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 [3ā€“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 equationšœ•2š‘¢(š‘„,š‘”)šœ•š‘”2=šœ•2š‘¢(š‘„,š‘”)šœ•š‘„2[]Ɨī€ŗš‘”āˆ’sin(š‘¢(š‘„,š‘”)),(š‘„,š‘”)āˆˆ0,10ī€»,,š‘‡(1.1) with the initial conditionsš‘¢ī€·š‘„,š‘”0ī€ø=š‘“0(š‘„),šœ•š‘¢ī€·šœ•š‘”š‘„,š‘”0ī€ø=š‘“1[](š‘„),š‘„āˆˆ0,1,(1.2) and the boundary conditionsš‘¢(0,š‘”)=š‘”0(š‘”),š‘¢(1,š‘”)=š‘”1(š‘”),š‘”ā‰„š‘”0.(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 [11ā€“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 š‘› be a fixed whole number, and š‘ƒš‘› is a sequence of polynomial, that is,š‘ƒš‘›=ī€½š‘ƒš‘›š‘˜ī€¾0š‘˜=š‘›,š‘ƒš‘›š‘˜ā‰”š‘ƒš‘›š‘˜(š‘„)=š‘›ī“š‘™=š‘˜šœš‘›š‘˜š‘™š‘„š‘™,(2.1) that satisfy the orthogonality relationshipī€œ10š‘ƒš‘›š‘˜š‘ƒš‘›š‘™īƒÆ1š‘‘š‘„=0,š‘˜ā‰ š‘™,(š‘˜+š‘™+1),š‘˜=š‘™,(2.2) and standardizationī€·šœsignš‘›š‘˜š‘›ī€ø=(āˆ’1)š‘›āˆ’š‘˜.(2.3) The coefficient šœš‘›š‘˜š‘™ of the polynomial š‘ƒš‘›š‘˜ is defined by requirements (2.2), (2.3), and the Gram-Schmidt orthogonalization procedure (without normalization).The explicit definition of the polynomial š‘ƒš‘›š‘˜ is as follows:š‘ƒš‘›š‘˜(š‘„)=š‘›āˆ’š‘˜ī“š‘—=0(āˆ’1)š‘—āŽ›āŽœāŽœāŽš‘—āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ š‘„š‘›āˆ’š‘˜š‘›+š‘˜+1+š‘—š‘›āˆ’š‘˜š‘˜+š‘—,š‘˜=0,1,ā€¦,š‘›.(2.4) This yields Rodrigueā€™s type representation as follows:š‘ƒš‘›š‘˜1(š‘„)=(1š‘›āˆ’š‘˜)!š‘„š‘˜+1š‘‘š‘›āˆ’š‘˜š‘‘š‘„š‘›āˆ’š‘˜ī€·š‘„š‘›+š‘˜+1(1āˆ’š‘„)š‘›āˆ’š‘˜ī€ø,š‘˜=0,1,ā€¦,š‘›,(2.5)

and the orthogonality relationship (2.2) is confirmed by applying (2.5). It also follows from (2.5) thatī€œ10š‘ƒš‘›š‘˜ī€œ(š‘„)š‘‘š‘„=10š‘„š‘›1š‘‘š‘„=.š‘›+1(2.6)This polynomial can be connected to a fixed set of the Jacobi polynomials š‘ƒš‘›(š›¼,š›½)(šœ‰) [15], that is,š‘ƒš‘›š‘˜=š‘„š‘›š‘ƒ(2š‘›,0)š‘˜āˆ’š‘›(1āˆ’2š‘„).(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 š‘¢š‘(š‘„,š‘”) to the exact solution š‘¢(š‘„,š‘”) can be written in the formš‘¢(š‘„,š‘”)ā‰…š‘ˆš‘(š‘„,š‘”)=š‘āˆ‘š‘›=0š‘Žš‘›(š‘”)š‘ƒš‘›0(š‘„),(3.1)

where š‘Žš‘› is the time-dependent quantities which must be determined. In this paper, we apply orthogonal polynomial š‘ƒš‘›š‘˜ for š‘˜=0. From (3.1), we getš‘ˆš‘š‘”š‘”=š‘ī“š‘›=0Ģˆš‘Žš‘›(š‘”)š‘ƒš‘›0š‘ˆ(š‘„),š‘š‘„š‘„=š‘ī“š‘›=0š‘Žš‘›Ģˆš‘ƒ(š‘”)š‘›0(š‘„).(3.2)

Using (3.2) and substituting in (1.1), we obtainš‘āˆ‘š‘›=0Ģˆš‘Žš‘›(š‘”)š‘ƒš‘›0ī€·š‘„š‘–ī€ø=š‘āˆ‘š‘›=0š‘Žš‘›Ģˆš‘ƒ(š‘”)š‘›0ī€·š‘„š‘–ī€øī€·š‘ˆāˆ’sinš‘ī€·š‘„š‘–,š‘”ī€øī€ø,š‘–=1,2,ā€¦,š‘āˆ’1,(3.3)

where Ģˆš‘Žš‘› and Ģˆš‘ƒš‘›0 denote the second-order derivatives of š‘Žš‘› and š‘ƒš‘›0 with respect to š‘” and š‘„, respectively, wherein the collocation points are as š‘„š‘–=š‘–/š‘,š‘–=0,1,ā€¦,š‘.

Equation (13) is rewritten as follows:š‘€Ģˆš“(š‘”)=š¾š“(š‘”)āˆ’š‘†1.(3.4)

For displaying this equation in matrix form, for š‘–=1,2,ā€¦,š‘āˆ’1, and š‘›=0,1,ā€¦,š‘, we define ī€ŗš‘Žš“(š‘”)=0(š‘”),š‘Ž1(š‘”),ā€¦,š‘Žš‘ī€»(š‘”)š‘‡,š‘€=š‘€in=š‘ƒš‘›0ī€·š‘„š‘–ī€ø,š¾=š¾in=Ģˆš‘ƒš‘›0ī€·š‘„š‘–ī€ø,š‘†1=ī€ŗī€·š‘ˆsin1ī€øī€·š‘ˆ,sin2ī€øī€·š‘ˆ,ā€¦,sinš‘āˆ’1ī€øī€»š‘‡,š‘ˆš‘–=š‘ˆš‘ī€·š‘„š‘–ī€ø.,š‘”(3.5)

The matrices š‘€ and š¾ are positive definite matrices of order (š‘āˆ’1)Ɨ(š‘āˆ’1). The system given in (3.4) consists of (š‘āˆ’1) equations in (š‘+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:š‘ˆš‘(0,š‘”)=š‘ī“š‘›=0š‘Žš‘›(š‘”)š‘ƒš‘›0(0)=š‘”0(š‘”),š‘”ā‰„š‘”0,š‘ˆš‘(1,š‘”)=š‘ī“š‘›=0š‘Žš‘›(š‘”)š‘ƒš‘›0(1)=š‘”1(š‘”),š‘”ā‰„š‘”0.(3.6)

From (3.6), we getš‘ī“š‘›=0Ģˆš‘Žš‘›(š‘”)š‘ƒš‘›0š‘‘(0)=2š‘”0(š‘”)š‘‘š‘”2,š‘”ā‰„š‘”0,š‘ī“š‘›=0Ģˆš‘Žš‘›(š‘”)š‘ƒš‘›0š‘‘(1)=2š‘”1(š‘”)š‘‘š‘”2,š‘”ā‰„š‘”0.(3.7)

Therefore, by using (3.6), and (3.7), a new system is obtained as follows:š‘€Ģˆš“(š‘”)=š¹(š‘”,š“(š‘”)),š¹(š‘”,š“(š‘”))=š¾š“(š‘”)āˆ’š‘†+šŗ(š‘”),(3.8)

where š“(š‘”), š¹(š‘”,š“(š‘”)), š‘†, and šŗ(š‘”) are vectors of order (š‘+1) with (š‘+1) components. The matrices š‘€ and š¾ are of order (š‘+1)Ɨ(š‘+1), which are defined as follows:āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽš‘š‘€=00(0)š‘10(0)ā‹Æš‘š‘0š‘(0)00ī€·š‘„1ī€øš‘10ī€·š‘„1ī€øā‹Æš‘š‘0ī€·š‘„1ī€øš‘ā‹®ā‹®ā‹±ā‹®00ī€·š‘„š‘āˆ’1ī€øš‘10ī€·š‘„š‘āˆ’1ī€øā‹Æš‘š‘0ī€·š‘„š‘āˆ’1ī€øš‘00(1)š‘10(1)ā‹Æš‘š‘0āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ(1),(3.9)š¾=00ā‹Æ0Ģˆš‘00ī€·š‘„1ī€øĢˆš‘10ī€·š‘„1ī€øā‹ÆĢˆš‘š‘0ī€·š‘„1ī€øā‹®ā‹®ā‹±ā‹®Ģˆš‘00ī€·š‘„š‘āˆ’1ī€øĢˆš‘10ī€·š‘„š‘āˆ’1ī€øā‹ÆĢˆš‘š‘0ī€·š‘„š‘āˆ’1ī€øāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ 00ā‹Æ0.(3.10) Also, the vectors š‘† and šŗ(š‘”) are defined as follows:ī€ŗī€·š‘ˆš‘†=0,sin1ī€øī€·š‘ˆ,sin2ī€øī€·š‘ˆ,ā€¦,sinš‘āˆ’1ī€øī€»,0š‘‡,īƒ¬š‘‘šŗ(š‘”)=2š‘”0(š‘”)š‘‘š‘”2š‘‘,0,ā€¦,0,2š‘”1(š‘”)š‘‘š‘”2īƒ­š‘‡.(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 š“0 and Ģ‡š“0. Using (3.1), we getš‘ˆš‘ī€·š‘„š‘–ī€ø=,0š‘ī“š‘›=0š‘Žš‘›ī€·š‘”0ī€øš‘ƒš‘›0ī€·š‘„š‘–ī€ø,š‘–=0,1,ā€¦,š‘,šœ•š‘ˆš‘ī€·š‘„šœ•š‘”š‘–ī€ø=,0š‘ī“š‘›=0Ģ‡š‘Žš‘›ī€·š‘”0ī€øš‘ƒš‘›0ī€·š‘„š‘–ī€ø,š‘–=0,1,ā€¦,š‘.(3.12)

Equation (3.12) is system of (š‘+1) equations in (š‘+1) unknowns. These equations can be written in the matrix form š‘€š“0=š‘1 and š‘€Ģ‡š“0=š‘2, where š‘1=ī€ŗš‘“0ī€·š‘„0ī€ø,ā€¦,š‘“0ī€·š‘„š‘ī€øī€»š‘‡,š‘2=ī€ŗš‘“1ī€·š‘„0ī€ø,ā€¦,š‘“1ī€·š‘„š‘ī€øī€»š‘‡.(3.13)

3.1. The DIRKN Method

For solving (3.8), we use DIRKN method [3, 16]:š‘€š‘„š‘–=š‘€š“š‘›+š‘š‘–Ģ‡š“Ī”š‘”š‘€š‘›+Ī”š‘”2š‘ āˆ‘š‘—=1š‘Žš‘–š‘—š¹ī€·š‘”š‘›+š‘š‘—Ī”š‘”,š‘„š‘—ī€ø,š‘–=1,ā€¦,š‘ ,š‘€š“š‘›+1=š‘€š“š‘›+š‘š‘–Ģ‡š“Ī”š‘”š‘€š‘›+Ī”š‘”2š‘ ī“š‘—=1š‘š‘–š¹ī€·š‘”š‘›+š‘š‘—Ī”š‘”,š‘„š‘—ī€ø,š‘€Ģ‡š“š‘›+1Ģ‡š“=š‘€š‘›+Ī”š‘”š‘ ī“š‘—=1š‘š‘–š¹ī€·š‘”š‘›+š‘š‘—Ī”š‘”,š‘„š‘—ī€ø,(3.14)

where š‘„š‘– is the internal stages, š“š‘›+1 and Ģ‡š“š‘›+1 represent approximations to š“(š‘”š‘›+1) and Ģ‡š“(š‘”š‘›+1), respectively. Also we defineĪ”š‘”=š‘”0+š‘—Ī”š‘”,š‘—=0,1,ā€¦,š‘€,(3.15)where Ī”š‘”=(š‘‡āˆ’š‘”0)/š‘€. The RKN method can be denoted in šµš‘¢š‘”š‘ā„Žš‘’š‘Ÿ,š‘  notation by the table of coefficients:š‘1š‘Ž11ā‹Æš‘Ž1š‘ ā‹®š‘ā‹®ā‹±ā‹®š‘ š‘Žš‘ 1ā‹Æš‘Žš‘ š‘ š‘1ā‹Æš‘š‘ š‘1ā‹Æš‘š‘ (3.16)

In the DIRKN method, š‘Žš‘–š‘—=0 as (š‘–<š‘—) and š‘Žš‘–š‘– are equal. Also, in the above table of coefficients, š‘  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 š‘„š‘–, š‘–=1,ā€¦,4, separately from (3.14) with Newton-Raphson iteration method. We iterate until the following criterion is satisfied:ā€–ā€–ī€·š‘„š‘–š‘˜+1āˆ’š‘„š‘˜š‘–ī€øā€–ā€–āˆž<10āˆ’12,(3.17) where š‘„š‘˜š‘– is the value of š‘„š‘– 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 šæ2 and šæāˆž error norms are defined as follows:šæ2=ā€–ā€–š‘¢āˆ’š‘ˆš‘ā€–ā€–=īƒ¬š‘ī“š‘–=0ī€·š‘¢š‘–āˆ’š‘ˆš‘š‘–ī€ø2īƒ­1/2,šæāˆž=ā€–ā€–š‘¢āˆ’š‘ˆš‘ā€–ā€–=max0ā‰¤š‘–ā‰¤š‘||š‘¢š‘–āˆ’Uš‘š‘–||.(4.1)

We choose Ī”š‘”=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 2.4ā€‰GHZ.

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

Experiment 2. In this experiment [3, 8], we consider (1.1) with š‘”0=0 and the initial conditions š‘“0š‘“(š‘„)=0,(4.4)14(š‘„)=āˆš1+š‘2īƒ©š‘„sechāˆš1+š‘2īƒŖ,(4.5) with the boundary conditions (1.3), which can be obtained from the following exact solution: īƒ©š‘š‘¢(š‘„,š‘”)=4arctanāˆ’1īƒ©sinš‘š‘”āˆš1+š‘2īƒŖīƒ©š‘„sechāˆš1+š‘2īƒŖīƒŖ,š‘=1.(4.6)
In Tables 3 and 4, we show the šæ2 and šæāˆž error norms, for several final times (š‘‡). As we see from Tables 3 and 4, for the given values of š‘, the šæāˆž error norm is less than the šæ2 error norm. Figure 2 shows four type plots at š‘”=0.5 of Experiment 2.

Experiment 3. In experiment [3, 9], we consider (1.1) with š‘”0=0 and the initial conditions š‘“0īƒ©īƒ©š‘„(š‘„)=4arctanš‘sinhāˆš1āˆ’š‘2š‘“īƒŖīƒŖ,(4.7)1(š‘„)=0,(4.8) with the boundary conditions (1.3), which can be obtained from the following exact solution: āŽ›āŽœāŽœāŽī‚€āˆšš‘¢(š‘„,š‘”)=4arctanš‘sinhš‘„/1āˆ’š‘2ī‚ī‚€āˆšcoshš‘š‘”/1āˆ’š‘2ī‚āŽžāŽŸāŽŸāŽ ,š‘=0.5.(4.9)
In Table 5, we represent the šæ2 and šæāˆž error norms for different values of š‘ at š‘”=0.5. With increasing š‘, we obtain better results, and these error norms decrease five orders in magnitude. Figure 3 shows four-type plots at š‘”=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 š‘=16 is more accurate than the other values of š‘. The proposed method has higher order of accuracy even for small values of š‘ (i.e., š‘=4). We obtain better results when we choose small value of Ī”š‘”, in comparison to those obtained in the literature.