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Year 2021, , 81 - 94, 01.06.2021
https://doi.org/10.36753/mathenot.778766

Abstract

References

  • [1] Abidi, F., Omrani, K.: The Homotopy Analysis Method for Solving the Fornberg–Whitham Equation and Comparison With Adomian’s Decomposition Method. Comput. Math. Appl. 59 (8), 2743-2750 (2010).
  • [2] Bariza, B., Akgul A., Inc M.: New approach for the Fornberg “Whitham type equations. J. Comput. Appl. Math., 312 (1), 13-26 (2017).
  • [3] Jafar, B., Eslami, M.: Approximate solutions for Fornberg-Whitham type equations. Internat. J. Numer. Methods Heat Fluid Flow, 22 (6), 803-812 (2012).
  • [4] Nuseir, A.S.: New exact solutions to the modified Fornberg-Whitham equation. Taiwanese J. Math., 16 (6), 2083-2091 (2012).
  • [5] Chen, A., Li, J., Deng, X., Huang, W.: Travelling Wave Solutions of the Fornberg–Whitham Equation. Appl. Math. Comput. 215 (8), 3068-3075 (2009).
  • [6] Fornberg, B., Whitham, G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. A Trans. R. Soc. London (1978).
  • [7] Gupta, P.K.,Singh, M.: Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation. Comput. Math. Appl. 61 (2), 250-254 (2011).
  • [8] Geiser, J.: Decomposition methods for differential equations: theory and applications. CRC Press. London (2009).
  • [9] He, B., Meng, Q., Li, S.: Explicit peakon and solitary wave solutions for the modified Fornberg–Whitham equation. Appl. Math. Comput. 217 (1), 1976-1982 (2010).
  • [10] Hesam, S., Nazemi, A., Haghbin, A.: Reduced Differential Transform Method for Solving the Fornberg–Whitham Type Equation. Int. J. Nonlinear Sci. 13 (2), 158-162 (2012).
  • [11] Lu, J.: An analytical approach to the Fornberg Whitham type equations by using the variational iteration method. Comput. Math. Appl., 61 (8), 2010-2013 (2011).
  • [12] Marchuk, G.I.: Some application of splitting-up methods to the solution of mathematical physics problems. Aplikace Matematiky. 13 (1), 103-132 (1968).
  • [13] Prenter, P.M.: Splines and Variational Methods. Wiley Publications. New York (1975).
  • [14] Ray, S., Gupta, A.K.: Numerical Investigation of Time- Fractional Modified Fornberg–Whitham Equation for Analysing the Behaviour of Water Waves. Appl. Math. Comput. 266 (1), 135-148 (2015).
  • [15] Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (3), 506-517 (1968).
  • [16] Zhou, J., Tian, L.: Type of Bounded Traveling Wave Solutions for the Fornberg–Whitham Equation. J. Math. Anal. Appl. 346 (1), 255-261 (2008).

Numerical Investigation of Modified Fornberg Whitham Equation

Year 2021, , 81 - 94, 01.06.2021
https://doi.org/10.36753/mathenot.778766

Abstract

The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms $L_{2}$ and $L_{\infty}$. The presentations of the numerical results are shown by graphics as well.

References

  • [1] Abidi, F., Omrani, K.: The Homotopy Analysis Method for Solving the Fornberg–Whitham Equation and Comparison With Adomian’s Decomposition Method. Comput. Math. Appl. 59 (8), 2743-2750 (2010).
  • [2] Bariza, B., Akgul A., Inc M.: New approach for the Fornberg “Whitham type equations. J. Comput. Appl. Math., 312 (1), 13-26 (2017).
  • [3] Jafar, B., Eslami, M.: Approximate solutions for Fornberg-Whitham type equations. Internat. J. Numer. Methods Heat Fluid Flow, 22 (6), 803-812 (2012).
  • [4] Nuseir, A.S.: New exact solutions to the modified Fornberg-Whitham equation. Taiwanese J. Math., 16 (6), 2083-2091 (2012).
  • [5] Chen, A., Li, J., Deng, X., Huang, W.: Travelling Wave Solutions of the Fornberg–Whitham Equation. Appl. Math. Comput. 215 (8), 3068-3075 (2009).
  • [6] Fornberg, B., Whitham, G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. A Trans. R. Soc. London (1978).
  • [7] Gupta, P.K.,Singh, M.: Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation. Comput. Math. Appl. 61 (2), 250-254 (2011).
  • [8] Geiser, J.: Decomposition methods for differential equations: theory and applications. CRC Press. London (2009).
  • [9] He, B., Meng, Q., Li, S.: Explicit peakon and solitary wave solutions for the modified Fornberg–Whitham equation. Appl. Math. Comput. 217 (1), 1976-1982 (2010).
  • [10] Hesam, S., Nazemi, A., Haghbin, A.: Reduced Differential Transform Method for Solving the Fornberg–Whitham Type Equation. Int. J. Nonlinear Sci. 13 (2), 158-162 (2012).
  • [11] Lu, J.: An analytical approach to the Fornberg Whitham type equations by using the variational iteration method. Comput. Math. Appl., 61 (8), 2010-2013 (2011).
  • [12] Marchuk, G.I.: Some application of splitting-up methods to the solution of mathematical physics problems. Aplikace Matematiky. 13 (1), 103-132 (1968).
  • [13] Prenter, P.M.: Splines and Variational Methods. Wiley Publications. New York (1975).
  • [14] Ray, S., Gupta, A.K.: Numerical Investigation of Time- Fractional Modified Fornberg–Whitham Equation for Analysing the Behaviour of Water Waves. Appl. Math. Comput. 266 (1), 135-148 (2015).
  • [15] Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (3), 506-517 (1968).
  • [16] Zhou, J., Tian, L.: Type of Bounded Traveling Wave Solutions for the Fornberg–Whitham Equation. J. Math. Anal. Appl. 346 (1), 255-261 (2008).
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Murat Yağmurlu 0000-0003-1593-0254

Ersin Yıldız This is me

Yusuf Uçar 0000-0003-1469-5002

Alaattin Esen 0000-0002-7927-5941

Publication Date June 1, 2021
Submission Date August 10, 2020
Acceptance Date December 16, 2020
Published in Issue Year 2021

Cite

APA Yağmurlu, M., Yıldız, E., Uçar, Y., Esen, A. (2021). Numerical Investigation of Modified Fornberg Whitham Equation. Mathematical Sciences and Applications E-Notes, 9(2), 81-94. https://doi.org/10.36753/mathenot.778766
AMA Yağmurlu M, Yıldız E, Uçar Y, Esen A. Numerical Investigation of Modified Fornberg Whitham Equation. Math. Sci. Appl. E-Notes. June 2021;9(2):81-94. doi:10.36753/mathenot.778766
Chicago Yağmurlu, Murat, Ersin Yıldız, Yusuf Uçar, and Alaattin Esen. “Numerical Investigation of Modified Fornberg Whitham Equation”. Mathematical Sciences and Applications E-Notes 9, no. 2 (June 2021): 81-94. https://doi.org/10.36753/mathenot.778766.
EndNote Yağmurlu M, Yıldız E, Uçar Y, Esen A (June 1, 2021) Numerical Investigation of Modified Fornberg Whitham Equation. Mathematical Sciences and Applications E-Notes 9 2 81–94.
IEEE M. Yağmurlu, E. Yıldız, Y. Uçar, and A. Esen, “Numerical Investigation of Modified Fornberg Whitham Equation”, Math. Sci. Appl. E-Notes, vol. 9, no. 2, pp. 81–94, 2021, doi: 10.36753/mathenot.778766.
ISNAD Yağmurlu, Murat et al. “Numerical Investigation of Modified Fornberg Whitham Equation”. Mathematical Sciences and Applications E-Notes 9/2 (June 2021), 81-94. https://doi.org/10.36753/mathenot.778766.
JAMA Yağmurlu M, Yıldız E, Uçar Y, Esen A. Numerical Investigation of Modified Fornberg Whitham Equation. Math. Sci. Appl. E-Notes. 2021;9:81–94.
MLA Yağmurlu, Murat et al. “Numerical Investigation of Modified Fornberg Whitham Equation”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 2, 2021, pp. 81-94, doi:10.36753/mathenot.778766.
Vancouver Yağmurlu M, Yıldız E, Uçar Y, Esen A. Numerical Investigation of Modified Fornberg Whitham Equation. Math. Sci. Appl. E-Notes. 2021;9(2):81-94.

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