Abstract
In this paper, we obtained the exact and solitary wave solutions of modified equal width wave equation by using Lie symmetry method. With the help of MAPLE software we obtained infinitesimal generators and commutation table. Lie symmetry transformation has been used for converting nonlinear partial differential equation into nonlinear ordinary differential equation. Then, we used tanh method and power series method for solving reduced nonlinear ordinary differential equations. Convergence of power series solution has also been shown.
Similar content being viewed by others
References
Karakoc, S.B.G., Geyikli, T.: Numerical solution of the modified equal width wave equation. Int. J. Differ. Equ. 2012, 1–15 (2012)
Essa, Y.M.A.: Multigrid method for the numerical soluton of the modified equal width wave equation. Appl. Math. 7, 1140–1147 (2016)
Gardner, L.R.T., Gardner, G.A.: Solitary waves of the regularised long-wave equation. J. Comput. Phys. 91(2), 441–459 (1990)
Gardner, L.R.T., Gardner, G.A.: Solitary waves of the equal width wave equation. J. Comput. Phys. 101(1), 218–223 (1992)
Morrison, P.J., Meiss, J.D., Cary, J.R.: Scattering of regularized-long-wave solitary waves. Physica D. Nonlinear Phenomena 11(3), 324–336 (1984)
Gardner, L.R.T., Gardner, G.A., Geyikli, T.: The boundary forced MKdV equation. J. Comput. Phys. 113(1), 5–12 (1994)
Abdulloev, Kh O., Bogolubsky, I.L., Makhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56(6), 427–428 (1976)
Geyikli, T., Karakoc, S.B.G.: Septic B-spline collocation method for numerical solution of modified equal width wave equation. Appl. Math. 2(6), 739–749 (2011)
Geyikli, T., Karakoc, S.B.G.: Petro-Galerkin method with cubic bsplines for solving the MEW equation. Bull. Belgian Math. Soc. Simon Stevin 19, 215–227 (2012)
Arora, R., Siddiqui, Md J., Singh, V.P.: Solution of modified equal width wave equation, its variant and non-homogeneous Burgers’ equation by RDT method. Am. J. Comput. Appl. Math. 1(2), 53–56 (2011)
Saka, B.: Algorithms for numerical solution of the modified equal width wave equation using collocation method. Math. Comput. Model. 45(9–10), 1117–2007 (2007)
Wazwaz, A.M.: The tanh and sine–cosine methods for a reliable treatment of the modified equal width wave equation and its variants. Commun. Nonlinear Sci. Numer. Simul. 11(2), 148–160 (2006)
Din, S.T.M., Yildirim, A., Berberler, M.E., Hosseini, M.M.: Numerical solution of the modified equal width wave equation. World Appl. Sci. J. 8(7), 792–798 (2010)
Hassan, H.N.: An accurate numerical solution for the modified equal width wave equation using the Fourier pseudo-spectral method. J. Appl. Math. Phys. Lett. A 4, 1054–1067 (2016)
Esen, A., Kutluay, S.: Solitary wave solutions of the modified equal width wave equation. Commun. Nonlinear Sci. Numer. Simul. 13(8), 1538–1546 (2008)
Wang, H., Chen, L., Wang, H.: Exact travelling wave solutions of the modified equal width wave equation via the dynamical system method. Nonlinear Anal. Differ. Equ. 4(1), 9–15 (2016)
Sahoo, S., Ray, S.S.: Lie symmetry analysis and exact solutions of (3+1) dimensional Yu–Toda–Sasa–Fukuyama equation in mathematical physics. Comput. Math. Appl. 73, 253–260 (2017)
Yang, S., Hua, C.: Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation. Appl. Math. Comput. 234, 579–583 (2014)
Wang, G., Kara, A.H., Fakhar, K., Guzman, J.V.: Group analysis. Exact solutions and conservation laws of a generalized fifth order KdV equation. Chaos Solitons Fractals 86, 8–15 (2016)
Kumar, M., Kumar, R., Kumar, A.: Some more similarity solutions of the (2+1)-dimensional BLP system. Comput. Math. Appl. 70, 212–221 (2015)
Kumar, M., Kumar, R., Kumar, A.: On similarity solutions of Zabolotskaya–Khokhlov equation. Comput. Math. Appl. 68, 454–463 (2014)
Olver, P.J.: Applications of Lie Groups to Differential Equations, pp. 30–130. Springer, New York (1993)
Bluman, G., Cheviakov, A.: Applications of Symmetry Methods to Partial Differential Equations. Appl. Math. Sci., vol. 168, p. 417. Springer, New York (2010)
Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations. Springer, New York (1974)
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)
Goyal, N., Wazwaz, A.M., Gupta, R.K.: Applications of maple software to derive exact solutions of generalized fifth-order Korteweg–De Vries equation with time-dependent coefficients. Rom. Rep. Phys. 68(1), 99–111 (2016)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)
Wazwaz, A.M.: Partial Differential Equation and Solitary Wave Theory. Nonlinear Physical Science. Springer, New York (2009)
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. China Machine Press, Beijing (2004)
Acknowledgements
The second author is thankful to the “University Grants Commission (UGC)” India for financial support to carry out her research work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arora, R., Chauhan, A. Lie Symmetry Reductions and Solitary Wave Solutions of Modified Equal Width Wave Equation. Int. J. Appl. Comput. Math 4, 122 (2018). https://doi.org/10.1007/s40819-018-0557-z
Published:
DOI: https://doi.org/10.1007/s40819-018-0557-z