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

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1 International Scholarly Research Network ISRN Applied Mathematics Volume, Article ID 4673, pages doi:.54//4673 Research Article The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial Zoleikha Soori and Azim Aminataei Department of Mathematics, K. N. Toosi University of Technology, P.O. Bo , Tehran, Iran Correspondence should be addressed to Azim Aminataei, ataei@kntu.ac.ir Received 7 September ; Accepted October Academic Editors: P. J. García Nieto and H.-T. Hu Copyright q Z. Soori and A. 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. 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.. Introduction The sine-gordon equation is a nonlinear hyperbolic partial differential equation PDE.Itwas originally considered in the nineteenth century in the course of surface of constant negative curvature. This equation attracted a lot of attention in the 97s due to the presence of soliton solutions,. 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 u, t t u, t sin u, t,, t, t,t,. with the initial conditions u, t f, u t, t f,,,.

2 ISRN Applied Mathematics and the boundary conditions u,t g t, u,t g t, t t..3 In the last decade, several numerical schemes have been developed for solving., 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. 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, 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 3. In this paper, we have proposed a numerical scheme for solving. using spectral method with a basis of a new orthogonal polynomial. This polynomial is introduced by Chelyshkov in 4. The outline of this paper is as follows. In Section, we introduce a new orthogonal polynomial. In Section 3, we apply spectral method for solving.. InSection 4, weshow 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 4. Let n be a fied whole number, and P n is a sequence of polynomial, that is, P n {P nk } k n, n P nk P nk τ nkl l, l k. that satisfy the orthogonality relationship, k/ l, P nk P nl d k l, k l,. and standardization sign τ nkn n k..3

3 ISRN Applied Mathematics 3 The coefficient τ nkl of the polynomial P nk is defined by requirements.,.3, andthe Gram-Schmidt orthogonalization procedure without normalization. The eplicit definition of the polynomial P nk is as follows: n k ( )( ) P nk j n k n k j k j, k,,...,n..4 j n k j This yields Rodrigue s type representation as follows: P nk n k! k d n k ( n k n k), k,,...,n,.5 d n k and the orthogonality relationship. is confirmed by applying.5. It also follows from.5 that P nk d n d n..6 This polynomial can be connected to a fied set of the Jacobi polynomials P α,β n ξ 5, that is, P nk n P n, k n The Proposed Method In this section, the introduced orthogonal polynomial is applied as the basis for spectral method to solve sine-gordon equation. The approimate solution u N, t to the eact solution u, t can be written in the form u, t U N, t N a n t P n, n 3. where a n is the time-dependent quantities which must be determined. In this paper, we apply orthogonal polynomial P nk for k. From 3., weget U N tt U N ä n t P n, n a n t P n. n 3.

4 4 ISRN Applied Mathematics Using 3. and substituting in., weobtain ä n t P n i N a n t P n i sin ( U N i,t ), i,,...,n, 3.3 n n where ä n and P n denote the second-order derivatives of a n and P n with respect to t and, respectively, wherein the collocation points are as i i/n, i,,...,n. Equation 3 is rewritten as follows: MÄ t KA t S. 3.4 For displaying this equation in matri form, for i,,...,n, and n,,...,n,we define A t a t,a t,...,a N t T, M M in P n i, K K in P n i, S sin U, sin U,...,sin U N T, U i U N i,t. 3.5 The matrices M and K are positive definite matrices of order N N. The system given in 3.4 consists of N equations in N 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.3, in the following forms: U N,t a n t P n g t, t t, n U N,t a n t P n g t, t t. n 3.6 From 3.6, weget ä n t P n d g t, t t dt, n ä n t P n d g t, t t dt. n 3.7 Therefore, by using 3.6, and 3.7, a new system is obtained as follows: MÄ t F t, A t, F t, A t KA t S G t, 3.8

5 ISRN Applied Mathematics 5 where A t, F t, A t, S,andG t are vectors of order N with N components. The matrices M and K are of order N N, which are defined as follows: p p p N p p p N M......, 3.9 p N p N p N N p p p N p p p N K p N p N p N N Also, the vectors S and G t are defined as follows: S, sin U, sin U,...,sin U N, T, [ ] d T g t G t,,...,, d g t. dt dt 3. The nonlinear second-order system of ODEs 3.8 can be solved numerically using the fourthorder A-stable DIRKN method 3, 6. From the initial conditions., we determine the initial vectors A and Ȧ.Using 3., weget U N i, a n t P n i, n U N i, ȧ n t P n i, t n i,,...,n, i,,...,n. 3. Equation 3. is system of N equations in N unknowns. These equations can be written in the matri form MA b and MȦ b, where b [ f,...,f N ] T, b [ f,...,f N ] T. 3.3

6 6 ISRN Applied Mathematics Table : Four-stage A-stable DIRKN method The DIRKN Method For solving 3.8, we use DIRKN method 3, 6 : MQ i MA n c i ΔtMȦ n Δt s a ij F ( ) t n c j Δt, Q j, j s MA n MA n c i ΔtMȦ n Δt b i F ( ) t n c j Δt, Q j, j s MȦ n MȦ n Δt b i F ( ) t n c j Δt, Q j, j i,...,s, 3.4 where Q i is the internal stages, A n and Ȧ t n, respectively. Also we define A n represent approimations to A t n and Δt t jδt, j,,...,m, 3.5 where Δt T t /M. The RKN method can be denoted in Butcher, s notation by the table of coefficients: c a a s c s a s a ss 3.6 b b s b b s In the DIRKN method, a ij 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 6, with algebraic fourth order, which is defined by the following table of coefficients for solving all eperiments. Using Table, we can calculate Q i, i,...,4, separately from 3.4 with Newton- Raphson iteration method. We iterate until the following criterion is satisfied: ( ) Q k i Q k i <, 3.7 where Q k i is the value of Q i at the kth iteration of the Newton-Raphson method.

7 ISRN Applied Mathematics 7 Table : The approimate solution for N 6 of Eperiment. T. T. T.3 T.4 T Numerical Eperiments In this section, we present the results of numerical eperiments using the method introduced in Section 3.TheL and L error norms are defined as follows: [ L u U N ( u i U N i i L u U N ma u i U N i i N ) ] /,. 4. We choose Δt /, for solving all eperiments. The computations associated with the eperiments discussed above were performed in Maple 3 on a PC with a CPU of.4 GHZ. Eperiment. In this eperiment 3, 6,, we consider. with t conditions and the initial f, f 4sech, 4. with the boundary conditions.3, which can be obtained from the following eact solution: u, t 4arctan sech t. 4.3 In Table, the approimate solution for several final times T with different locations is shown. Figure shows two type plots at t.5 ofeperiment. Eperiment. In this eperiment 3, 8, we consider. with t and the initial conditions f, 4.4 ( ) 4 f sech, c c 4.5

8 8 ISRN Applied Mathematics u Eact Approimate a Error b Figure : a The eact and approimate solutions for N 6 are shown. b The L error norm for N 6 is shown. Table 3: The L error norm for different values of N of Eperiment. T N 4 N 8 N with the boundary conditions.3, which can be obtained from the following eact solution: u, t 4arctan ( ( ) ( )) ct c sin sech, c. 4.6 c c In Tables 3 and 4, we show the L and L error norms, for several final times T. As we see from Tables 3 and 4, for the given values of N, thel error norm is less than the L error norm. Figure shows four type plots at t.5ofeperiment.

9 ISRN Applied Mathematics u Approimate Eact a Error b N c N d Figure : a The eact and approimate solutions for N 6 are shown. b The L error norm for N 6 is shown. c The L error norm for different values of N is shown. d The L error norm for different values of N is shown.

10 ISRN Applied Mathematics Table 4: The L error norm for different values of N of Eperiment. T N 4 N 8 N Table 5: Error norms for different values of N of Eperiment 3. N L error norm L error norm Eperiment 3. In eperiment 3, 9, we consider. with t and the initial conditions f 4arctan ( ( )) c sinh, 4.7 c f, 4.8 with the boundary conditions.3, which can be obtained from the following eact solution: c sinh (/ ) c u, t 4arctan cosh (ct/ ), c c In Table 5, we represent the L and L error norms for different values of N at t.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.5ofeperiment Conclusion In this paper, we have presented the spectral method for solving one-dimensional nonlinear sine-gordon equation using a new orthogonal polynomial. Numerical eperiments show that the spectral method is an efficient one, and the results for value of N 6 is more accurate

11 ISRN Applied Mathematics Approimate Eact Error a b N N c d Figure 3: a The eact and approimate solutions for N 6 are shown. b The L error norm for N 6 is shown. c The L error norm for different values of N is shown. d The L error norm for different values of N is shown.

12 ISRN Applied Mathematics 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 A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/ CRC, Boca Raton, Fla, USA, 4. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, The Netherlands, A. Mohebbi and M. Dehghan, High-order solution of one-dimensional sine-gordon equation using compact finite difference and DIRKN methods, Mathematical and Computer Modelling, vol. 5, no. 5-6, pp ,. 4 A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structure, vol. 8 of Oford Tets in Applied and Engineering Mathematics, Oford University Press, Oford, UK, nd edition, 3. 5 T. Dauois and M. Peyrard, Physics of Solitons, Cambridge University Press, 6. 6 M. Dehghan and A. Shokri, A numerical method for one-dimensional nonlinear sine-gordon equation using collocation and radial basis functions, Numerical Methods for Partial Differential Equations, vol. 4, no., pp , 8. 7 M. Dehghan and D. Mirzaei, The boundary integral equation approach for numerical solution of the one-dimensional sine-gordon equation, Numerical Methods for Partial Differential Equations, vol. 4, no. 6, pp , 8. 8 A. G. Bratsos, A third order numerical scheme for the two-dimensional sine-gordon equation, Mathematics and Computers in Simulation, vol. 76, no. 4, pp. 7 8, 7. 9 A. G. Bratsos, A numerical method for the one-dimensional sine-gordon equation, Numerical Methods for Partial Differential Equations, vol. 4, no. 3, pp , 8. M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-gordon equation using the radial basis functions, Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 7 75, 8. M. Lakestani and M. Dehghan, Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation, Computer Physics Communications, vol. 8, no. 8, pp. 39 4,. M. Dehghan and F. Fakhar-Izadi, The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Mathematical and Computer Modelling, vol. 53, no. 9-, pp ,. 3 A. H. A. Ali, Chebyshev collocation spectral method for solving the RLW equation, International Nonlinear Science, vol. 7, no., pp. 3 4, 9. 4 V. S. Chelyshkov, Alternative orthogonal polynomials and quadratures, Electronic Transactions on Numerical Analysis, vol. 5, p. 7 6 electronic, 6. 5 A. Imani, A. Aminataei, and A. Imani, Collocation method via Jacobi polynomials for solving nonlinear ordinary differential equations, International Mathematics and Mathematical Sciences, vol., Article ID 67385, pages,. 6 J. M. Franco and I. Gómez, Accuracy and linear stability of RKN methods for solving second-order stiff problems, Applied Numerical Mathematics, vol. 59, no. 5, pp , 9.

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