Abstract
In this article, one dimensional non-linear sine-Gordon equation has been studied. We propose a new method based on collocation of cubic B spline to find the numerical solution of non-linear sine-Gordon equation with Dirichlet boundary conditions. The method involves high order perturbations of the classical cubic spline collocation method at the partition nodes. The existence and uniqueness of the numerical solution has been proved. The method produces optimal fourth order accurate solutions. Stability analysis of the method has been done. The numerical experiments confirm the expected accuracy. We compare the results obtained by our method with those by other techniques for some already existing examples in the literature.
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References
Ablowitz MJ, Herbst BM, Schober C (1996) On the numerical solution of the sine-Gordon equation I, integrable discretizations and homoclinic manifolds. J Comput Phys 26:299–314
Ablowitz MJ, Herbst BM, Schober C (1997) On the numerical solution of sine-Gordon equation II, performance of numerical schemes. J Comput Phys 131:354–367
Bratsos AG (2005) An explicit scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2:189–211
Bratsos AG (2008) A numerical method for the one-dimensional sine-Gordon equation. Numer Methods Differ Equ 24:833–844
Bratsos AG (2008) A forth order numerical scheme for the one-dimensional sine-Gordon equation. Int J Comput Math 85(7):1083–1095
Bratsos AG, Twizell EH (1996) The solution of the sine-Gordon equation using method of lines. Int J Comput Math 61(3–4):271–292
Bratsos AG, Twizell EH (1998) A family of parametric finite-difference methods for the solution of the sine-Gordon equation. Appl Math Comput 93(2–3):117–137
de Boor C (1978) Practical guide to splines. Springer, New York
Dehghan M, Mirzaei D (2008) The boundary integral approach for numerical solution of the one-dimensional sine-Gordon equation. Numer Method Part Differ Equ 24(6):1405–1415
Dehghan M, Shokri A (2008) A numerical method for one-dimensional non-linear sine-Gordon equation using collocation and radial basis functions. Numer Methods Part Differ Equ 24(2):687–698
Djijel K, Price WG, Twizell EH (1995) Numerical solutions of a damped sine-Gordon equation in TWo space variables. J Eng Math 24(4):347–369
Guo BY, Pascual PJ, Roriguez MJ (1986) Numerical solution of sine-Gordon equation. Appl Math Comput 18:1–14
Jiang ZW, Wang RH (2012) Numerical solution of sine-Gordon equation using high accuracy multiquadric quasi-interpolation. Appl Math Comput 218:7711–7716
Khaliq AQM, Abukhocliar B, Sheng Q, Ismail MS (2000) A predictor-corrector scheme for the sine-Gordon equation. Numer Methods Part Differ Equ 16:133–146
Lucas TR (1974) Error bounds for interpolating cubic splines under various end conditions. SIAM J Numer Anal 11(3):569–584
Mittal RC, Bhatia R (2014) Numerical solution of non-linear sine-Gordon equation by modified cubic B-spline collocation method. Int J Part Differ Equ 2014
Mohanty RK (2014) New high accuracy super stable alternating direction implicit methods for two and three dimensional hyperbolic damped wave equations. Results Phys 4:156–163
Mohebbi A, Dehghan M (2010) High order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN method. Math Comput Model 51:537–549
Ramos JI (2001) The sine-Gordon equation in finite line. Appl Math Comput 124:45–93
Rashidinia J, Mohammadi R (2011) Tension spline solution of non-linear sine-Gordon. Numer Algorithms 56(1):129–142
Singh S, Singh S, Arora R (2017) Numerical solution of second order one-dimensional hyperbolic equation by exponential b-spline collocation method. Numer Anal Appl 7:164–176
Singh S, Singh S, Aggarwal A (2017) Fourth-order cubic B-spline collocation method for hyperbolic telegraph equation. Math Sci. https://doi.org/10.1007/s40096-021-00428-y
Strauss WA, Vazquez L (1978) Numerical solution of a non-linear Klein–Gordon equation. J Comput Phys 28:271–278
Wei GW (2000) Discrete convolution for the sine-Gordon equation. Phys D 137(3–4):247–259
Zheng C (2007) Numerical solution to the sine-Gordon equation defined on the whole real axis. Siam J Sci Comput 29(6):2494–2506
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Communicated by Jose Alberto Cuminato.
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Singh, S., Singh, S. & Aggarwal, A. Cubic B-spline method for non-linear sine-Gordon equation. Comp. Appl. Math. 41, 382 (2022). https://doi.org/10.1007/s40314-022-02092-x
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DOI: https://doi.org/10.1007/s40314-022-02092-x