#### 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 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 with the initial conditions and the boundary conditions 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 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 Gram-Schmidt 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 sine-Gordon equation.

The approximate solution to the exact solution can be written in the form

where is the time-dependent 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 second-order 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 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 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 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 , , separately from (3.14) with Newton-Raphson iteration method. We iterate until the following criterion is satisfied:
where is the value of at the *k*th 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 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.

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*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.

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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 four-type plots at of Experiment 3.

**(a)**

**(b)**

**(c)**

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#### 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 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.