International Scholarly Research Notices

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International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 462731 | 12 pages | https://doi.org/10.5402/2012/462731

The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial

Zoleikha Soori1 and Azim Aminataei 1

1Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

Academic Editor: P. J. García Nieto
Received17 Sep 2011
Accepted12 Oct 2011
Published02 Feb 2012

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 [3–5], 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[]×𝑡−sin(𝑢(𝑥,𝑡)),(𝑥,𝑡)∈0,10,,𝑇(1.1) with the initial conditions𝑢𝑥,𝑡0=𝑓0(𝑥),𝜕𝑢𝜕𝑡𝑥,𝑡0=𝑓1[](𝑥),𝑥∈0,1,(1.2) and the boundary conditions𝑢(0,𝑡)=𝑔0(𝑡),𝑢(1,𝑡)=𝑔1(𝑡),𝑡≥𝑡0.(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 [11–13].

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,𝑃𝑛𝑘≡𝑃𝑛𝑘(𝑥)=𝑛𝑙=𝑘𝜏𝑛𝑘𝑙𝑥𝑙,(2.1) that satisfy the orthogonality relationship10𝑃𝑛𝑘𝑃𝑛𝑙1𝑑𝑥=0,𝑘≠𝑙,(𝑘+𝑙+1),𝑘=𝑙,(2.2) and standardization𝜏sign𝑛𝑘𝑛=(−1)𝑛−𝑘.(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)=𝑛−𝑘𝑗=0(−1)𝑗⎛⎜⎜⎝𝑗⎞⎟⎟⎠⎛⎜⎜⎝⎞⎟⎟⎠𝑥𝑛−𝑘𝑛+𝑘+1+𝑗𝑛−𝑘𝑘+𝑗,𝑘=0,1,…,𝑛.(2.4) This yields Rodrigue’s type representation as follows:𝑃𝑛𝑘1(𝑥)=(1𝑛−𝑘)!𝑥𝑘+1𝑑𝑛−𝑘𝑑𝑥𝑛−𝑘𝑥𝑛+𝑘+1(1−𝑥)𝑛−𝑘,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)𝑘−𝑛(1−2𝑥).(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)=𝑁∑𝑛=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 obtain𝑁∑𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑥𝑖=𝑁∑𝑛=0𝑎𝑛̈𝑃(𝑡)𝑛0𝑥𝑖𝑈−sin𝑁𝑥𝑖,𝑡,𝑖=1,2,…,𝑁−1,(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,…,𝑁−1, 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,…,sin𝑁−1𝑇,𝑈𝑖=𝑈𝑁𝑥𝑖.,𝑡(3.5)

The matrices 𝑀 and 𝐾 are positive definite matrices of order (𝑁−1)×(𝑁−1). The system given in (3.4) consists of (𝑁−1) 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),𝑡≥𝑡0,𝑈𝑁(1,𝑡)=𝑁𝑛=0𝑎𝑛(𝑡)𝑃𝑛0(1)=𝑔1(𝑡),𝑡≥𝑡0.(3.6)

From (3.6), we get𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑑(0)=2𝑔0(𝑡)𝑑𝑡2,𝑡≥𝑡0,𝑁𝑛=0̈𝑎𝑛(𝑡)𝑃𝑛0𝑑(1)=2𝑔1(𝑡)𝑑𝑡2,𝑡≥𝑡0.(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)00𝑥1𝑝10𝑥1⋯𝑝𝑁0𝑥1𝑝⋮⋮⋱⋮00𝑥𝑁−1𝑝10𝑥𝑁−1⋯𝑝𝑁0𝑥𝑁−1𝑝00(1)𝑝10(1)⋯𝑝𝑁0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝(1),(3.9)𝐾=00⋯0̈𝑝00𝑥1̈𝑝10𝑥1⋯̈𝑝𝑁0𝑥1⋮⋮⋱⋮̈𝑝00𝑥𝑁−1̈𝑝10𝑥𝑁−1⋯̈𝑝𝑁0𝑥𝑁−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠00⋯0.(3.10) Also, the vectors 𝑆 and 𝐺(𝑡) are defined as follows:𝑈𝑆=0,sin1𝑈,sin2𝑈,…,sin𝑁−1,0𝑇,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+Δ𝑡2𝑠∑𝑗=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=(𝑇−𝑡0)/𝑀. The RKN method can be denoted in 𝐵𝑢𝑡𝑐ℎ𝑒𝑟,𝑠 notation by the table of coefficients:𝑐1𝑎11⋯𝑎1𝑠⋮𝑐⋮⋱⋮𝑠𝑎𝑠1⋯𝑎𝑠𝑠𝑏1⋯𝑏𝑠𝑏1⋯𝑏𝑠(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+1−𝑄𝑘𝑖‖‖∞<10−12,(3.17) where 𝑄𝑘𝑖 is the value of 𝑄𝑖 at the kth iteration of the Newton-Raphson method.

0.65820.24
0.93000.1864849679520.24
0.0700−3.1486640861952.9456280221830.24
0.34181.491936101409−1.8163048528350.1030070016890.24
0.1085722379060.0127646089720.1695869477700.209076205352
0.3176484432580.1823515567420.1823515567420.317648443258
Table 1 
Four-stage A-stable DIRKN 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=‖‖𝑢−𝑈𝑁‖‖=𝑁𝑖=0𝑢𝑖−𝑈𝑁𝑖21/2,𝐿∞=‖‖𝑢−𝑈𝑁‖‖=max0≤𝑖≤𝑁||𝑢𝑖−U𝑁𝑖||.(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.

𝑥 𝑇 = 0 . 1 𝑇 = 0 . 2 𝑇 = 0 . 3 𝑇 = 0 . 4 𝑇 = 0 . 5
0.0 0.3986746100 0.7895822394 1.1658271779 1.5220255084 1.8545904360
0.2 0.3908822100 0.7744385442 1.1441349052 1.4948183070 1.8229915644
0.4 0.3689530692 0.7311732829 1.0827641117 1.4175301638 1.7328129513
0.6 0.3366233389 0.6685450323 0.9914459040 1.3016971244 1.5965446689
0.8 0.2985244887 0.5937600941 0.8826302804 1.1624500568 1.4310481369
1.0 0.2588597329 0.5155692281 0.7680835230 1.0145556439 1.2534069046
Table 2 
The approximate solution for 𝑁=16 of Experiment 1.
(a)
(a)
(b)
(b)
Figure 1 
(a) The exact and approximate solutions for 𝑁 = 1 6 are shown. (b) The 𝐿 ∞ error norm for 𝑁 = 1 6 is shown.

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+𝑐2𝑥sech√1+𝑐2,(4.5) with the boundary conditions (1.3), which can be obtained from the following exact solution: 𝑐𝑢(𝑥,𝑡)=4arctan−1sin𝑐𝑡√1+𝑐2𝑥sech√1+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.

T 𝑁 = 4 𝑁 = 8 𝑁 = 1 6
0.1 5 . 0 2 × 1 0 − 6 9 . 1 1 × 1 0 − 9 9 . 5 6 × 1 0 − 1 5
0.2 5 . 3 7 × 1 0 − 5 1 . 0 0 × 1 0 − 8 1 . 3 5 × 1 0 − 1 3
0.3 2 . 0 5 × 1 0 − 4 3 . 6 7 × 1 0 − 7 2 . 2 2 × 1 0 − 1 3
0.4 5 . 0 0 × 1 0 − 4 8 . 3 7 × 1 0 − 6 9 . 1 0 × 1 0 − 1 3
0.5 9 . 4 6 × 1 0 − 4 1 . 2 9 × 1 0 − 6 3 . 4 3 × 1 0 − 1 2
0.6 1 . 5 0 × 1 0 − 3 1 . 5 2 × 1 0 − 6 1 . 7 5 × 1 0 − 1 1
0.7 2 . 1 0 × 1 0 − 3 1 . 5 8 × 1 0 − 6 1 . 2 9 × 1 0 − 1 0
0.8 2 . 6 5 × 1 0 − 3 1 . 6 5 × 1 0 − 6 8 . 0 8 × 1 0 − 1 0
0.9 3 . 0 8 × 1 0 − 3 1 . 9 4 × 1 0 − 6 4 . 9 4 × 1 0 − 9
1.0 3 . 3 7 × 1 0 − 3 2 . 1 3 × 1 0 − 6 2 . 2 6 × 1 0 − 8
Table 3 
The 𝐿2 error norm for different values of 𝑁 of Experiment 2.
𝑇 𝑁 = 4 𝑁 = 8 𝑁 = 1 6
0.1 1 . 3 0 × 1 0 − 6 2 . 3 7 × 1 0 − 9 2 . 6 8 × 1 0 − 1 5
0.2 9 . 7 9 × 1 0 − 6 1 . 8 3 × 1 0 − 8 2 . 2 3 × 1 0 − 1 4
0.3 2 . 9 8 × 1 0 − 5 5 . 3 6 × 1 0 − 8 2 . 9 7 × 1 0 − 1 4
0.4 6 . 1 0 × 1 0 − 5 9 . 1 4 × 1 0 − 8 2 . 3 1 × 1 0 − 1 3
0.5 9 . 8 2 × 1 0 − 5 1 . 0 0 × 1 0 − 7 7 . 9 4 × 1 0 − 1 3
0.6 1 . 3 3 × 1 0 − 4 1 . 0 0 × 1 0 − 7 2 . 3 8 × 1 0 − 1 2
0.7 1 . 5 7 × 1 0 − 4 1 . 0 0 × 1 0 − 7 1 . 2 9 × 1 0 − 1 1
0.8 1 . 6 3 × 1 0 − 4 1 . 0 0 × 1 0 − 7 1 . 1 4 × 1 0 − 1 0
0.9 1 . 6 3 × 1 0 − 4 1 . 1 6 × 1 0 − 7 8 . 0 3 × 1 0 − 1 0
1.0 1 . 8 0 × 1 0 − 4 1 . 1 6 × 1 0 − 7 4 . 5 9 × 1 0 − 9
Table 4 
The 𝐿∞ error norm for different values of 𝑁 of Experiment 2.
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
Figure 2 
(a) The exact and approximate solutions for 𝑁 = 1 6 are shown. (b) The 𝐿 ∞ error norm for 𝑁 = 1 6 is shown. (c) The 𝐿 2 error norm for different values of 𝑁 is shown. (d) The 𝐿 ∞ error norm for different values of 𝑁 is shown.

Experiment 3. In experiment [3, 9], we consider (1.1) with 𝑡0=0 and the initial conditions 𝑓0𝑥(𝑥)=4arctan𝑐sinh√1−𝑐2𝑓,(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/1−𝑐2√cosh𝑐𝑡/1−𝑐2⎞⎟⎟⎠,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.

𝑁 𝐿 2 error norm 𝐿 ∞ error norm
4 9 . 1 3 × 1 0 − 3 7 . 3 6 × 1 0 − 4
6 5 . 2 9 × 1 0 − 4 3 . 7 7 × 1 0 − 5
8 4 . 1 3 × 1 0 − 5 2 . 9 4 × 1 0 − 6
10 9 . 3 1 × 1 0 − 7 7 . 3 0 × 1 0 − 8
12 2 . 0 0 × 1 0 − 7 3 . 8 2 × 1 0 − 8
14 3 . 2 3 × 1 0 − 8 7 . 2 1 × 1 0 − 9
16 1 . 8 7 × 1 0 − 8 3 . 9 4 × 1 0 − 9
Table 5 
Error norms for different values of 𝑁 of Experiment 3.
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
Figure 3 
(a) The exact and approximate solutions for 𝑁 = 1 6 are shown. (b) The 𝐿 ∞ error norm for 𝑁 = 1 6 is shown. (c) The 𝐿 2 error norm for different values of 𝑁 is shown. (d) The 𝐿 ∞ error norm for different values of 𝑁 is shown.

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.

References

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Copyright © 2012 Zoleikha Soori and Azim Aminataei. 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.

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