Research Article  Open Access
The Spectral Method for Solving SineGordon Equation Using a New Orthogonal Polynomial
Abstract
We have presented a numerical scheme for solving onedimensional nonlinear sineGordon equation. We apply the spectral method with a basis of a new orthogonal polynomial which is orthogonal over the interval with weighting function one. The results show the accuracy and efficiency of the proposed method.
1. Introduction
The sineGordon 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 sineGordon 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 onedimensional nonlinear sineGordon equation with the initial conditions and the boundary conditions In the last decade, several numerical schemes have been developed for solving (1.1), for instance, highorder solution of onedimensional sineGordon 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 onedimensional sineGordon equation and a thirdorder numerical scheme for the twodimensional sineGordon equation in [8, 9], respectively. Also, in [10], a numerical scheme using radial basis function for the solution of twodimensional sineGordon 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 be a fixed whole number, and is a sequence of polynomial, that is, that satisfy the orthogonality relationship and standardization The coefficient of the polynomial is defined by requirements (2.2), (2.3), and the GramSchmidt orthogonalization procedure (without normalization).The explicit definition of the polynomial is as follows: This yields Rodrigue’s type representation as follows:
and the orthogonality relationship (2.2) is confirmed by applying (2.5). It also follows from (2.5) thatThis polynomial can be connected to a fixed set of the Jacobi polynomials [15], that is,
3. The Proposed Method
In this section, the introduced orthogonal polynomial is applied as the basis for spectral method to solve sineGordon equation.
The approximate solution to the exact solution can be written in the form
where is the timedependent quantities which must be determined. In this paper, we apply orthogonal polynomial for . From (3.1), we get
Using (3.2) and substituting in (1.1), we obtain
where and denote the secondorder derivatives of and with respect to and , respectively, wherein the collocation points are as ,.
Equation (13) is rewritten as follows:
For displaying this equation in matrix form, for , and , we define
The matrices and are positive definite matrices of order . The system given in (3.4) consists of equations in 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:
From (3.6), we get
Therefore, by using (3.6), and (3.7), a new system is obtained as follows:
where , , , and are vectors of order with components. The matrices and are of order , which are defined as follows: Also, the vectors and are defined as follows: The nonlinear secondorder system of ODEs (3.8) can be solved numerically using the fourthorder Astable DIRKN method [3, 16]. From the initial conditions (1.2), we determine the initial vectors and . Using (3.1), we get
Equation (3.12) is system of equations in unknowns. These equations can be written in the matrix form and , where
3.1. The DIRKN Method
For solving (3.8), we use DIRKN method [3, 16]:
where is the internal stages, and represent approximations to and , respectively. Also we definewhere . The RKN method can be denoted in notation by the table of coefficients:
In the DIRKN method, as and are equal. Also, in the above table of coefficients, is the number of stages. We use the fourstage Astable 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 , , separately from (3.14) with NewtonRaphson iteration method. We iterate until the following criterion is satisfied: where is the value of at the kth iteration of the NewtonRaphson method.

4. Numerical Experiments
In this section, we present the results of numerical experiments using the method introduced in Section 3. The and error norms are defined as follows:
We choose , 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 and the initial conditions with the boundary conditions (1.3), which can be obtained from the following exact solution: In Table 2, the approximate solution for several final times () with different locations is shown. Figure 1 shows two type plots at of Experiment 1.

(a)
(b)
Experiment 2. In this experiment [3, 8], we consider (1.1) with and the initial conditions
with the boundary conditions (1.3), which can be obtained from the following exact solution:
In Tables 3 and 4, we show the and error norms, for several final times (). As we see from Tables 3 and 4, for the given values of , the error norm is less than the error norm. Figure 2 shows four type plots at of Experiment 2.


(a)
(b)
(c)
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Experiment 3. In experiment [3, 9], we consider (1.1) with and the initial conditions
with the boundary conditions (1.3), which can be obtained from the following exact solution:
In Table 5, we represent the and error norms for different values of at . With increasing , we obtain better results, and these error norms decrease five orders in magnitude. Figure 3 shows fourtype plots at of Experiment 3.

(a)
(b)
(c)
(d)
5. Conclusion
In this paper, we have presented the spectral method for solving onedimensional nonlinear sineGordon equation using a new orthogonal polynomial. Numerical experiments show that the spectral method is an efficient one, and the results for value of is more accurate than the other values of . The proposed method has higher order of accuracy even for small values of (i.e., ). We obtain better results when we choose small value of , in comparison to those obtained in the literature.
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Copyright
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.