Explicit Travelling Wave Solutions to Nonlinear Partial Differential Equations Arise in Mathematical Physics and Engineering

Muktarebatul Jannah; Tarikul Islam; Armina Akter

doi:10.38032/jea.2021.01.008

Muktarebatul Jannah Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh
Tarikul Islam Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh
Armina Akter Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh

DOI: https://doi.org/10.38032/jea.2021.01.008

Keywords: The Rational (G'/G)-Expansion Method, Nonlinear Partial Differential Equation, Complex Transformation, Exact Solution

Abstract

To describe the interior phenomena of the mysterious problems around the real world, non-linear partial differential equations (NLPDEs) plays a substantial role, for which construction of analytic solutions of those is most important. This paper stands for a goal to find fresh and wide-ranging solutions to some familiar NLPDEs namely the non-linear cubic Klein-Gordon (cKG) equation and the non-linear Benjamin-Ono (BO) equation. A wave variable transformation is made use to convert the mentioned equations into ordinary differential equations. To acquire the desired precise exact travelling wave solutions to the above-stated equations, the rational -expansion method is employed. Consequently, three types of equipped solutions are successfully come out in the forms of hyperbolic, trigonometric and rational functions in a compatible way. To analyse the physical problems arisen relating to nonlinear complex dynamical systems, our obtained solutions might be most helpful. So far we know, these achieved solutions are different than those in the literature. The applied method is efficient and reliable which might further be used to find different and novel solutions to many other NLPDEs successfully in research field.

References

Wazwaz, A.M., 2002. Partial differential equations: methods and applications. AA Balkema.

Helal, M.A. and Mehanna, M.S., 2006. A comparison between two different methods for solving KdV–Burgers equation. Chaos, Solitons & Fractals, 28(2), pp.320-326. DOI: https://doi.org/10.1016/j.chaos.2005.06.005

Parkes, E.J. and Duffy, B.R., 1996. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Computer Physics Communications, 98(3), pp.288-300. DOI: https://doi.org/10.1016/0010-4655(96)00104-X

Abdou, M.A., 2007. The extended tanh method and its applications for solving nonlinear physical models. Applied mathematics and computation, 190(1), pp.988-996. DOI: https://doi.org/10.1016/j.amc.2007.01.070

Kudryashov, N.A., 1990. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Physics Letters A, 147(5-6), pp.287-291. DOI: https://doi.org/10.1016/0375-9601(90)90449-X

Kudryashov, N.A., 2005. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals, 24(5), pp.1217-1231. DOI: https://doi.org/10.1016/j.chaos.2004.09.109

Xu, G., 2006. An elliptic equation method and its applications in nonlinear evolution equations. Chaos, Solitons & Fractals, 29(4), pp.942-947. DOI: https://doi.org/10.1016/j.chaos.2005.08.058

Parkes, E.J., Duffy, B.R. and Abbott, P.C., 2002. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Physics Letters A, 295 (5-6), pp.280-286. DOI: https://doi.org/10.1016/S0375-9601(02)00180-9

He, J.H. and Wu, X.H., 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), pp.700-708.

Naher, H., Abdullah, F.A. and Akbar, M.A., 2012. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. Journal of Applied Mathematics, 2012. DOI: https://doi.org/10.1155/2012/575387

Khan, K., Akbar, M. A. and Alam, M. N., 2013. "Travelling wave solutions of the nonlinear Drinfel’d-Shokolov-Wilson equation and modified Benjamin-Bona-Mahony equations," Journal of the Egyptian Mathematical Society, 21, pp. 233-240. DOI: https://doi.org/10.1016/j.joems.2013.04.010

Islam, M.H., Khan, K., Akbar, M.A. and Salam, M.A., 2014. Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation. SpringerPlus, 3(1), pp.1-9. DOI: https://doi.org/10.1186/2193-1801-3-105

Wang, M., Li, X. and Zhang, J., 2008. The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), pp.417-423. DOI: https://doi.org/10.1016/j.physleta.2007.07.051

Ebadi, G. and Biswas, A., 2011. The G′ G method and topological soliton solution of the K (m, n) equation. Communications in Nonlinear Science and Numerical Simulation, 16(6), pp.2377-2382. DOI: https://doi.org/10.1016/j.cnsns.2010.09.009

Biswas, A., 2009. Solitary wave solution for the generalized Kawahara equation. Applied Mathematics Letters, 22(2), pp.208-210. DOI: https://doi.org/10.1016/j.aml.2008.03.011

Biswas, A., Petković, M.D. and Milović, D., 2010. Topological and non-topological exact soliton solution of the power law KdV equation. Communications in Nonlinear Science and Numerical Simulation, 15(11), pp.3263-3269. DOI: https://doi.org/10.1016/j.cnsns.2009.12.008

Chun, C. and Sakthivel, R., 2010. Homotopy perturbation technique for solving two-point boundary value problems–comparison with other methods. Computer Physics Communications, 181(6), pp.1021-1024. DOI: https://doi.org/10.1016/j.cpc.2010.02.007

Sakthivel, R., Chun, C. and Lee, J., 2010. New travelling wave solutions of Burgers equation with finite transport memory. Zeitschrift für Naturforschung A, 65(8-9), pp.633-640. DOI: https://doi.org/10.1515/zna-2010-8-903

He, J.H. and Wu, X.H., 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), pp.700-708. DOI: https://doi.org/10.1016/j.chaos.2006.03.020

Ma, W.X., Huang, T. and Zhang, Y., 2010. A multiple exp-function method for nonlinear differential equations and its application. Physica Scripta, 82(6), p.065003. DOI: https://doi.org/10.1088/0031-8949/82/06/065003

Zheng, B., 2011. A new Bernoulli sub-ODE method for constructing traveling wave solutions for two nonlinear equations with any order. UPB Sci. Bull., Series A, 73(3), pp.85-94.

Zheng, B., 2012. Soling a nonlinear evolution equation by a proposed Bernoulli sub-ODE method. In Int. Conf. Image Vis. Comput.

Abbasbandy, S., 2010. Homotopy analysis method for the Kawahara equation. Nonlinear Analysis: Real World Applications, 11(1), pp.307-312. DOI: https://doi.org/10.1016/j.nonrwa.2008.11.005

Molabahrami, A. and Khani, F., 2009. The homotopy analysis method to solve the Burgers–Huxley equation. Nonlinear Analysis: Real World Applications, 10(2), pp.589-600. DOI: https://doi.org/10.1016/j.nonrwa.2007.10.014

Molliq, Y., Noorani, M.S.M. and Hashim, I., 2009. Variational iteration method for fractional heat-and wave-like equations. Nonlinear Analysis: Real World Applications, 10(3), pp.1854-1869. DOI: https://doi.org/10.1016/j.nonrwa.2008.02.026

Mohyud-Din, S.T., Noor, M.A. and Noor, K.I., 2009. Modified Variational Iteration Method for Solving Sine-Gordon Equations, World Applied Sciences Journal, 6(7), pp. 999-1004.

Lee, J. and Sakthivel, R., 2014. Exact travelling wave solutions of a variety of Boussinesq-like equations.," Chinese Journal of Physics, 52(3), pp. 939-957.

Ganji, D.D., 2006. The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A, 355(4-5), pp.337-341. DOI: https://doi.org/10.1016/j.physleta.2006.02.056

Ganji, D.D., Afrouzi, G.A. and Talarposhti, R.A., 2007. Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations. Physics Letters A, 368(6), pp.450-457. DOI: https://doi.org/10.1016/j.physleta.2006.12.086

Wang, M., 1995. Solitary wave solutions for variant Boussinesq equations. Physics letters A, 199(3-4), pp.169-172. DOI: https://doi.org/10.1016/0375-9601(95)00092-H

Ablowitz, M.J., Ablowitz, M.A., Clarkson, P.A. and Clarkson, P.A., 1991. Solitons, nonlinear evolution equations and inverse scattering (Vol. 149). Cambridge University Press, UK. DOI: https://doi.org/10.1017/CBO9780511623998

Rogers, C. and Shadwick, W.F., 1982. Bäcklund transformations and their applications. Academic Press, New York.

Jianming, L., Jie, D. and Wenjun, Y., 2011, January. Bäcklund transformation and new exact solutions of the Sharma-Tasso-Olver equation. In Abstract and Applied Analysis (Vol. 2011). Hindawi. DOI: https://doi.org/10.1155/2011/935710

Seadawy, A.R., 2015. Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I. Computers & Mathematics with Applications, 70(4), pp.345-352. DOI: https://doi.org/10.1016/j.camwa.2015.04.015

Seadawy, A.R., 2016. Three-dimensional nonlinear modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Computers & Mathematics with Applications, 71(1), pp.201-212. DOI: https://doi.org/10.1016/j.camwa.2015.11.006

Arshad, M., Seadawy, A., Lu, D. and Wang, J., 2016. Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations. Results in Physics, 6, pp.1136-1145. DOI: https://doi.org/10.1016/j.rinp.2016.11.043

Seadawy, A.R., El-Kalaawy, O.H. and Aldenari, R.B., 2016. Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability. Applied Mathematics and Computation, 280, pp.57-71. DOI: https://doi.org/10.1016/j.amc.2016.01.014

Islam, M., Akbar, M.A. and Azad, A.K., 2015. A Rational (G'/G)-expansion method and its application to the modified KdV-Burgers equation and the (2+ l)-dimensional Boussinesq equationn. Nonlinear Studies, 22(4).

Islam, M.T., Akbar, M.A. and Azad, M.A.K., 2017. Multiple closed form wave solutions to the KdV and modified KdV equations through the rational (G′/G)-expansion method. Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, pp.160-168. DOI: https://doi.org/10.1016/j.jaubas.2017.06.004

Islam, M.T., Ali, M.A. and Hasan, M.R., Closed Form Wave Solutions to the Nonlinear Partial Differential Equations via the Rational)/'(GG-Expansion Method. International Journal for Research in Applied Science & Engineering Technology, 6(2), pp 1519-1525.

Akbar, M.A., Ali, N.H.M. and Islam, M.T., 2019. Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Mathematics, 4(3), pp.397-411. DOI: https://doi.org/10.3934/math.2019.3.397