Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies. 1. Introduction In this paper we consider the one-dimensional sine-Gordon equation with initial conditions The Dirichlet boundary conditions are given by The nonlinear sine-Gordon equation arises in many different applications such as propagation of fluxion in Josephson junctions [1], differential geometry, stability of fluid motion, nonlinear physics, and applied sciences [2]. The sine-Gordon equation (1) is a particular case of Klein-Gordon equation, which plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [3], given by where is a nonlinear force and is a constant. In the literature several schemes have been developed for the numerical solution of sine-Gordon equation. Ben-Yu et al. [4] proposed two difference schemes; Bratsos and Twizell [5] used method of lines to transform the initial/boundary value problem associated with (1) into a first order nonlinear initial value problem. Mohebbi and Dehghan [6] presented a combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method. Kuang and Lu [7] proposed two classes of finite difference method for generalized sine-Gordon equation; Bratsos and Twizell [8] presented a family of finite difference method, in which time and space derivatives are replaced by finite-difference approximations and then the equation is converted into a linear algebraic system. Wei [9] used the discrete singular convolution algorithm for the integration of (1). A variational iteration method to obtain approximate analytical solution of the sine-Gordon equation without any discretization has been developed by Batiha et al. [10]. Zheng [11] presented a numerical solution of sine-Gordon
References
[1]
J. K. Perring and T. H. R. Skyrme, “A model unified field equation,” Nuclear Physics, vol. 31, pp. 550–555, 1962.
[2]
A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, “Theory and applications of the sine-Gordon equation,” La Rivista del Nuovo Cimento, vol. 1, no. 2, pp. 227–267, 1971.
[3]
M. Dehghan and A. Shokri, “Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 400–410, 2009.
[4]
G. Ben-Yu, P. J. Pascual, M. J. Rodriguez, and L. Vázquez, “Numerical solution of the sine-Gordon equation,” Applied Mathematics and Computation, vol. 18, no. 1, pp. 1–14, 1986.
[5]
A. G. Bratsos and E. H. Twizell, “The solution of the sine-Gordon equation using the method of lines,” International Journal of Computer Mathematics, vol. 61, no. 3-4, pp. 271–292, 1996.
[6]
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. 51, no. 5-6, pp. 537–549, 2010.
[7]
J. X. Kuang and L. H. Lu, “Two classes of finite-difference methods for generalized sine-Gordon equations,” Journal of Computational and Applied Mathematics, vol. 31, no. 3, pp. 389–396, 1990.
[8]
A. G. Bratsos and E. H. Twizell, “A family of parametric finite-difference methods for the solution of the sine-Gordon equation,” Applied Mathematics and Computation, vol. 93, no. 2-3, pp. 117–137, 1998.
[9]
G. W. Wei, “Discrete singular convolution for the sine-Gordon equation,” Physica D, vol. 137, no. 3-4, pp. 247–259, 2000.
[10]
B. Batiha, M. S. M. Noorani, and I. Hashim, “Numerical solution of sine-Gordon equation by variational iteration method,” Physics Letters A, vol. 370, no. 5-6, pp. 437–440, 2007.
[11]
C. Zheng, “Numerical solution to the sine-Gordon equation defined on the whole real axis,” SIAM Journal on Scientific Computing, vol. 29, no. 6, pp. 2494–2506, 2007.
[12]
A. G. Bratsos, “A fourth order numerical scheme for the one-dimensional sine-Gordon equation,” International Journal of Computer Mathematics, vol. 85, no. 7, pp. 1083–1095, 2008.
[13]
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. 24, no. 2, pp. 687–698, 2008.
[14]
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. 24, no. 6, pp. 1405–1415, 2008.
[15]
J. Rashidinia and R. Mohammadi, “Tension spline solution of nonlinear sine-Gordon equation,” Numerical Algorithms, vol. 56, no. 1, pp. 129–142, 2011.
[16]
M. Li-Min and W. Zong-Min, “A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation,” Chinese Physics B, vol. 18, no. 8, pp. 3099–3103, 2009.
[17]
Z.-W. Jiang and R.-H. Wang, “Numerical solution of one-dimensional Sine-Gordon equation using high accuracy multiquadric quasi-interpolation,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7711–7716, 2012.
[18]
D. Kaya, “A numerical solution of the sine-Gordon equation using the modified decomposition method,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 309–317, 2003.
[19]
M. Uddin, S. Haq, and G. Qasim, “A meshfree approach for the numerical solution of nonlinear sine-Gordon equation,” International Mathematical Forum, vol. 7, no. 21–24, pp. 1179–1186, 2012.
[20]
S. Gottlieb, C.-W. Shu, and E. Tadmor, “Strong stability-preserving high-order time discretization methods,” SIAM Review, vol. 43, no. 1, pp. 89–112 (electronic), 2001.
[21]
R. C. Mittal and R. Bhatia, “Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method,” Applied Mathematics and Computation, vol. 220, pp. 496–506, 2013.
