QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION

Reza Abazarı; Kenan Yıldırım

QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION

Abstract

In this paper, the quintic B–spline method is employed to calculatenumerical solution of the initial-boundary value problem of Rosenau–Burgersequation. This scheme is based on the Crank–Nicolson formulation for time integration and quintic B–spline functions for space integration. The unconditional stability of the method is proved using Von–Neumann approach. A priori bound and the error estimates of the approximate solutions are discussed with a numerical example.

Keywords

References

[1] Behnam Sepehrian, Mahmood Lashani. A numerical Solution of the Burgers equation using quintic B-splines. In: Proceedings of the world congress on engineering 2008, vol. III, WCE; UK: London, 2008.
[2] Dag I, Saka B, Irk D. Galerkin method for the numerical solution of the RLW equation using quintic B-splines. Comput. Appl. Math, 190 (2006) 532–547.
[3] M.A. Park, On the Rosenau equation, Math. Appl. Comput. 9 (1990) 145–152.
[4] P. Rosenau, A quasi-continuous description of a nonlinear transmission line, Phys. Scripta 34 (1986) 827–829.
[5] P. Rosenau, Dynamics of dense discrete systems, Progr. Theor. Phys. 79 (1988) 1028–1042.
[6] S.K. Chung, Finite difference approximate solutions for the Rosenau equation, Appl. Anal. 69 (1–2)(1998) 149–156.
[7] S.K. Chung, A.K. Pani, Numerical methods for the Rosenau equation, Appl. Anal. 77 (2001) 351–369.
[8] S.A. Manickam, A.K. Pani, S.K. Chung, A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation, Numer. Meth. Partial Diff. Eq. 14 (1998) 695–716.

[9] Y.D. Kim, H.Y. Lee, The convergence of finite element Galerkin solution for the Rosenau equation, Korean J. Comput. Appl. Math. 5 (1998) 171–180.
[10] Zaki SI. A quintic B-spline finite elements scheme for the KdVB equation. Comput. Meth. Appl. Eng. 188 (2000) 121–134.
[11] Gardner GA, Gardner LRT, Ali A. H. A., Modelling solitons of the Korteweg-de Vries equation with quintic B-splines, U.C.N.W. Math., Preprint; 1990.
[12] Mittal RC, Arora G. Quintic B-spline collocation method for numerical solution of the KuramotoSivashinsky equation, Commun. Nonlinear Sci. Numer. Simulat, 15(10 (2010) 2798–2808.
[13] Aurelian Bejancu, Gradient superconvergence for a class of semi-cardinal interpolation schemes with cubic and quintic B- splines, Appl. Math. Comput, 308(1) (2017) 142–148.
[14] Ram KishunLodhi, Hradyesh KumarMishra, Quintic B-spline method for solving second order linear and nonlinear singularly perturbed two-point boundary value problems, Comput. Appl. Math, 319(1)(2017) 170–187.
[15] Alper Korkmaz, Idris Dag, Quartic and quintic B-spline methods for advectiondiffusion equation, Appl. Math. Comput, 274(1) (2016) 208–219.
[16] Hepson, O.E., Korkmaz, A., Dag, I., Numerical solutions of the Gardner equation by extended form of the cubic B-splines, Pramana - J. Phys, (2018) 91: 59. https://doi.org/10.1007/s12043-018-1631-0.
[17] B. Hu, Y. Xu, J. Hu, Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation, Appl. Math. Comput, 204, (2008), 311– 316.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Reza Abazarı This is me
0000-0003-0125-2958
Iran

Kenan Yıldırım This is me
0000-0002-4471-3964
Türkiye

Publication Date

September 1, 2020

Submission Date

December 13, 2018

Acceptance Date

June 20, 2019

Published in Issue

Year 2019 Volume: 37 Number: 3

IZ

https://izlik.org/JA72HU78HM

Cite

RIS / Bibtex

APA

Abazarı, R., & Yıldırım, K. (2020). QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION. Sigma Journal of Engineering and Natural Sciences, 37(3), 967-979. https://izlik.org/JA72HU78HM

AMA

1.Abazarı R, Yıldırım K. QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION. SIGMA. 2020;37(3):967-979. https://izlik.org/JA72HU78HM

Chicago

Abazarı, Reza, and Kenan Yıldırım. 2020. “QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences 37 (3): 967-79. https://izlik.org/JA72HU78HM.

EndNote

Abazarı R, Yıldırım K (September 1, 2020) QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION. Sigma Journal of Engineering and Natural Sciences 37 3 967–979.

IEEE

[1]R. Abazarı and K. Yıldırım, “QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION”, SIGMA, vol. 37, no. 3, pp. 967–979, Sept. 2020, [Online]. Available: https://izlik.org/JA72HU78HM

ISNAD

Abazarı, Reza - Yıldırım, Kenan. “QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences 37/3 (September 1, 2020): 967-979. https://izlik.org/JA72HU78HM.

JAMA

1.Abazarı R, Yıldırım K. QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION. SIGMA. 2020;37:967–979.

MLA

Abazarı, Reza, and Kenan Yıldırım. “QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION”. Sigma Journal of Engineering and Natural Sciences, vol. 37, no. 3, Sept. 2020, pp. 967-79, https://izlik.org/JA72HU78HM.

Vancouver

1.Reza Abazarı, Kenan Yıldırım. QUINTIC B-SPLINE METHOD FOR NUMERICAL SOLUTION OF THE ROSENAU-BURGERS EQUATION. SIGMA [Internet]. 2020 Sep. 1;37(3):967-79. Available from: https://izlik.org/JA72HU78HM