Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions
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İhsan Çelikkaya
Abstract
In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] H. R. Marasi and A. P. Aqdam, “Homotopy analysis method and homotopy Padè approximants for solving the Fornberg-Whitham equation,” Eurasian Math. J., vol. 6, no. 1, pp. 65–75, 2015.Search in Google Scholar
[2] G. B. Whitham, “Variational methods and applications to water waves,” Proc. R. Soc. London, Ser. A, vol. 299, pp. 6–25, 1967.10.1098/rspa.1967.0119Search in Google Scholar
[3] B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Phil. Trans. R. Soc. Lond. A., vol. 289, pp. 373–404, 1978.10.1098/rsta.1978.0064Search in Google Scholar
[4] B. He, Q. Meng, and S. Li, “Explicit peakon and solitary wave solutions for the modified Fornberg–Whitham equation,” Appl. Math. Comput., vol. 217, no. 5, pp. 1976–1982, 2010. https://doi.org/10.1016/j.amc.2010.06.055.Search in Google Scholar
[5] J. Lu, “An analytical approach to the Fornberg–Whitham type equations by using the variational iteration method,” Comput. Math. Appl., vol. 61, pp. 2010–2013, 2011. https://doi.org/10.1016/j.camwa.2010.08.052.Search in Google Scholar
[6] B. Boutarfa, A. Akgül, and M. Inc, “New approach for the Fornberg–Whitham type equations,” J. Comput. Appl. Math., vol. 312, pp. 13–26, 2017. https://doi.org/10.1016/j.cam.2015.09.016.Search in Google Scholar
[7] G. Hörmann and H. Okamoto, “Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation,” Discrete Contin. Dyn. Syst., vol. 39, pp. 4455–4469, 2019. https://doi.org/10.3934/dcds.2019182.Search in Google Scholar
[8] M. Dehghan and J. M. Heris, “Study of the wave-breaking’s qualitative behavior of the Fornberg-Whitham equation via quasi-numeric approaches,” Internat. J. Numer. Methods Heat Fluid Flow, vol. 22, no. 5, pp. 537–553, 2012. https://doi.org/10.1108/09615531211231235.Search in Google Scholar
[9] S. Hesam, A. R. Nazemi, and A. Haghbin, “Reduced differential transform method for solving the Fornberg-Whitham type equation,” Int. J. Nonlinear Sci., vol. 13, no. 2, pp. 158–162, 2012.Search in Google Scholar
[10] E. Az-Zo’bi, “Peakon and solitary wave solutions for the modified Fornberg-Whitham equation using simplest equation method,” Int. J. Math. Comput. Sci., vol. 14, no. 3, pp. 635–645, 2019.Search in Google Scholar
[11] S. Li and M. song, “Kink-like wave and compacton-like wave solutions for a two-component Fornberg-Whitham equation,” Abstr. Appl. Anal., vol. 2014, pp. 1–13, 2014. https://doi.org/10.1155/2014/621918.Search in Google Scholar
[12] J. Zhou and L. Tian, “A type of bounded traveling wave solutions for the Fornberg-Whitham equation,” J. Math. Anal. Appl., vol. 346, pp. 255–261, 2008. https://doi.org/10.1016/j.jmaa.2008.05.055.Search in Google Scholar
[13] A. Chen, J. Li, and W. Huang, “Single peak solitary wave solutions for the Fornberg-Whitham equation,” Appl. Anal., vol. 91, no. 3, pp. 587–600, 2012. https://doi.org/10.1080/00036811.2010.550577.Search in Google Scholar
[14] M. A. Ramadan and M. S. Al-Luhaibi, “New iterative method for solving the Fornberg-Whitham equation and comparison with homotopy perturbation transform method,” Br. J. Math. Comput. Sci., vol. 4, no. 9, pp. 1213–1227, 2014. https://doi.org/10.9734/bjmcs/2014/8534.Search in Google Scholar
[15] F. Abidi and K. Omrani, “Numerical solutions for the nonlinear Fornberg-Whitham equation by He’s methods,” Int. J. Mod. Phys. B, vol. 25, pp. 4721–4732, 2011. https://doi.org/10.1142/S0217979211059516.Search in Google Scholar
[16] A. Chen, J. Li, X. Deng, and W. Huang, “Travelling wave solutions of the Fornberg-Whitham equation,” Appl. Math. Comput., vol. 215, no. 8, pp. 3068–3075, 2009. https://doi.org/10.1016/j.amc.2009.09.057.Search in Google Scholar
[17] J. Biazar and M. Eslami, “Approximate solutions for Fornberg-Whitham type equations,” Int. J. Numer. Methods Heat Fluid Flow, vol. 22, no. 6, pp. 803–812, 2012. https://doi.org/10.1108/09615531211244934.Search in Google Scholar
[18] F. Abidi and K. Omrani, “The homotopy analysis method for solving the Fornberg–Whitham equation and comparison with Adomian’s decomposition method,” Comput. Math. Appl., vol. 59, no. 8, pp. 2743–2750, 2010. https://doi.org/10.1016/j.camwa.2010.01.042.Search in Google Scholar
[19] M. Yağmurlu, E. Yıldız, Y. Uçar, and A. Esen, “Numerical investigation of modified Fornberg Whitham equation,” Mathematical Sciences and Applications e-Notes, vol. 9, no. 2, pp. 81–94, 2021. https://doi.org/10.36753/mathenot.778766.Search in Google Scholar
[20] D. L. Logan, A First Course in the Finite Element Method, 4th ed. Canada, Thomson, 2007.Search in Google Scholar
[21] S. S. Rao, The Finite Element Method in Engineering, 5th ed. Oxford, Elsevier/Butterworth Heinemann, 2011.10.1016/B978-1-85617-661-3.00001-5Search in Google Scholar
[22] G. Walz, “Identities for trigonometric B-splines with an application to curve design,” BIT, vol. 37, pp. 189–201, 1997. https://doi.org/10.1007/bf02510180.Search in Google Scholar
[23] P. Keskin, “Trigonometric B-Spline Solutions of the RLW Equation,” Doctoral Dissertation, Department of Mathematics & Computer, 2017.Search in Google Scholar
[24] J. VonNeumann and R. D. Richtmyer, “A method for the numerical calculation of hydrodynamic shocks,” J. Appl. Phys., vol. 21, pp. 232–237, 1950. https://doi.org/10.1063/1.1699639.Search in Google Scholar
[25] CERC. Shore Protection Manual, Vicksburg, MS, Coastal Engineering Research Center, US Corps of Engineer, 1984.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical solutions for variable-order fractional Gross–Pitaevskii equation with two spectral collocation approaches
- Threshold dynamics of an HIV-1 model with both virus-to-cell and cell-to-cell transmissions, immune responses, and three delays
- Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model
- On modeling column crystallizers and a Hermite predictor–corrector scheme for a system of integro-differential equations
- On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale
- A new self adaptive Tseng’s extragradient method with double-projection for solving pseudomonotone variational inequality problems in Hilbert spaces
- Solitary waves of the RLW equation via least squares method
- Optical solitons and stability regions of the higher order nonlinear Schrödinger’s equation in an inhomogeneous fiber
- Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces
- Modified inertial subgradient extragradient method for equilibrium problems
- Propagation of dark-bright soliton and kink wave solutions of fluidized granular matter model arising in industrial applications
- Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative
- Investigation of nonlinear fractional delay differential equation via singular fractional operator
- Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions
- The (3 + 1)-dimensional Wazwaz–KdV equations: the conservation laws and exact solutions
- Stability and ψ-algebraic decay of the solution to ψ-fractional differential system
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- Numerical simulation of particulate matter propagation in an indoor environment with various types of heating
- Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces
- Stability analysis and abundant closed-form wave solutions of the Date–Jimbo–Kashiwara–Miwa and combined sinh–cosh-Gordon equations arising in fluid mechanics
- Contrasting effects of prey refuge on biodiversity of species
