝑘(𝑥)=𝑛𝑙=𝑘𝜏𝑛𝑘𝑙𝑥𝑙,(2.1) that satisfy the orthogonality relationship10𝑃𝑛𝑘𝑃𝑛𝑙1𝑑𝑥=0,𝑘𝑙,(𝑘+𝑙+1),𝑘=𝑙,(2.2) and standardization𝜏sign𝑛𝑘𝑛=(1)𝑛𝑘.(2.3) The coefficient 𝜏𝑛𝑘𝑙 of the polynomial 𝑃𝑛𝑘 is defined by requirements (2.2), (2.3), and the Gram-Schmidt orthogonalization procedure (without normalization).The explicit definition of the polynomial 𝑃𝑛𝑘 is as follows:𝑃𝑛𝑘(𝑥)=𝑛𝑘𝑗=0(1)𝑗𝑗𝑥𝑛𝑘𝑛+𝑘+1+𝑗𝑛𝑘𝑘+𝑗,𝑘=0,1,,𝑛.(2.4) This yields Rodrigue’s type representation as follows:𝑃𝑛𝑘1(𝑥)=(1𝑛𝑘)!𝑥𝑘+1𝑑𝑛𝑘𝑑𝑥𝑛𝑘𝑥𝑛+𝑘+1(1𝑥)𝑛𝑘,𝑘=0,1,,𝑛,(2.5)

and the orthogonality relationship (2.2) is confirmed by applying (2.5). It also follows from (2.5) that10𝑃𝑛𝑘(𝑥)𝑑𝑥=10𝑥𝑛1𝑑𝑥=.𝑛+1(2.6)This polynomial can be connected to a fixed set of the Jacobi polynomials 𝑃𝑛(𝛼,𝛽)(𝜉) [15], that is,𝑃𝑛𝑘=𝑥𝑛𝑃(2𝑛,0)𝑘𝑛(12𝑥).(2.7)

3. The Proposed Method

In this section, the introduced orthogonal polynomial is applied as the basis for spectral method to solve sine-Gordon equation.

The approximate solution 𝑢𝑁(𝑥,𝑡) to the exact solution 𝑢(𝑥,𝑡) can be written in the form𝑢(𝑥,𝑡)𝑈𝑁(𝑥,𝑡)=𝑁𝑛=0𝑎𝑛(𝑡)𝑃𝑛0(𝑥),(3.1)

where 𝑎𝑛 is the time-dependent quantities which must be determined. In this paper, we apply orthogonal polynomial 𝑃𝑛𝑘 for 𝑘=0. From (3.1), we get𝑈𝑁𝑡𝑡=𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑈(𝑥),𝑁𝑥𝑥=𝑁𝑛=0𝑎𝑛̈𝑃(𝑡)𝑛0(𝑥).(3.2)

Using (3.2) and substituting in (1.1), we obtain𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑥𝑖=𝑁𝑛=0𝑎𝑛̈𝑃(𝑡)𝑛0𝑥𝑖𝑈sin𝑁𝑥𝑖,𝑡,𝑖=1,2,,𝑁1,(3.3)

where ̈𝑎𝑛 and ̈𝑃𝑛0 denote the second-order derivatives of 𝑎𝑛 and 𝑃𝑛0 with respect to 𝑡 and 𝑥, respectively, wherein the collocation points are as 𝑥𝑖=𝑖/𝑁,𝑖=0,1,,𝑁.

Equation (13) is rewritten as follows:𝑀̈𝐴(𝑡)=𝐾𝐴(𝑡)𝑆1.(3.4)

For displaying this equation in matrix form, for 𝑖=1,2,,𝑁1, and 𝑛=0,1,,𝑁, we define 𝑎𝐴(𝑡)=0(𝑡),𝑎1(𝑡),,𝑎𝑁(𝑡)𝑇,𝑀=𝑀in=𝑃𝑛0𝑥𝑖,𝐾=𝐾in=̈𝑃𝑛0𝑥𝑖,𝑆1=𝑈sin1𝑈,sin2𝑈,,sin𝑁1𝑇,𝑈𝑖=𝑈𝑁𝑥𝑖.,𝑡(3.5)

The matrices 𝑀 and 𝐾 are positive definite matrices of order (𝑁1)×(𝑁1). The system given in (3.4) consists of (𝑁1) equations in (𝑁+1) unknowns. To obtain a unique solution for this system, we need two additional equations. For this, we add two boundary conditions, which are given by (1.3), in the following forms:𝑈𝑁(0,𝑡)=𝑁𝑛=0𝑎𝑛(𝑡)𝑃𝑛0(0)=𝑔0(𝑡),𝑡𝑡0,𝑈𝑁(1,𝑡)=𝑁𝑛=0𝑎𝑛(𝑡)𝑃𝑛0(1)=𝑔1(𝑡),𝑡𝑡0.(3.6)

From (3.6), we get𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑑(0)=2𝑔0(𝑡)𝑑𝑡2,𝑡𝑡0,𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑑(1)=2𝑔1(𝑡)𝑑𝑡2,𝑡𝑡0.(3.7)

Therefore, by using (3.6), and (3.7), a new system is obtained as follows:𝑀̈𝐴(𝑡)=𝐹(𝑡,𝐴(𝑡)),𝐹(𝑡,𝐴(𝑡))=𝐾𝐴(𝑡)𝑆+𝐺(𝑡),(3.8)

where 𝐴(𝑡), 𝐹(𝑡,𝐴(𝑡)), 𝑆, and 𝐺(𝑡) are vectors of order (𝑁+1) with (𝑁+1) components. The matrices 𝑀 and 𝐾 are of order (𝑁+1)×(𝑁+1), which are defined as follows:𝑝𝑀=00(0)𝑝10(0)𝑝𝑁0𝑝(0)00𝑥1𝑝10𝑥1𝑝𝑁0𝑥1𝑝00𝑥𝑁1𝑝10𝑥𝑁1𝑝𝑁0𝑥𝑁1𝑝00(1)𝑝10(1)𝑝𝑁0(1),(3.9)𝐾=000̈𝑝00𝑥1̈𝑝10𝑥1̈𝑝𝑁0𝑥1̈𝑝00𝑥𝑁1̈𝑝10𝑥𝑁1̈𝑝𝑁0𝑥𝑁1000.(3.10) Also, the vectors 𝑆 and 𝐺(𝑡) are defined as follows:𝑈𝑆=0,sin1𝑈,sin2𝑈,,sin𝑁1,0𝑇,𝑑𝐺(𝑡)=2𝑔0(𝑡)𝑑𝑡2𝑑,0,,0,2𝑔1(𝑡)𝑑𝑡2𝑇.(3.11) The nonlinear second-order system of ODEs (3.8) can be solved numerically using the fourth-order A-stable DIRKN method [3, 16]. From the initial conditions (1.2), we determine the initial vectors 𝐴0 and ̇𝐴0. Using (3.1), we get𝑈𝑁𝑥𝑖=,0𝑁𝑛=0𝑎𝑛𝑡0𝑃𝑛0𝑥𝑖,𝑖=0,1,,𝑁,𝜕𝑈𝑁𝑥𝜕𝑡𝑖=,0𝑁𝑛=0̇𝑎𝑛𝑡0𝑃𝑛0𝑥𝑖,𝑖=0,1,,𝑁.(3.12)

Equation (3.12) is system of (𝑁+1) equations in (𝑁+1) unknowns. These equations can be written in the matrix form 𝑀𝐴0=𝑏1 and 𝑀̇𝐴0=𝑏2, where 𝑏1=𝑓0𝑥0,,𝑓0𝑥𝑁𝑇,𝑏2=𝑓1𝑥0,,𝑓1𝑥𝑁𝑇.(3.13)

3.1. The DIRKN Method

For solving (3.8), we use DIRKN method [3, 16]:𝑀𝑄𝑖=𝑀𝐴𝑛+𝑐𝑖̇𝐴Δ𝑡𝑀𝑛+Δ𝑡2𝑠𝑗=1𝑎𝑖𝑗𝐹𝑡𝑛+𝑐𝑗Δ𝑡,𝑄𝑗,𝑖=1,,𝑠,𝑀𝐴𝑛+1=𝑀𝐴𝑛+𝑐𝑖̇𝐴Δ𝑡𝑀𝑛+Δ𝑡2𝑠𝑗=1𝑏𝑖𝐹𝑡𝑛+𝑐𝑗Δ𝑡,𝑄𝑗,𝑀̇𝐴𝑛+1̇𝐴=𝑀𝑛+Δ𝑡𝑠𝑗=1𝑏𝑖𝐹𝑡𝑛+𝑐𝑗Δ𝑡,𝑄𝑗,(3.14)

where 𝑄𝑖 is the internal stages, 𝐴𝑛+1 and ̇𝐴𝑛+1 represent approximations to 𝐴(𝑡𝑛+1) and ̇𝐴(𝑡𝑛+1), respectively. Also we defineΔ𝑡=𝑡0+𝑗Δ𝑡,𝑗=0,1,,𝑀,(3.15)where Δ𝑡=(𝑇𝑡0)/𝑀. The RKN method can be denoted in 𝐵𝑢𝑡𝑐𝑒𝑟,𝑠 notation by the table of coefficients:𝑐1𝑎11𝑎1𝑠𝑐𝑠𝑎𝑠1𝑎𝑠𝑠𝑏1𝑏𝑠𝑏1𝑏𝑠(3.16)

In the DIRKN method, 𝑎𝑖𝑗=0 as (𝑖<𝑗) and 𝑎𝑖𝑖 are equal. Also, in the above table of coefficients, 𝑠 is the number of stages. We use the four-stage A-stable DIRKN method [16], with algebraic fourth order, which is defined by the following table of coefficients for solving all experiments.

Using Table 1, we can calculate 𝑄𝑖, 𝑖=1,,4, separately from (3.14) with Newton-Raphson iteration method. We iterate until the following criterion is satisfied:𝑄𝑖𝑘+1𝑄𝑘𝑖<1012,(3.17) where 𝑄𝑘𝑖 is the value of 𝑄𝑖 at the kth iteration of the Newton-Raphson method.

4. Numerical Experiments

In this section, we present the results of numerical experiments using the method introduced in Section 3. The 𝐿2 and 𝐿 error norms are defined as follows:𝐿2=𝑢𝑈𝑁=𝑁𝑖=0𝑢𝑖𝑈𝑁𝑖21/2,𝐿=𝑢𝑈𝑁=max0𝑖𝑁||𝑢𝑖U𝑁𝑖||.(4.1)

We choose Δ𝑡=1/1000, for solving all experiments.

The computations associated with the experiments discussed above were performed in Maple 13 on a PC with a CPU of 2.4 GHZ.

Experiment 1. In this experiment [3, 6, 10], we consider (1.1) with 𝑡0=0 and the initial conditions 𝑓0𝑓(𝑥)=0,1(𝑥)=4sech(𝑥),(4.2) with the boundary conditions (1.3), which can be obtained from the following exact solution: 𝑢(𝑥,𝑡)=4arctan(sech(𝑥)t).(4.3) In Table 2, the approximate solution for several final times (𝑇) with different locations is shown. Figure 1 shows two type plots at 𝑡=0.5 of Experiment 1.

Experiment 2. In this experiment [3, 8], we consider (1.1) with 𝑡0=0 and the initial conditions 𝑓0𝑓(𝑥)=0,(4.4)14(𝑥)=1+𝑐2𝑥sech1+𝑐2,(4.5) with the boundary conditions (1.3), which can be obtained from the following exact solution: 𝑐𝑢(𝑥,𝑡)=4arctan1sin𝑐𝑡1+𝑐2𝑥sech1+𝑐2,𝑐=1.(4.6)
In Tables 3 and 4, we show the 𝐿2 and 𝐿 error norms, for several final times (𝑇). As we see from Tables 3 and 4, for the given values of 𝑁, the 𝐿 error norm is less than the 𝐿2 error norm. Figure 2 shows four type plots at 𝑡=0.5 of Experiment 2.

Experiment 3. In experiment [3, 9], we consider (1.1) with 𝑡0=0 and the initial conditions 𝑓0𝑥(𝑥)=4arctan𝑐sinh1𝑐2𝑓,(4.7)1(𝑥)=0,(4.8) with the boundary conditions (1.3), which can be obtained from the following exact solution: 𝑢(𝑥,𝑡)=4arctan𝑐sinh𝑥/1𝑐2cosh𝑐𝑡/1𝑐2,𝑐=0.5.(4.9)
In Table 5, we represent the 𝐿2 and 𝐿 error norms for different values of 𝑁 at 𝑡=0.5. With increasing 𝑁, we obtain better results, and these error norms decrease five orders in magnitude. Figure 3 shows four-type plots at 𝑡=0.5 of Experiment 3.

5. Conclusion

In this paper, we have presented the spectral method for solving one-dimensional nonlinear sine-Gordon equation using a new orthogonal polynomial. Numerical experiments show that the spectral method is an efficient one, and the results for value of 𝑁=16 is more accurate than the other values of 𝑁. The proposed method has higher order of accuracy even for small values of 𝑁 (i.e., 𝑁=4). We obtain better results when we choose small value of Δ𝑡, in comparison to those obtained in the literature.