On the numerical solution of the Klein-Gordon equation by exponential cubic B-spline collocation method

Abstract

In the present study, we solve initial boundary value problem constructed on nonlinear Klein-Gordon equation. The collocation method on exponential cubic B-spline functions forming a set of basis for the functions defined in the same interval is set up for the numerical approach. The efficiency and validity of the proposed method are determined by computing the error between the numerical and the analytical solutions and relative change of the conserved quantities.

Keywords

• Klein-Gordon Equation,exponential cubic B-spline,collocation,wave motion

References

1. Whitham G.B., Linear and Nonlinear Waves, John Wiley & Sons Inc. New York, 1999.
2. Kim, J. J., & Hong, W. P. (2004). New solitary-wave solutions for the generalized reaction Duffing model and their dynamics. Zeitschrift für Naturforschung A, 59(11), 721-728.
3. Kragh, H. (1984). Equation with the many fathers. The Klein-Gordon equation in 1926. American Journal of Physics, 52(11), 1024-1033.
4. Ablowitz M., Nonlinear Dispersive Waves, Cambridge Press, Cambridge, 2011.
5. Galehouse, D. C. (1981). Geometrical derivation of the Klein-Gordon equation. International Journal of Theoretical Physics, 20(6), 457-479.
6. Schechter, M. (1976). The Klein-Gordon equation and scattering theory. Annals of Physics, 101(2), 601-609.
7. Weder, R. A. (1978). Scattering theory for the Klein-Gordon equation. Journal of Functional Analysis, 27(1), 100-117.
8. Lundberg, L. E. (1973). Spectral and scattering theory for the Klein-Gordon equation. Communications in Mathematical Physics, 31(3), 243-257.

9. Tsukanov, V. D. (1990). Motion of a Klein-Gordon kink in an external field. Theoretical and Mathematical Physics, 84(3), 930-933.
10. Chambers, L. G. (1966). Derivation of solutions of the Klein-Gordon equation from solutions of the wave equation. Proceedings of the Edinburgh Mathematical Society (Series 2), 15(02), 125-129.
11. Fleischer, W., & Soff, G. (1984). Bound state solutions of the Klein-Gordon equation for strong potentials. Zeitschrift für Naturforschung A, 39(8), 703-719.
12. Burt, P. B., & Reid, J. L. (1976). Exact solution to a nonlinear Klein-Gordon equation. Journal of Mathematical Analysis and Applications, 55(1), 43-45.
13. Sharma, A. S., & Buti, B. (1976). Envelope solitons and holes for sine-Gordon and non-linear Klein-Gordon equations. Journal of Physics A: Mathematical and General, 9(11), 1823.
14. Burt, P. B. (1974). Solitary waves in nonlinear field theories. Physical Review Letters, 32(19), 1080.
15. Huang, D. J., & Zhang, H. Q. (2005). The extended first kind elliptic sub-equation method and its application to the generalized reaction Duffing model. Physics Letters A, 344(2), 229-237.
16. Akter, J., & Akbar, M. A. (2015). Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in Physics, 5, 125-130.
17. Adomian, G. (1996). Nonlinear Klein-Gordon equation. Applied Mathematics Letters, 9(3), 9-10.
18. Kudryavtsev, A. E. (1975). Solitonlike solutions for a Higgs scalar field. Institute of Theoretical and Experimental Physics.
19. Strauss, W., & Vazquez, L. (1978). Numerical solution of a nonlinear Klein-Gordon equation. Journal of Computational Physics, 28(2), 271-278.
20. Jiménez, S., & Vázquez, L. (1990). Analysis of four numerical schemes for a nonlinear Klein-Gordon equation. Applied Mathematics and Computation, 35(1), 61-94.
21. Dehghan, M., & Shokri, A. (2009).Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. Journal of Computational and Applied Mathematics, 230(2), 400-410.
22. Sarboland, M., Aminataei, A. (2015). Numerical Solution of the Nonlinear Klein-Gordon Equation Using Multiquadric Quasi-interpolation Scheme. Universal Journal of Applied Mathematics, 3(3), 40-49.
23. Zahra, W. K., Ouf, W. A., & El-Azab, M. S. (2016). A robust uniform B-spline collocation method for solving the generalized PHI-four equation. Applications and Applied Mathematics, 11(1), 384-396.
24. Rashidinia, J., Ghasemi, M., & Jalilian, R. (2010). Numerical solution of the nonlinear Klein-Gordon equation. Journal of Computational and Applied Mathematics, 233(8), 1866-1878.
25. Rashidinia, J., Mohammadi, R., (2010). Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Computer Physics Communications, 181, 78-91.
26. Lakestani, M., Dehghan, M., (2010). A Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation. Computer Physics Communications, 181, 1392-1401.
27. Khuri, S. A., Sayfy, A., (2010). A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation. Applied Mathematics and Computation, 216, 1047-1056.
28. McCartin, B.J., Theory of exponential splines. Journal of Approximation Theory, 661, 1-23, 1991.
29. Korkmaz, A., & Dag, I. (2013). Cubic B-spline differential quadrature methods and stability for Burgers' equation. Engineering Computations, 30(3), 320-344.
30. Mohammadi, R., Exponential B-Spline Solution of Convection-Diffusion Equations. Applied Mathematics, 4, 933-944, 2013.
31. Mohammadi, R., Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation Chin. Phys. B, 24 5, 050206, 2015.
32. Ersoy, O., & Dag, I., The exponential cubic B-spline algorithm for Korteweg-de Vries Equation, Advances in Numerical Analysis, Article ID 367056, 2015.
33. Ersoy, O., & Dag, I. (2016). The Exponential Cubic B-Spline Collocation Method for the Kuramoto-Sivashinsky Equation. Filomat, 30 3, 853-861.
34. Dag, I., & Ersoy, O. (2016). The exponential cubic B-spline algorithm for Fisher equation. Chaos, Solitons & Fractals, 86, 101-106.
35. Ersoy, O., & Dag, I. (2015). Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms. Open Physics, 13(1).
36. Zaki, S.I., Gardner L.R.T., Gardner G.A., 1997, Numerical simulations of Klein-Gordon solitary wave interactions Il Nuovo Cimento, 112B, N.7.
37. A. Boz, Solution of Klein Gordon Equation using B-Spline Collocation Methods. PhD Thesis, 2006.