A HYBRID CUBIC B-SPLINE APPROACH FOR THE NUMERICAL TREATMENT OF THE SINE-GORDON EQUATION

Iqra safdar; Muhammad Amin; Tanveer Abbas; Mubashar Shahzad

doi:10.71146/kjmr897

Authors

Iqra safdar Faculty of Sciences, The Superior University Lahore, Pakistan Author
Muhammad Amin Faculty of Sciences, The Superior University Lahore, Pakistan Author
Tanveer Abbas Department of Mathematics, Government College University, Faisalabad, Pakistan. Author
Mubashar Shahzad Department of Mathematics, Government College University, Faisalabad, Pakistan. Author

DOI:

https://doi.org/10.71146/kjmr897

Keywords:

Hybrid cubic B-Spline, system of equations, Sine-Gordon equation, spatial variable, finite difference scheme

Abstract

In this article, a hybrid cubic B-spline collocation method has been proposed to find the numerical solutions of a well-known nonlinear Sine-Gordon equation. The finite difference scheme has been employed to discretize the time derivative, whereas cubic B-spline functions are used for spatial discretization. The efficiency of the applied method is checked through some test problems. Numerical outcomes are compared with the exact solutions available in the literature which illustrate the effectiveness of the proposed method. It can easily be concluded that the results obtained are reliable and consistent with those found in earlier research.

Downloads

Download data is not yet available.

References

Ahmad, N. R. (2025). The impact of fintech startups on financial innovation and stability in Pakistan’s evolving financial landscape. International Journal of Financial Innovation, 8(1), 112–126.

[1] M. Shaheen, M. Abbas, F. A. Abdullah, and Y. S. Hamed, “A new numerical technique for the solution of time-fractional nonlinear Klein-Gordon equation involving Atangana-Baleanu derivative using cubic B-spline functions,” Open Phys., vol. 23, no. 1, Jan. 2025, doi: 10.1515/PHYS-2025-0131/PDF.

[2] G. Ben-Yu, P. J. Pascual, M. J. Rodriguez, and L. Vázquez, “Numerical solution of the sine-Gordon equation,” Appl. Math. Comput., vol. 18, no. 1, pp. 1–14, 1986, doi: 10.1016/0096-3003(86)90025-1.

[3] M. Amin, M. Abbas, M. K. Iqbal, and D. Baleanu, “Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions,” Front. Phys., vol. 8, no. September, pp. 1–13, 2020, doi: 10.3389/fphy.2020.00288.

[4] R. M. Ganji, H. Jafari, M. Kgarose, and A. Mohammadi, “Numerical solutions of time-fractional Klein–Gordon equations by clique polynomials,” Alex. Eng. J., vol. 60, no. 5, pp. 4563–4571, Oct. 2021, doi: 10.1016/j.aej.2021.03.026.

[5] A. Mohebbi and M. Dehghan, “High order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN method,” Math Comput Model, vol. 51, no. 5–6, pp. 537–549, Mar. 2010, doi: 10.1016/j.mcm.2009.11.015.

[6] C. Lu, W. Huang, and J. Qiu, “An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation,” J. Sci. Comput., vol. 74, no. 1, pp. 122–144, 2018, doi: 10.1007/s10915-017-0427-6.

[7] K. Djidjeli, W. G. Price, and E. H. Twizell, “Numerical solutions of a damped sine-Gordon equation in TWo space variables,” J Eng Math, vol. 24, no. 4, pp. 347–369, Jul. 1995, doi: 10.1007/bf00042761.

[8] N. A. Kudryashov, “Exact solutions of the generalized Kuramoto-Sivashinsky equation,” Phys. Lett. A, vol. 147, no. 5–6, pp. 287–291, Jul. 1990, doi: 10.1016/0375-9601(90)90449-X.

[9] B. Batiha, M. S. M. Noorani, and I. Hashim, “Approximate analytical solution of the coupled sine-Gordon equation using the variational iteration method,” Phys. Scr., vol. 76, no. 5, pp. 445–448, Nov. 2007, doi: 10.1088/0031-8949/76/5/007.

[10] R. C. Mittal and R. Bhatia, “Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method,” Int. J. Partial Differ. Equations, vol. 2014, pp. 1–8, Aug. 2014, doi: 10.1155/2014/343497.

[11] A. G. Bratsos, “A fourth order numerical scheme for the one-dimensional sine-Gordon equation,” Int. J. Comput. Math., vol. 85, no. 7, pp. 1083–1095, Jul. 2008, doi: 10.1080/00207160701473939.

[12] A. Shokri and M. Dehghan, “A meshless method using the radial basis functions for numerical solution of the regularized long wave equation,” Numer. Methods Partial Differ. Equ., vol. 26, no. 4, pp. 807–825, 2010, doi: 10.1002/num.20457.

[13] J. Rashidinia and R. Mohammadi, “Tension spline solution of non-linear sine-Gordon,” Numer Algorithms, vol. 56, no. 1, pp. 129–142, Jan. 2011, doi: 10.1007/s11075-010-9377-x.

[14] Z. W. Jiang and R. H. Wang, “Numerical solution of sine-Gordon equation using high accuracy multiquadric quasi-interpolation,” Appl Math Comput, vol. 218, no. 15, pp. 7711–7716, Apr. 2012, doi: 10.1016/j.amc.2011.12.095.

[15] M. Uddin, S. Haq, and Siraj-ul-Islam, “A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations,” Appl. Math. Comput., vol. 212, no. 2, pp. 458–469, Jun. 2009, doi: 10.1016/j.amc.2009.02.037.

[16] M. Dehghan and A. Shokri, “A numerical method for one-dimensional nonlinear sine-gordon equation using collocation and radial basis functions,” Numer. Methods Partial Differ. Equ., vol. 24, no. 2, pp. 687–698, 2008, doi: 10.1002/NUM.20289.

[17] M. Dehghan and A. Shokri, “Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions,” J. Comput. Appl. Math., vol. 230, no. 2, pp. 400–410, Aug. 2009, doi: 10.1016/j.cam.2008.12.011.

[18] University of Texas, “No TitleΕΛΕΝΗ,” Afghanisan Physiogr. Map, vol. 31, no. 2, pp. 1–106, 2009.

[19] A. G. Bratsos, “A fourth order numerical scheme for the one-dimensional sine-Gordon equation,” Int. J. Comput. Math., vol. 85, no. 7, pp. 1083–1095, Jul. 2008, doi: 10.1080/00207160701473939.