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

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


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 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 onedimensional 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.

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 P n is a sequence of polynomial, that is, that satisfy the orthogonality relationship 1 0 and standardization sign τ nkn −1 n−k .

2.3
The coefficient τ nkl of the polynomial P nk is defined by requirements 2.2 , 2.3 , and the Gram-Schmidt orthogonalization procedure without normalization .The explicit definition of the polynomial P nk 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 that This polynomial can be connected to a fixed set of the Jacobi polynomials P α,β n ξ 15 , that is, 2.7

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 u N x, t to the exact solution u x, t can be written in the form where a n is the time-dependent quantities which must be determined.In this paper, we apply orthogonal polynomial P nk for k 0. From 3.1 , we get

ISRN Applied Mathematics
Using 3.2 and substituting in 1.1 , we obtain where än and Pn0 denote the second-order derivatives of a n and P n0 with respect to t and x, respectively, wherein the collocation points are as x i i/N, i 0, 1, . . ., N. Equation 13 is rewritten as follows: For displaying this equation in matrix form, for i 1, 2, . . ., N − 1, and n 0, 1, . . ., N, we define

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:

3.6
From 3.6 , we get

3.7
Therefore, by using 3.6 , and 3.7 , a new system is obtained as follows: 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: Also, the vectors S and G t are defined as follows:

3.11
The nonlinear second-order system of ODEs 3.8 can be solved numerically using the fourthorder A-stable DIRKN method 3, 16 .From the initial conditions 1.2 , we determine the initial vectors A 0 and Ȧ0 .Using 3.1 , we get ȧn t 0 P n0 x i , 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 MA 0 b 1 and M Ȧ0 b 2 , where

3.14
where Q i is the internal stages, A n 1 and Ȧn 1 represent approximations to A t n 1 and Ȧ t n 1 , respectively.Also we define Δt t 0 jΔt, j 0, 1, . . ., M, 3.15 where Δt T − t 0 /M.The RKN method can be denoted in Butcher , s notation by the table of coefficients: In the DIRKN method, a ij 0 as i < j and a ii 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 Q i , i 1, . . ., 4, separately from 3.14 with Newton-Raphson iteration method.We iterate until the following criterion is satisfied: where Q k i is the value of Q i at the kth iteration of the Newton-Raphson method.

Numerical Experiments
In this section, we present the results of numerical experiments using the method introduced in Section 3. The L 2 and L ∞ error norms are defined as follows:

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 t 0 0 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 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 t 0 0 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 L 2 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 L 2 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 t 0 0 and the initial conditions with the boundary conditions 1.3 , which can be obtained from the following exact solution: 4.9 In Table 5, we represent the L 2 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.

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

Figure 1 :
Figure 1: a The exact and approximate solutions for N 16 are shown.b The L ∞ error norm for N 16 is shown.

Figure 2 :
Figure 2: a The exact and approximate solutions for N 16 are shown.b The L ∞ error norm for N 16 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.

9 dFigure 3 :
Figure 3: a The exact and approximate solutions for N 16 are shown.b The L ∞ error norm for N 16 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.

Table 2 :
The approximate solution for N 16 of Experiment 1.

Table 3 :
The L 2 error norm for different values of N of Experiment 2.

Table 4 :
The L ∞ error norm for different values of N of Experiment 2.

Table 5 :
Error norms for different values of N of Experiment 3.