Authors:
M.Mijanur Rahman,M. A. Habib,H. M. Shahadat Ali,M. Mamun Miah ,DOI NO:
https://doi.org/10.26782/jmcms.2019.02.00022Keywords:
The generalized Kudryashov method, Couple Boiti-Leon-Pempinelli equations, DSSH equation, fourth-order nonlinear AKNS equation,travelingwave solution,exact solution,Abstract
The present study deals with the applicability and effectiveness of the algorithm of generalized Kudryashov method (GKM), which is one of the most workable methods to constitute the exact traveling wave solutions of non-linear evolution equations (NLEEs) in physical and mathematical science. The recent paper, we enucleated this method for each of the following Couple Boiti-Leon-Pempinelli equations system, DSSH equation and fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The prominent competence of this method is to naturalize the way of solving systems of NLEEs. Moreover, we can see that when the parameters are ascribed to the particular values, obtain solitary wave solution from the exact travelling wave solution. The obtained new solutions have a wide range of inflictions in the field of physics and other areas of applied science. To perceive the physical phenomena, we have plotted coupled with some 2𝐷 and 3𝐷 graphical patterns of analytic solutions obtained in this study by using computer programming wolfram Mathematica. The worked-out solutions ascertained that the suggested method is effectual, simple and direct and can be exerted to several types of nonlinear systems of partial differential equations.Refference:
.A. Akbulut, M. Kaplan, F. Tascan, “The investigation of exact solutions of nonlinear partial differential equations by using exp (–𝜑𝜉)method”, Optik-Int. J. Light Elect. Optics, Vol.: 132, pp.: 382-387, 2017.
II.A. Ali, A. R. Seadawy, D. Lu, “Computational methods and traveling waves solutions for the fourth-order nonlinear Ablowitz-Kaup-Neweel-Segur water wave dynamical equation via two methods and its application”,Open Phys., Vol.: 16, Issue: 1, pp.: 219-226, 2018.
III.A. Bekir, A. Boz, “Applicationof He’s exp-function method for nonlinear evolution equations”,Comp. Math. Appl., Vol.: 58, Issue: 11-12, pp.: 2286-2293, 2009.
IV.A. Bekir, M. Kaplan, “Exponential rational function method for solving nonlinear equations arising in various physical models”,Chinese J. Phys., Vol.: 54, Issue: 3, pp.: 365-370, 2016.
V.A. Bekir, M. Kaplan, O. Guner, “A novel modified simple equation method and its application to some nonlinear evolution equation system”, 2ndInt. Conf. Analy. Appl. Math., Vol.:1611, Issue: 30, 2014.
VI.A. K. M. K. S. Hossain, M. A. Akbar, “Closed form solutions of two nonlinear equation via the enhanced (𝐺′/𝐺)-expansion method”, Cogent Math., Vol.: 4, ID: 1355958, 2017.
VII.A. M. Wazwaz,”A Sine-Cosine method for handling nonlinear wave equations”,Math. Comp. Modell., Vol.: 40, Issue: 5-6, pp.: 499-508, 2004.
VIII.A. M. Wazwaz,”The simplified Hirota’s method for studying three extended higher-order KdV-type equation”,J. Ocean Eng. Sci., Vol.: 1, Issue: 2, pp.:181-185, 2016.
IX.A. Sonmezoglu, M. Ekici, A. H. Arnous, Q. Zhou, H. Triki, S. P. Moshokoa, M. Z. Ullah, A. Biswas, M. Belic, “Embedded solitons with 𝑥2and𝑥3nonlinear susceptibilities by extended trial equation method”,Optik –Int. J. Light Elec. Optics,Vol.: 154, pp.: 1-9, 2018.
X.D. Kumar, A. R. Seadawy, A. K. Joardar, “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology”,Chinese J. Phys., Vol.: 56, Issue: 1, pp.: 75-85, 2018.
XI.D. U. Zhong, B. O. Tian, X. Y. Xie,J. Chai, X. Y. Wu, “Backlund transformation and soliton solutions in terms of the wronskian for the Khdomtsev-petviashvili-based system in fluid dynamics”,Pramana J. Phys., Vol.: 90, Issue: 45, pp.:1-6, 2018.
XII.E. Fan, “Extended tanh method and its application to nonlinear equations”,Phys. Lett. A, Vol.: 277, Issue: 4-5, pp.: 212-218, 2000.
XIII.E. Fan, J. Zhang, “Application of the Jacobi elliptic function method to special type nonlinear equations”,Phys. Lett. A, Vol.: 305, Issue: 6, pp.: 383-392, 2002.
XIV.E. M. E. Zayed, A. G. A. Nowehy, “Solitons and other exact solutions for a class of nonlinear Schrodinger type equation”,Optik-Int. J. Light Elec. Optics, Vol.: 130, pp.:1295-1311, 2017.
XV.E. M. E. Zayed, K. A. E. Alurrfi, “Homogeneous balance method and itsapplication for finding the exact solution for nonlinear evolution equation”,Ital. J. Pure Appl. Math., Vol.: 33, pp.: 307-318, 2014.
XVI.F. Mahmud, M. Samsuzzoha, M. A. Akbar, “The generalized Kudryashov method to obtain exact traveling wave solutions of thePHI-four equation and the Fishers equation”, Results Phys., Vol.: 7, pp.: 4296-4302, 2017.
XVII.H. Bulut, S. S. Atas, H. M. Baskonus, “Some novel exponential function structures to the Cahn-Allen equation”,Cogent Phys., Vol.: 3, ID: 1240886, 2016.
XVIII.J. L. Zhang, M. L. Wang, Y. M. Wang, Z. D. Fang, “The improved F-expansion method and its applications”,Phys. Letters A, Vol.: 350, Issue:1-2, pp.:103-109, 2006.
XIX.K. A. Gepreel, T. A. Nofal, A. A. Alasmari, “Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method”,J. Egyp. Math. Soci., Vol.: 25, Issue: 4, pp.: 438-444, 2017.
XX.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 equation”, Int. Schol. Resear. Notices, Vol.: 2013, ID:146704, 2013.
XXI.K. Khan, M. A. Akbar, “The exp(–𝜑𝜉)-expansion method for finding travelling wave solution of Vakhnenko-Parkes equation”, Int. J. Dyna. Syst. Diff. Equ., Vol.: 5, Issue: 1, pp.: 72, 2014.
XXII.K. R. Raslan, T. S. E. Danfal, K. A. Ali, “New exact solutions of coupled generalized regularized long wave equation”,J. Egyp. Math. Soci., Vol.: 25, Issue: 4, pp.: 400-405, 2017.
XXIII.L. Jianbin, K.Yang, “The extended F-expansion method and exact solution of nonlinear PDEs”,Chaos Solit. Frac., Vol.: 22, Issue:1, pp.:111-121, 2004.
XXIV.M. A. Akbar, N. H. M. Ali, “An ansatz for solving nonlinear partial differential equation in mathematical physics”,Springer Plus, Vol.: 5, Issue: 24, pp.:1-13, 2016.
XXV.M. A. Khater, A. R. Seadawy, D.Lu, “Elliptic and solitary wave solutions for the Bogoyavlenskii equations system: Couple Boiti-Leon-Pempinelli equations system and time fractional Cahn-allen equation”,Results Phys., Vol.: 7, pp.: 2325-2333, 2017.
XXVI.M. K. Elboree, “The Jacobi elliptic function method and its application for two component BKP hieracy equations”,Comp. Math. Appl., Vol.: 62, Issue: 12, pp.: 4402-4414, 2011.
XXVII.M. koparan, M. Kaplan, A. Bekir, O. Guner, “A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations”,AIP Conference Proc., Vol.: 1798, Issue: 1, 2017.
XXVIII.M. M. Kabir, A. Khajeh, E. Aghdam, A. Y. Koma, “Modified Kudryashov method for finding exact solitary wave solutions of higher order nonlinear equations”,Math. Meth. Appl. Sci., Vol.: 34, Issue: 2, pp.: 213-219, 2011.
XXIX.M. N. Alam, M. A. Akbar, “Some new exact traveling wave solutions to the Simplified MCH equation and the (1+1)-dimensional combined KdV-mKdV equations”,J. Assoc. Arab Univ. Basic Appl. Sci., Vol.: 17, pp.: 6-13, 2015.
XXX.M. S. Islam, K. Khan, M. A. Akbar, “Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equation”, J. Egypt. Math. Soci., Vol.: 25, Issue: 1, pp.: 13-18, 2017.
XXXI.N. Ahmed, S. Bibi, U. Khan, S. T. Mohyud-Din, “A new modification in the exponential rational function method for nonlinear fractional differential equation”,Eur. Phys. J. Plus, Vol.: 133, Issue: 45, pp.: 1-11, 2018.
XXXII.N. A. Kudryashov, “One method for finding exact solutions of nonlinear differential equations”,Commun. Nonl. Sci. Num. Simul., Vol.: 17, Issue: 6, pp.: 2248-2253, 2012.
XXXIII.O. Guner, A. Bekir, “Solving nonlinear space-time fractional differential equation via ansatz method”,Comp. Meth. Diff. Equ., Vol.: 6, Issue: 1, pp.: 1-11, 2018.
XXXIV.Q. Feng, B. Zheng, B., “Traveling wave solutions for three nonlinear equations by (𝐺′/𝐺)-expansion method”,WSEAS Trans. Comp., Vol.: 9, Issue: 3, pp.: 225-234, 2010.
XXXV.S. Bibi, S. T. Mohyud-Din, U. Khan, N. Ahmed, “Khater method for non-linear Sharma Tasso-Olever (STO) equation of fractional order”, Results in Phys., Vol.: 7, pp.:4440-4450, 2017.
XXXVI.S. T. Demiray,Y. Pandir, H. Bulut, “Generalized Kudryashov method for time fractional differential equations”,Abst. Appl. Analy., Vol.: 2014, ID: 901540, 2014.
XXXVII.S. T. Mohyud-Din, A. Yildirim, M. M. Hosseini, “Variational iteration method for initial and boundary value problems using He’s polynomials”, Int. J. Diff. Equ., ID 426213, 2010.
XXXVIII.W. Malfliet, “The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations”,J. Comp. Appl. Math., Vol.: 164-165, Issue: 1, pp.: 529-541, 2004.
XXXIX.W. M. Taha, M. S. M. Noorani, “Application of the (𝐺′/𝐺)-expansion method to the generalized Fisher’s equation and modified equal width equation”,J. Assoc. Arab Univ. Basic Appl. Sci., Vol.:15, pp.: 82-89, 2014.
XL.Y. Pandir, Y. Gurefe, E. Misirli, “The extended trial equation method for some Time-Fractional differential equation”,Disc. Dyna. Nature Soci., Vol.: 2013, ID: 491359, 2013
M. Mijanur Rahman, M. A. Habib, H. M. Shahadat Ali, M. Mamun Miah View Download