Exact wave solutions to the simplified modified Camassa-Holm equation in mathematical physics

Md. Nurul Islam; Md. Asaduzzaman; Md. Shajib Ali; Md. Nurul Islam; Md. Asaduzzaman; Md. Shajib Ali

doi:10.3934/math.2020003

AIMS Mathematics

2020, Volume 5, Issue 1: 26-41. doi: 10.3934/math.2020003

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Research article

Exact wave solutions to the simplified modified Camassa-Holm equation in mathematical physics

Department of Mathematics, Islamic University, Kushtia-7003, Bangladesh

Received: 04 August 2019 Accepted: 11 October 2019 Published: 17 October 2019
MSC : 35C07, 35C08, 35K99, 35P09

In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.
- nonlinear evolution equations,
- simplified modified Camassa-Holm equation,
- travelling transformation,
- modified simple equation method
Citation: Md. Nurul Islam, Md. Asaduzzaman, Md. Shajib Ali. Exact wave solutions to the simplified modified Camassa-Holm equation in mathematical physics[J]. AIMS Mathematics, 2020, 5(1): 26-41. doi: 10.3934/math.2020003

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Abstract

In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.

References

[1]	H. Resat, L. Petzold, M. F. Pettigrew, Kinetic modeling of biological system, Comput. Syst. Biol., 541 (2009), 311-335. doi: 10.1007/978-1-59745-243-4_14
[2]	J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs, Am. J. Math. Anal., 3 (2013), 14-19.
[3]	M. Kaplan, A. Bekir, A. Akbulut, et al. The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015), 1374-1383.
[4]	B. Lu, The first integral method for some time fractional differential equations, J. Math. Appl., 395 (2012), 684-693.
[5]	M. N. Alam, M. A. Akbar, The new approach of the generalized (G'/G)-expansion method for nonlinear evolution equations, Ain Shams Eng. J., 5 (2014), 595-603. doi: 10.1016/j.asej.2013.12.008
[6]	M. N. Islam, M. A. Akbar, New exact wave solutions to the space-time fractional coupled Burgers equations and the space-time fractional foam drainage equation, Cogent Phys., 5 (2018), 1422957.
[7]	A. Bekir, O. Guner, Exact solutions of nonlinear fractional differential equation by (G'/G)-expansion method, Chin. Phys. B, 22 (2013), 110202.
[8]	A. Bekir, O. Guner, The (G'/G)-expansion method using modified Riemann-Liouville derivative for some space-time fractional differential equations, Ain Shams Eng. J., 5 (2014), 959-965. doi: 10.1016/j.asej.2014.03.006
[9]	A. M. Wazwaz, The variational iteration method for analytic treatment for linear and nonlinear ODEs, Appl. Math. Comput., 212 (2007), 120-134.
[10]	A. M. Wazwaz, The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl., 54 (2007), 933-939. doi: 10.1016/j.camwa.2006.12.039
[11]	N. Faraz, Y. Khan, A. Yildirim, Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, J. King Saud Univ.-Sci., 23 (2011), 77-81. doi: 10.1016/j.jksus.2010.06.010
[12]	Y. Khan, N. Faraz, Application of modified Laplace decomposition method for solving boundary layer equation, J. King Saud Univ. Sci., 23 (2011), 115-119. doi: 10.1016/j.jksus.2010.06.018
[13]	A. M. Wazwaz, The modified decomposition method and Pade approximants for a boundary layer equation in unbounded domain, Appl. Math. Comput., 177 (2006), 737-744.
[14]	B. Zheng, Q. Feng, The Jacobi elliptic equation method for solving fractional partial differential equations, Abst. Appl. Anal., 2014 (2014), 249071.
[15]	S. M. Ege, E. Misirli, Solutions of space-time fractional foam drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method, Int. J. Res. Advent Tech., 2 (2014), 384-388.
[16]	G. W. Wang, T. Z. Xu, The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations, Rom. J. Phys., 59 (2014), 636-645.
[17]	S. Gupta, D. Kumar, J. Singh, Application of He's homotopy perturbation method for solving nonlinear wave-like equations with variable coefficients, Int. J. Adv. Appl. Math. Mech., 1 (2013), 65-79.
[18]	S. Gupta, J. Singh, D. Kumar, Application of homotopy perturbation transform method for solving time-dependent functional differential equations, Int. J. Nonlin. Sci., 16 (2013), 37-49.
[19]	K. Khan, M. A. Akbar, N. H. M. Ali, The modified simple equation method for exact and solitary wave solutions of nonlinear evolution equation: The GZK-BBM equation and right-handed non commutative Burgers equations, ISRN Math. Phys., 2013 (2013), 146704.
[20]	A. Bekir, M. Kaplan, O. Guner, A novel modified simple equation method and its application to some nonlinear evolution equation system, AIP Conf. Proc., 1611 (2015), 30-36.
[21]	M. N. Islam, M. A. Akbar, Close form exact solutions to the higher dimensional fractional Schrodinger equation via the modified simple equation method, J. Appl. Math. Phys., 6 (2018) 90-102.
[22]	J. Akther, M. A. Akbar, Solitary wave solution to two nonlinear evolution equations via the modified simple equation method, New Trends Math. Sci., 4 (2016), 12-26. doi: 10.20852/ntmsci.2016422033
[23]	A. M. Wazwaz, The tanh-function method for travelling wave solutions of nonlinear equations, Appl. Math. Comput., 154 (2004), 713-723.
[24]	M. Dehghan, A finite difference method for a non-local boundary value problem for two dimensional heat equations, Appl. Math. Comput., 112 (2000), 133-142.
[25]	B. C. Shin, M. T. Darvishi, A. Barati, Some exact and new solutions of the Nizhnik-Novikov-Vesselov equation using the exp-function method, Comput. Math. Appl., 58 (2009), 2147-2151. doi: 10.1016/j.camwa.2009.03.006
[26]	C. Q. Dai, J. F. Zhang, Application of He's exp-function method to the stochastic mKdV equation, Int. J. Nonlin. Sci. Numer. Simulat., 10 (2009), 675-680.
[27]	W. M. Zhang, L. X. Tian, Generalized solitary solution and periodic solution of the combined KdV-mKdV equation with variable coefficients using the exp-function method, Int. J. Nonlin. Sci. Numer. Simulat., 10 (2009), 711-715.
[28]	Y. M. Zhao, F-expansion method and its application for finding new exact solutions to the Kudryashov-Sinelshchikov equation, J. Appl. Math., 2013 (2013), 895760.
[29]	A. Filiz, M. Ekici, A. Sonmezoglu, F-expansion method and new exact solutions of the Schrodinger-KdV equation, Sci. World J., 2014 (2014), 534063.
[30]	R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Frac. Calculus Appl. Anal., 12 (2009), 299-318.
[31]	A. J. M. Jawad, The sine-cosine function method for the exact solutions of nonlinear partial differential equations, IOSR J. Math., 13 (2012), 186-191.
[32]	A. Hossein, S. A. Refahi, R. Hadi, Exact solutions for the fractional differential equations by using the first integral method, Nonlin. Eng., 4 (2015), 15-22.
[33]	A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352 (2006), 500-504. doi: 10.1016/j.physleta.2005.12.036
[34]	A. Ali, M. A. Iqbal, S. T. Mohyud-Din, Traveling wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Hol equation exp(-φ(η))-Expansion method, Egypt. J. Basic Appl. Sci., 3 (2016), 134-140. doi: 10.1016/j.ejbas.2016.01.001
[35]	A. Irshad, M. Usman, S. T. Mohyud-Din, Exp-function method for simplified modified Camassa Holm equation, Int. J. Moden. Math. Sci., 4 (2012), 146-155.
[36]	R. T. Redi, A. A. Anulo, Some new traveling wave solutions of modified Camassa Holm equation by the improved (G'/G)-expansion method, Math. Comput. Sci., 3 (2018), 23-45. doi: 10.11648/j.mcs.20180301.14

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