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Comment on: “The modified extended tanh-function method for solving Burgers-type equations”. (English) Zbl 07457339

Summary: In this letter, we analyze the paper by A. A. Soliman [“The modified extended tanh-function method for solving Burgers-type equations”, ibid. 361, No. 2, 394–404 (2006; doi:10.1016/j.physa.2005.07.008)]. Using the modified extended tanh-function (METF) method, Soliman have found exact “solutions” of the Burgers type equations including one-dimensional Burgers and coupled Burgers equations. In this comment, we show that none of these solutions satisfy the corresponding Burgers equations. In addition, we provide the corrected exact solutions by correcting some Soliman’s mistakes and errors.

MSC:

82-XX Statistical mechanics, structure of matter
Full Text: DOI

References:

[1] Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fractals, 24, 1217-1231 (2005) · Zbl 1069.35018
[2] Kudryashov, N. A., Exact solitary waves of the Fisher equation, Phys. Lett. A, 342, 99-106 (2005) · Zbl 1222.35054
[3] Kudryashov, N. A.; Loguinova, N. B., Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205, 396-402 (2008) · Zbl 1168.34003
[4] Vitanov, N.; Dimitrova, Z.; Kantz, H., Modified method of simplest equation and its application to nonlinear PDEs, Appl. Math. Comput., 216, 2587-2595 (2010) · Zbl 1195.35272
[5] Vitanov, N. K., Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling wave solutions for a class of PDEs with polynomial nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 15, 2050-2060 (2010) · Zbl 1222.35062
[6] Soliman, A., The modified extended tanh-function method for solving Burgers-type equations, Physica A, 361, 394-404 (2006)
[7] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277, 212-218 (2000) · Zbl 1167.35331
[8] Jiwari, R., A hybrid numerical scheme for the numerical solution of the Burgers’ equation, Comput. Phys. Comm., 188, 59-67 (2015) · Zbl 1344.65082
[9] Khater, A.; Temsah, R.; Hassan, M., A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math., 222, 333-350 (2008) · Zbl 1153.65102
[10] Kutluay, S.; Bahadir, A.; Ozdes, A., Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods, J. Comput. Appl. Math., 103, 251-256 (2007) · Zbl 0942.65094
[11] Liao, W., An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput., 206, 755-764 (2008) · Zbl 1157.65438
[12] Liao, W.; Zhu, J., Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation, Int. J. Comput. Math., 88, 2575-2590 (2011) · Zbl 1252.65141
[13] Piao, X.; Kim, S.; Kim, P., A new time stepping method for solving one dimensional Burgers’ equations, Kyungpook Math. J., 52, 327-346 (2012) · Zbl 1286.65134
[14] Esipov, S., Coupled Burgers equations: A model of polydispersive sedimentation, Phys. Rev. E., 52, 4, 3718-37711 (1995)
[15] Bak, S.; Kim, P.; Kim, D., A semi-Lagrangian approach for numerical simulation of coupled Burgers’ equations, Commun. Nonlinear Sci. Numer. Simul., 69, 31-44 (2019) · Zbl 1524.35327
[16] Bashan, A., A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number, Physica A, 545, Article 123755 pp. (2020)
[17] Bhatt, H.; Khaliq, A., Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Commun., 200, 117-138 (2016) · Zbl 1351.35167
[18] Lai, H.; Ma, C., A new lattice Bolthmann model for solving the coupled viscous Burgers’ equation, Physica A, 395, 445-457 (2014) · Zbl 1395.76070
[19] Mittal, R.; Arora, G., Numerical solution of the coupled viscous Burgers’ equation, Commun. Nonlinear Sci. Numer. Simul., 16, 1304-1313 (2011) · Zbl 1221.65264
[20] Nee, J.; Duan, J., Limit set of trajectorues of the coupled viscous Burgers’ equations, Appl. Math. Lett., 11, 1, 57-61 (1998) · Zbl 1076.35537
[21] Hopf, E., The partial differential equation \(u_t + u u_x = \mu u_{x x}\), Comm. Pure Appl. Math., 3, 201-230 (1950) · Zbl 0039.10403
[22] Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9, 225-236 (1951) · Zbl 0043.09902
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