Document Type : Research Paper

**Authors**

Department of Mathematics, Cochin University of Science and Technology, Kerala, India.

**Abstract**

In this article, we discuss the numerical solution of the nonlinear Sine-Gordon equation in one and two dimensions and its coupled form. A differential quadrature technique based on a modified set of cubic B-splines has been used. The chosen modification possesses the optimal accuracy order four in the spatial domain. The spatial derivatives are approximated by the differential quadrature technique, where the weight coefficients are calculated using this set of modified cubic B-splines. This approximation will lead to the discretization of the problem in the spatial domain that gives a system of first-order ordinary differential equations. This system is then solved using the SSP-RK54 scheme to progress the solution to the next time level. The convergence of this numerical scheme solely depends on the differential quadrature and is found to give a stable solution. The order of convergence is calculated and is observed to be four. The entire computation is performed up to a large time level with an efficient speed. It is found that the computed solution is in good agreement with the exact one and the error comparison with similar works in the literature indicates the scheme outperforms.

**Keywords**

- [1] J. Argyris, M. Haase, and J. C. Heinrich, Finite element approximation to two-dimensional sine-gordon solitons, Computer methods in applied mechanics and engineering, 86(1) (1991), 1-26.
- [2] G. Arora, B. K. Singh, Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method, Applied Mathematics and Computation, 224 (2013), 166-177.
- [3] A. Babu, B. Han, and N. Asharaf, Numerical solution of the viscous burgers’ equation using localized differential quadrature method, Partial Differential Equations in Applied Mathematics, 4 (2021), 100044.
- [4] A. Babu, B. Han, and N. Asharaf, Numerical solution of the hyperbolic telegraph equation using cubic B-spline- based differential quadrature of high accuracy, Computational Methods for Differential Equations, 10(4) (2022), 837-859.
- [5] A. Barone, F. Esposito, C. Magee, and A. Scott, Theory and applications of the sine-gordon equation, La Rivista del Nuovo Cimento (1971-1977), 1(2) (1971), 227-267.
- [6] A. G. Bratsos, A fourth order numerical scheme for the one-dimensional sine-gordon equation, International Journal of Computer Mathematics, 85(7) (2008), 1083-1095.
- [7] E. Bour, Th´eorie de la d´eformation des surfaces, Gauthier-Villars, 1891.
- [8] R. J. Cheng and K. M. Liew, Analyzing two-dimensional sine–Gordon equation with the mesh-free reproducing kernel particle Ritz method, Computer methods in applied mechanics and engineering, 245 (2012), 132-143.
- [9] O. Davydov and R. Schaback, Minimal numerical differentiation formulas, Numerische Mathematik, 140(3) (2018), 555–592.
- [10] M. Dehghan and A. Shokri, A numerical method for one-dimensional nonlinear sine-gordon equation using collo- cation and radial basis functions, Numerical Methods for Partial Differential Equations: An International Journal 24(2) (2008), 687-698.
- [11] 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, 79(3) (2008), 700-715.
- [12] 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: An International Jour- nal, 24(6) (2008), 1405-1415.
- [13] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-gordon and Klein–Gordon equations, Engineering Analysis with Boundary Elements, 50 (2015), 412-434.
- [14] K. Djidjeli, W. G. Price, and E. H. Twizell, Numerical solutions of a damped sine-Gordon equation in two space variables, Journal of Engineering Mathematics, 29(4) (1995), 347-369.
- [15] M. J. Huntul, N. Dhiman, and M. Tamsir, Reconstructing an unknown potential term in the third-order pseudo- parabolic problem, Computational and Applied Mathematics, 40(4) (2021), 1-18.
- [16] M. Ilati and M. Dehghan, The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-gordon equations, Engineering Analysis with Boundary Elements, 52 (2015), 99-109.
- [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, 218(15) (2012), 7711-7716.
- [18] R. Jiwari, S. Pandit, and R. C. Mittal, Numerical simulation of two-dimensional sine-gordon solitons by differential quadrature method, Computer Physics Communications, 183(3) (2012), 600-616.
- [19] R. Jiwari and J. Yuan, A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes, Journal of mathematical Chemistry, 52(6) (2014), 1535-1551.
- [20] R. Jiwari, Lagrange interpolation and modified cubic b-spline differential quadrature methods for solving hyperbolic partial differential equations with dirichlet and neumann boundary conditions, Computer Physics Communications, 193 (2015), 55-65.
- [21] D. Kaya, A numerical solution of the sine-gordon equation using the modified decomposition method, Applied Mathematics and Computation, 143(2-3) (2003), 309-317.
- [22] X. Li, S. Zhang, Y. Wang, and H. Chen, Analysis and application of the element-free Galerkin method for nonlinear sine-gordon and generalized sinh-Gordon equations, Computers & Mathematics with Applications 71(8) (2016), 1655-1678.
- [23] D. Li, H. Lai, and C. Lin, Mesoscopic simulation of the two-component system of coupled sine-gordon equations with lattice boltzmann method, Entropy 21(6) (2019), 542.
- [24] M. Li-Min and W. Zong-Min, A numerical method for one-dimensional nonlinear sine-gordon equation using multiquadric quasi-interpolation, Chinese Physics B, 18(8) (2009), 3099.
- [25] M. Lotfi and A. Alipanah, Legendre spectral element method for solving sine-Gordon equation, Advances in Dif- ference Equations, 2019(1) (2019), 1-15.
- [26] R. Mittal and R. Jain, Numerical solutions of nonlinear burgers’ equation with modified cubic b-splines collocation method, Applied Mathematics and Computation, 218(15) (2012), 7839-7855.
- [27] R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear fisher’s reaction–diffusion equation with modified cubic b-spline collocation method, Mathematical Sciences, 7(1) (2013), 1-10.
- [28] R. Mittal and R. Bhatia, Numerical solution of nonlinear sine-gordon equation by modified cubic b-spline collo- cation method, International Journal of Partial Differential Equations, 2014.
- [29] A. H. Msmali, M. Tamsir, and A. A. H. Ahmadini, Crank-Nicolson-DQM based on cubic exponential B-splines for the approximation of nonlinear Sine-Gordon equation, Ain Shams Engineering Journal, 12(4) (2021), 4091-4097.
- [30] B. Pekmen and M. Tezer-Sezgin. Differential quadrature solution of nonlinear Klein–Gordon and sine-gordon equations, Computer Physics Communications, 183(8) (2012), 1702-1713.
- [31] J. Perring and T. H. R. Skyrme, A model unified field equation, Selected Papers, With Commentary, Of Tony Hilton Royle Skyrme, 1994, 216-221.
- [32] S. S. Ray, A numerical solution of the coupled sine-gordon equation using the modified decomposition method, Applied mathematics and computation, 175(2) (2006), 1046-1054.
- [33] W. Shao and X. Wu, The numerical solution of the nonlinear Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method, Computer Physics Communications, 185(5) (2014), 1399-1409.
- [34] Y. Shen and Y. O. El-Dib, A periodic solution of the fractional sine-gordon equation arising in architectural engineering, Journal of Low Frequency Noise, Vibration and Active Control, 40(2) (2021), 683-691.
- [35] Q. Sheng, A. Q. M. Khaliq, and D. A. Voss, Numerical simulation of two-dimensional sine-gordon solitons via a split cosine scheme, Mathematics and Computers in Simulation, 68(4) (2005), 355-373.
- [36] C. Shu, Differential Quadrature and Its Application in Engineering, Springer Science & Business Media, 2012.
- [37] H. S. Shukla, M. Tamsir, V. K. Srivastava, and J. Kumar, Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method, AIP Advances, 4(11) (2014), 117134.
- [38] H. S. Shukla, M. Tamsir, and V. K. Srivastava, Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method, AIP Advances, 5(1) (2015), 017121.
- [39] H. S. Shukla, M. Tamsir, V. K. Srivastava, and M. M. Rashidi, Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation, Modern Physics Letters B, 30(11) (2016), 1650110.
- [40] H. S. Shukla and M. Tamsir, Numerical solution of nonlinear sine–gordon equation by using the modified cubic B-spline differential quadrature method, Beni-Suef University journal of basic and applied sciences, 7(4) (2018), 359-366.
- [41] A. Taleei and M. Dehghan, A pseudo-spectral method that uses an overlapping multidomain technique for the numerical solution of sine-Gordon equation in one and two spatial dimensions, Mathematical Methods in the Applied Sciences, 37(13) (2014), 1909-1923.
- [42] M. Uddin, S. Haq, and G. Qasim, A meshfree approach for the numerical solution of nonlinear sine-gordon equation, International Mathematical Forum, 7(24) (2012), 1179-1186.
- [43] G. B. Whitham, Linear and nonlinear waves, John Wiley & Sons, 2011.
- [44] F. Yin, T. Tian, J. Song, and M. Zhu, Spectral methods using Legendre wavelets for nonlinear Klein sine-gordon equations, Journal of computational and applied mathematics, 275 (2015), 321-334.

April 2023

Pages 369-386

**Receive Date:**09 February 2022**Revise Date:**17 August 2022**Accept Date:**15 September 2022**First Publish Date:**17 September 2022