The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs

M. A. Habib; H. M. Shahadat Ali; M. Mamun Miah; M. Ali Akbar; M. A. Habib; H. M. Shahadat Ali; M. Mamun Miah; M. Ali Akbar

doi:10.3934/math.2019.3.896

AIMS Mathematics

2019, Volume 4, Issue 3: 896-909. doi: 10.3934/math.2019.3.896

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Research article

The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs

1.
Department of Applied Mathematics, Noakhali Science and Technology University, Bangladesh
2.
Department of Mathematics, Khulna University of Engineering & Technology, Bangladesh
3.
Department of Applied Mathematics, University of Rajshahi, Bangladesh

Received: 14 April 2019 Accepted: 12 July 2019 Published: 25 July 2019
MSC : 34A08, 35R11

In this work, we construct closed form traveling wave solutions to some nonlinear evolution equations (NLEEs) associated with mathematical physics. This work implements the well-established generalized Kudryashov method (gKM) to compute new closed form traveling wave solutions to the Burgers-Huxley equation, the mKdV equation and the first extended fifth order nonlinear equation. Furthermore, in this investigation, we discuss the achieved results in details and portrayed some 2D and 3D figures with the aid of symbolic computation package like Mathematica. The worked-out results ascertained that the suggested generalized form of the Kudryashov method is a simple, efficient and reliable technique to deal with other kinds of NLEEs.
Citation: M. A. Habib, H. M. Shahadat Ali, M. Mamun Miah, M. Ali Akbar. The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs[J]. AIMS Mathematics, 2019, 4(3): 896-909. doi: 10.3934/math.2019.3.896

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Abstract

In this work, we construct closed form traveling wave solutions to some nonlinear evolution equations (NLEEs) associated with mathematical physics. This work implements the well-established generalized Kudryashov method (gKM) to compute new closed form traveling wave solutions to the Burgers-Huxley equation, the mKdV equation and the first extended fifth order nonlinear equation. Furthermore, in this investigation, we discuss the achieved results in details and portrayed some 2D and 3D figures with the aid of symbolic computation package like Mathematica. The worked-out results ascertained that the suggested generalized form of the Kudryashov method is a simple, efficient and reliable technique to deal with other kinds of NLEEs.

References

[1]	M. Wang, Y. Zhou, Z. Li, Application of homogeneous balance method to exact solutions of non-linear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67-75. doi: 10.1016/0375-9601(96)00283-6
[2]	M. Younis, A. Zafar, The modified simple equation method for solving the non-linear Phi-four equation, Int. J. Innov. Appl. Stud., 2 (2013), 661-664.
[3]	W. W. Li, Y. Tian, Z. Zhang, F-expansion method and its application for finding new exact solutions to the Sine-Gordon and Sinh-Gordon equations, Appl. Math. Comput., 219 (2012), 1135-1143.
[4]	S. Zhang, T. Xia, An improved generalized F-expansion method and its application to the (2+1) dimensional KdV equations, Commun. Nonlinear Sci., 13 (2008), 1294-1301. doi: 10.1016/j.cnsns.2006.12.008
[5]	H. M. Baskonus, New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics, Nonlinear Dynam., 86 (2016), 177-183. doi: 10.1007/s11071-016-2880-4
[6]	C. Cattani, T. A. Sulaiman, H. M. Baskonus, et al. Solitons in an inhomogeneous Murnaghan's rod, Eur. Phys. J. Plus, 133 (2018), 228.
[7]	C. Cattani, T. A. Sulaiman, H. M. Baskonus, et al. On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel's-Sokolov systems, Opt. Quant. Electron., 50 (2018), 138.
[8]	O. A. Ilhan, A. Esen, H. Bulut, et al. Singular solitons in the pseudo-parabolic model arising in nonlinear surface waves, Results Phys., 12 (2019), 1712-1715. doi: 10.1016/j.rinp.2019.01.059
[9]	A. M. Wazwaz, Multiple soliton solutions for the Boussinesq equation, Appl. Math. Comput., 192 (2007), 479-486.
[10]	A. M. Wazwaz, The tanh method for traveling wave solutions of non-linear equations, Appl. Math. Comput., 154 (2004), 713-723.
[11]	A. Bekir, New exact traveling wave solutions of some complex nonlinear equations, Commun. Nonlinear Sci., 14 (2009), 1069-1077. doi: 10.1016/j.cnsns.2008.05.007
[12]	M. Mirzazadeh, M. E. Slami, E. Zerrad, et al. Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoulli's equation approach, Nonlinear Dynam., 81 (2015), 1933-1949. doi: 10.1007/s11071-015-2117-y
[13]	M. Kaplan, A. Bekir, M. N. Ozer, Solving non-linear evolution equation system using two different methods, Open Phys., 13 (2015), 383-388.
[14]	M. M. Miah, H. M. S. Ali, M. A. Ali, et al. Some applications of the (G′/G, 1/G)-expansion method to find new exact solutions of NLEEs, Eur. Phys. J. Plus, 132 (2017), 252.
[15]	M. M. Miah, H. M. S. Ali, M. A. Ali, An investigation of abundant traveling wave solutions of complex nonlinear evolution equations: The perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation, Cog. Math., 3 (2016), 1277506.
[16]	A. Neirameh, New analytical solutions for the coupled nonlinear Maccari's system, Alex. Eng. J., 55 (2016), 2839-2847. doi: 10.1016/j.aej.2016.07.007
[17]	A. J. M. Jawad, M. J. Abu-Alshaeer, A. Biswas, et al. Hamiltonian perturbation of optical solitons with parabolic law non-linearity using three integration methodologies, Optik, 160 (2018), 248-254. doi: 10.1016/j.ijleo.2018.01.104
[18]	X. Lu, S. T. Chen, W. X. Ma, Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation, Nonlinear Dynam., 86 (2016), 523-534. doi: 10.1007/s11071-016-2905-z
[19]	X. Lu, W. X. Ma, Y. Zhou, et al. Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation, Comput. Math. Appl., 71 (2016), 1560-1567. doi: 10.1016/j.camwa.2016.02.017
[20]	X. Lu, W. X. Ma, S. T. Chen, et al. A note on rational solutions to a Hirota-Satsuma-like equation, Appl. Math. Lett., 58 (2016), 13-18. doi: 10.1016/j.aml.2015.12.019
[21]	J. H. He, M. A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Soliton. Fract., 34 (2007), 1421-1429. doi: 10.1016/j.chaos.2006.05.072
[22]	Y. Y. Wang, Y. P. Zhang, C. Q. Dai, RE-study on localized structures based on variable separation solutions from the modified tanh-function method, Nonlinear Dynam., 83 (2016), 1331-1339. doi: 10.1007/s11071-015-2406-5
[23]	E. Fan, Extended Tanh-Function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218. doi: 10.1016/S0375-9601(00)00725-8
[24]	J. Manafian, M. F. Aghdaei, M. Khalilian, et al. Application of the generalized (G′/G) expansion method for non-linear PDEs to obtaining soliton wave solution, Optik, 135 (2017), 395-406. doi: 10.1016/j.ijleo.2017.01.078
[25]	L. N. Gao, X. Y. Zhao, Y. Y. Zi, et al. Resonant behavior of multiple wave solutions to a Hirota bilinear equation, Comput. Math. Appl., 72 (2016), 1225-1229. doi: 10.1016/j.camwa.2016.06.008
[26]	L. N. Gao, Y. Y. Zi, Y. H. Yin, et al. Backlund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation, Nonlinear Dynam., 89 (2017), 2233-2240. doi: 10.1007/s11071-017-3581-3
[27]	Y. H. Yin, W. X. Ma, J. G. Liu, et al. Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction, Comput. Math. Appl., 76 (2018), 1275-1283. doi: 10.1016/j.camwa.2018.06.020
[28]	M. Kaplan, K. Hosseini, Investigation of exact solutions for the Tzitzéica type equations in nonlinear optics, Optik, 154 (2018), 393-397. doi: 10.1016/j.ijleo.2017.08.116
[29]	A. R. Adem, B. Muatjetjeja, Conservation laws and exact solutions for a 2D Zakharov-Kuznetsov equation, Appl. Math. Lett., 48 (2015), 109-117. doi: 10.1016/j.aml.2015.03.019
[30]	H. Naher, F. A. Abdullah, M. A. Akbar, Generalized and improved (G′/G)-expansion method for (3+1) dimensional modified KdV-Zakharov-Kuznetsev equation, PLOS One, 8 (2013), 64618.
[31]	H. M. Baskonus, Complex Soliton Solutions to the Gilson-Pickering Model, Axioms, 8 (2019), 18.
[32]	H. M. Baskonus, New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation, AIP Conference Proceedings, 1798 (2017), 020018.
[33]	H. Naher, F. A. Abdullah, New traveling wave solutions by the extended generalized Riccati equation mapping method of the (2+1)-dimensional evolution equation, J. Appl. Math., 2012 (2012), 486458.
[34]	A. Yokus, Comparison of Caputo and conformable derivatives for time-fractional Korteweg-de Vries equation via the finite difference method, Int. J. Mod. Phys. B, 32 (2018), 1850365.
[35]	A. Yokus, Numerical solution for space and time fractional order Burger type equation, Alex. Eng. J., 57 (2018), 2085-2091. doi: 10.1016/j.aej.2017.05.028
[36]	A. Yokus, H. Bulut, On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9 (2018), 18-23.
[37]	K. Hosseini, D. Kumar, M. Kaplan, et al. New exact traveling wave solutions of the unstable nonlinear Schrodinger equations, Commun. Theor. Phys., 68 (2017), 761-767. doi: 10.1088/0253-6102/68/6/761
[38]	S. M. Ege, E. Misirli, The modified Kudryashov method for solving some Fractional-order nonlinear equations, Adv. Differ. Equ-Ny, 2014 (2014), 135.
[39]	K. Hosseini, P. Mayeli, D. Kumar, New exact solutions of the coupled Sine-Gordon equations in nonlinear optics using the modified Kudryashov method, J. Mod. Optic., 65 (2018), 361-364. doi: 10.1080/09500340.2017.1380857
[40]	K. Hosseini, P. Mayeli, R. Ansari, Modified Kudryashov method for solving the conformable Time-Fractional Klein-Gordon equations with Quadratic and cubic nonlinearities, Optik, 130 (2017), 737-742. doi: 10.1016/j.ijleo.2016.10.136
[41]	A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable Time-Fractional parabolic equation with Exponential nonlinearity using reliable methods, Opt. Quant. Electron., 49 (2017), 278.
[42]	K. Hosseini, E. Y. Bejarbaneh, A. Bekir, et al. New exact solutions of some nonlinear evolution equations of pseudo parabolic type, Opt. Quant. Electron., 49 (2017), 241.
[43]	F. Mahmud, M. Samsuzzoha, M. A. Akbar, The generalized kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results Phys., 7 (2017), 4296-4302. doi: 10.1016/j.rinp.2017.10.049
[44]	S. Bibi, N. Ahmed, U. Khan, et al. Some new exact solitary wave solutions of the van der Walls model arising in nature, Results Phys., 9 (2018), 648-655. doi: 10.1016/j.rinp.2018.03.026
[45]	M. Kaplan, A. Bekir, A. Akbulut, A generalized kudryashov method to some nonlinear evolution equations in mathematical physics, Nonlinear Dynam., 85 (2016), 2843-2850. doi: 10.1007/s11071-016-2867-1
[46]	S. T. Demiray, Y. Pandir, H. Bulut, Generalized Kudryashov method for Time-fractional differential equations, Abstr. Appl. Anal., 2014 (2014), 901540.
[47]	K. A. Gepreel, T. A. Nofal, A. A. Alasmari, Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method, Journal of the Egyptian Mathematical Society, 25 (2017), 438-444. doi: 10.1016/j.joems.2017.09.001
[48]	M. Koparan, M. Kaplan, A. Bekir, et al. A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations, AIP Conference Proceedings, 1798 (2017), 020082.
[49]	H. Kheiri, M. R. Moghaddam, V. Vafaei, Application of the (G'/G)-expansion method for the Burgers, Burgers-Huxley and modified Burgers-KdV equations, Pramana, 76 (2011), 831-842. doi: 10.1007/s12043-011-0070-y
[50]	M. L. Wang, X. Li, J. Wang, The (G'/G)-expansion method and traveling wave solutions of nonlinear evolutions in mathematical physics, Phys. Lett. A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051
[51]	A. K. M. Kazi Sazzad Hossain, M. A. Akbar, Closed-form solutions of two nonlinear equations via the enhanced (G'/G)-expansion method, Cog. Math., 4 (2017), 1355958.
[52]	A. M. Wazwaz, Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations, Appl. Math. Comput., 195 (2008), 754-761.

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