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Yıl 2016, Cilt: 4 Sayı: 4, 12 - 26, 31.12.2016

Jesmin Akter M. Ali Akbar

Öz

Kaynakça

  • S.T. Mohyud-Din, A. Yildirim and S. Sariaydin, Numerical soliton solutions of the improved Boussinesq equation, Int. J. Numer. Methods Heat Fluid Flow, 21(7) (2011) 822-827.
  • S.T. Mohyud-Din, A. Yildirim and G. Demirli, Analytical solution of wave system in Rn with coupling controllers, Int. J. Numer. Methods Heat Fluid Flow, 21(2) (2011) 198-205..
  • S.T. Mohyud-Din, A. Yildirim and S. Sariaydin, Numerical soliton solution of the Kaup–Kupershmidt equation, Int. J. Numer. Methods Heat Fluid Flow, 21(3) (2011) 272-281.
  • A.T. Ali, New generalized Jacobi elliptic function rational expansion method, J. Comput. Appl. Math., 235 (2011) 4117-4127.
  • J.Weiss , M.Tabor and G.Carnevale , The Painleve property for partial differential equations, J. Math. Phys., 24 (1983) 522.
  • R. Sassaman, A. Heidari and A. Biswas, Topological and non-topological solitons of nonlinear Klein-Gordon equations by He’s semi-inverse variational principle, J. Frank. Ins., 347 (2010) 1148-1157.
  • R. Sassaman and A. Biswas, Soliton solution of the generalized Klein-Gordon equation by semi-inverse variational principle, Math. Eng. Sci. and Aeros., 2 (2011) 99-104.
  • J.H. He, Variational iteration method for delay differential equations. Commun Nonlinear Sci. Numer. Simulat., 2(4) (1997) 235-236.
  • M.A. Abdou and A.A. Soliman, New applications of variational iteration method. Phys. D, 211 (1-2) (2005) 1-8.
  • S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method. Internat. J. Numer. Meth. Engr., 70 (2007) 876-881.
  • A.S. Arife and A. Yildirim, New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step. World Appl. Sci. J., 13(10) (2011) 2186-2190.
  • C. Rogers and W.F. Shadwick, Backlund Transformations, Academic Press, New York, 1982.
  • T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-One equation, Phys. Lett. A, 74 (1979)173-176.
  • M.L.Wang and X.Z. Li, Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos,Solitons and Fract., 24 (2005) 1257-1268.
  • Z. Yan and H. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham Broer-Kaup equation in shallow water, Phys. Lett. A, vol. 285 no.5-6 (2001) 355-362.
  • M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995) 169-172.
  • E.M.E. Zayed, H.A. Zedan and K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations, Chaos, Solitons Fract., 22 (2004) 285-303.
  • M. Inc and D.J. Evans, On traveling wave solutions of some nonlinear evolution equations, Int. J. Comput. Math., 81 (2004)191-202.
  • K, Khan and M.A. Akbar, Application of exp(−j(x ))-expansion Method to Find the Exact Solutions of Modified Benjamin-Bona-Mahony Equation, World Appl. Sci. J, 24(10) (2013) 1373-1377.
  • J.H. He and X.H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708.
  • J. Akter and M.A. Akbar, The exp(−j(x ))- expansion method for exact solutions to the nonlinear KdV Equation and the (2+1)dimensional Zakharov-Kuznetsov (ZK) equations, Elixir Appl. Math., 75 (2014) 27684-27692
  • M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models, Appl. Math. Comput., 190 (2007) 988-996.
  • E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000) 212-218.
  • W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992) 650- 654.
  • M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge, 1991,
  • A.M. Wazwaz, A sine-cosine method for handing nonlinear wave equations. Math. Comput., 40 (2004) 499-508.
  • S .Bibi, S.T.Mohyud-Din (2013) Traveling wave solutions of KdVs using sine-cosine method. J. Asso. Arab Univ. Basic and Appl. Sci., doi: org/10.1016/j.jaubas.2013.03.006 (in press).
  • J.H. He and X.H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708.
  • M.A. Akbar , N.H.M. Ali, New Solitary and Periodic Solutions of Nonlinear Evolution Equation by Exp-function Method, World Appl. Sci. J., 17(12) (2012) 1603-1610.
  • H. Naher, A.F. Abdullah and M.A.Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, J. Appl. Math., Article ID 575387, (2012) 14 pages. doi: 10.1155/2012/575387.
  • J.H. He, An elementary introduction to recently developed asymptotic methods and nano-mechanics in textile engineering, Int. J. Mod. Phys. B, 22(21) (2008) 3487-3578.
  • R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973) 805-810.
  • R. Hirota and J. Satsuma, Soliton solution of a coupled KdV equation, Phys. Lett. A, 85 (1981) 407-408.
  • Sirendaoreji, New exact travelling wave solutions for the Kawahara and modified Kawahara equations, Chaos Solitons Fract., 19(2004) 147–150.
  • J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, Berlin, 2006.
  • A. Biswas, G. Ebadi, M. Fessak, A. G. Johnpillai, S. Johnson, E. V. Krishnan and A. Yildirim, solutions of the perturbed Klein-Gordon equations, Iranian Journal of Science and Technology, Transaction A., 36 (2012) 431-452.
  • A. Biswas, C. Zony and E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation. Appl. Math. Comput ., 203 (2008) 153-156.
  • R. Sassaman and A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 3239-3249.
  • A. Biswas, A. Yildirim, T. Hayat, O. M. Aldossary and R. Sassaman, Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity, Proceedings of the Romanian Academy, Series A, 13 (2012) 32-41.
  • M. Wang, X. Li and J. Zhang, The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008) 417-423.
  • E.M.E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G′/G)-expansion method,J. Appl. Math. Inform., 28 (2010) 383-395.
  • M.A.Akbar, N.H.M. Ali and E.M.E. Zayed, Abundant exact traveling wave solutions of the generalized Bretherton equation via (G′/G)-expansion method, Commun. Theor. Phys., 57 (2012) 173-178.
  • M.A. Akbar, N.H.M. Ali and S.T. Mohyud-Din, Some new exact traveling wave solutions to the (3+1)-dimensional Kadomtsev–Petviashvili equation, World Appl. Sci. J., 16(11) (2012) 1551-1558.
  • E.M.E. Zayed and K.A. Gepreel, The (G′/G)-expansion method for finding the traveling wave solutions of nonlinear partial differential equations in mathematical physics, J. Math. Phys., 50 (2009) 013502-013514.
  • M.A. Akbar and N.H.M. Ali, The alternative (G′/G)-expansion method and its applications to nonlinear partial differential equations, Int. J. Phys. Sci., 6(35)(2011) 7910-7920.
  • A.R. Shehata, The traveling wave solutions of the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg Landau equation using the modified (G′/G)-expansion method, Appl. Math. Comput., 217 (2010) 1-10.
  • M.A. Akbar, N.H.M. Ali and S.T. Mohyud-Din, The alternative (G′/G)-expansion method with generalized Riccati equation: application to fifth order (1+1)-dimensional Caudrey–Dodd–Gibbon equation, Int. J. Phys. Sci., 7(5) (2012) 743-752.
  • O. F. Gozukizil and T. Aydemir, Exact traveling wave solutions for some nonlinear parrrtial differential equations by using the (G′/G)-expansion method.
  • M. Alquran and A. Qawasmeh, Soliton sulutions of shallow water wave equations by means of (G′/G)-expansion method, J.Appl. Ana.Comput., 4 (2014) 221-229

Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method

Yıl 2016, Cilt: 4 Sayı: 4, 12 - 26, 31.12.2016

Jesmin Akter M. Ali Akbar

Öz

In this article, we investigate two essential nonlinear evolution equations namely modified dispersive water wave equations  and the Whitham-Broer-Kaup model for dispersive long waves in the shallow water small-amplitude regime by using the modified  simple equation (MSE) method. The obtained solutions with parameters expose that the method is incredibly prominent and effective    mathematical tool for solving nonlinear evolution equations (NLEEs) in mathematical physics, applied mathematics and engineering.   When the parameters have taken special values the solitary wave solutions are attained from the exact solutions. In addition, this   procedure reduces the size of calculations.

Anahtar Kelimeler

Modified simple equation method nonlinear evolution equations solitary wave solutions modified dispersive water wave equations

Kaynakça

  • S.T. Mohyud-Din, A. Yildirim and S. Sariaydin, Numerical soliton solutions of the improved Boussinesq equation, Int. J. Numer. Methods Heat Fluid Flow, 21(7) (2011) 822-827.
  • S.T. Mohyud-Din, A. Yildirim and G. Demirli, Analytical solution of wave system in Rn with coupling controllers, Int. J. Numer. Methods Heat Fluid Flow, 21(2) (2011) 198-205..
  • S.T. Mohyud-Din, A. Yildirim and S. Sariaydin, Numerical soliton solution of the Kaup–Kupershmidt equation, Int. J. Numer. Methods Heat Fluid Flow, 21(3) (2011) 272-281.
  • A.T. Ali, New generalized Jacobi elliptic function rational expansion method, J. Comput. Appl. Math., 235 (2011) 4117-4127.
  • J.Weiss , M.Tabor and G.Carnevale , The Painleve property for partial differential equations, J. Math. Phys., 24 (1983) 522.
  • R. Sassaman, A. Heidari and A. Biswas, Topological and non-topological solitons of nonlinear Klein-Gordon equations by He’s semi-inverse variational principle, J. Frank. Ins., 347 (2010) 1148-1157.
  • R. Sassaman and A. Biswas, Soliton solution of the generalized Klein-Gordon equation by semi-inverse variational principle, Math. Eng. Sci. and Aeros., 2 (2011) 99-104.
  • J.H. He, Variational iteration method for delay differential equations. Commun Nonlinear Sci. Numer. Simulat., 2(4) (1997) 235-236.
  • M.A. Abdou and A.A. Soliman, New applications of variational iteration method. Phys. D, 211 (1-2) (2005) 1-8.
  • S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method. Internat. J. Numer. Meth. Engr., 70 (2007) 876-881.
  • A.S. Arife and A. Yildirim, New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step. World Appl. Sci. J., 13(10) (2011) 2186-2190.
  • C. Rogers and W.F. Shadwick, Backlund Transformations, Academic Press, New York, 1982.
  • T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-One equation, Phys. Lett. A, 74 (1979)173-176.
  • M.L.Wang and X.Z. Li, Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos,Solitons and Fract., 24 (2005) 1257-1268.
  • Z. Yan and H. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham Broer-Kaup equation in shallow water, Phys. Lett. A, vol. 285 no.5-6 (2001) 355-362.
  • M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995) 169-172.
  • E.M.E. Zayed, H.A. Zedan and K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations, Chaos, Solitons Fract., 22 (2004) 285-303.
  • M. Inc and D.J. Evans, On traveling wave solutions of some nonlinear evolution equations, Int. J. Comput. Math., 81 (2004)191-202.
  • K, Khan and M.A. Akbar, Application of exp(−j(x ))-expansion Method to Find the Exact Solutions of Modified Benjamin-Bona-Mahony Equation, World Appl. Sci. J, 24(10) (2013) 1373-1377.
  • J.H. He and X.H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708.
  • J. Akter and M.A. Akbar, The exp(−j(x ))- expansion method for exact solutions to the nonlinear KdV Equation and the (2+1)dimensional Zakharov-Kuznetsov (ZK) equations, Elixir Appl. Math., 75 (2014) 27684-27692
  • M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models, Appl. Math. Comput., 190 (2007) 988-996.
  • E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000) 212-218.
  • W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992) 650- 654.
  • M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge, 1991,
  • A.M. Wazwaz, A sine-cosine method for handing nonlinear wave equations. Math. Comput., 40 (2004) 499-508.
  • S .Bibi, S.T.Mohyud-Din (2013) Traveling wave solutions of KdVs using sine-cosine method. J. Asso. Arab Univ. Basic and Appl. Sci., doi: org/10.1016/j.jaubas.2013.03.006 (in press).
  • J.H. He and X.H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708.
  • M.A. Akbar , N.H.M. Ali, New Solitary and Periodic Solutions of Nonlinear Evolution Equation by Exp-function Method, World Appl. Sci. J., 17(12) (2012) 1603-1610.
  • H. Naher, A.F. Abdullah and M.A.Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, J. Appl. Math., Article ID 575387, (2012) 14 pages. doi: 10.1155/2012/575387.
  • J.H. He, An elementary introduction to recently developed asymptotic methods and nano-mechanics in textile engineering, Int. J. Mod. Phys. B, 22(21) (2008) 3487-3578.
  • R. Hirota, Exact envelope soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973) 805-810.
  • R. Hirota and J. Satsuma, Soliton solution of a coupled KdV equation, Phys. Lett. A, 85 (1981) 407-408.
  • Sirendaoreji, New exact travelling wave solutions for the Kawahara and modified Kawahara equations, Chaos Solitons Fract., 19(2004) 147–150.
  • J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, Berlin, 2006.
  • A. Biswas, G. Ebadi, M. Fessak, A. G. Johnpillai, S. Johnson, E. V. Krishnan and A. Yildirim, solutions of the perturbed Klein-Gordon equations, Iranian Journal of Science and Technology, Transaction A., 36 (2012) 431-452.
  • A. Biswas, C. Zony and E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation. Appl. Math. Comput ., 203 (2008) 153-156.
  • R. Sassaman and A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 3239-3249.
  • A. Biswas, A. Yildirim, T. Hayat, O. M. Aldossary and R. Sassaman, Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity, Proceedings of the Romanian Academy, Series A, 13 (2012) 32-41.
  • M. Wang, X. Li and J. Zhang, The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008) 417-423.
  • E.M.E. Zayed, Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G′/G)-expansion method,J. Appl. Math. Inform., 28 (2010) 383-395.
  • M.A.Akbar, N.H.M. Ali and E.M.E. Zayed, Abundant exact traveling wave solutions of the generalized Bretherton equation via (G′/G)-expansion method, Commun. Theor. Phys., 57 (2012) 173-178.
  • M.A. Akbar, N.H.M. Ali and S.T. Mohyud-Din, Some new exact traveling wave solutions to the (3+1)-dimensional Kadomtsev–Petviashvili equation, World Appl. Sci. J., 16(11) (2012) 1551-1558.
  • E.M.E. Zayed and K.A. Gepreel, The (G′/G)-expansion method for finding the traveling wave solutions of nonlinear partial differential equations in mathematical physics, J. Math. Phys., 50 (2009) 013502-013514.
  • M.A. Akbar and N.H.M. Ali, The alternative (G′/G)-expansion method and its applications to nonlinear partial differential equations, Int. J. Phys. Sci., 6(35)(2011) 7910-7920.
  • A.R. Shehata, The traveling wave solutions of the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg Landau equation using the modified (G′/G)-expansion method, Appl. Math. Comput., 217 (2010) 1-10.
  • M.A. Akbar, N.H.M. Ali and S.T. Mohyud-Din, The alternative (G′/G)-expansion method with generalized Riccati equation: application to fifth order (1+1)-dimensional Caudrey–Dodd–Gibbon equation, Int. J. Phys. Sci., 7(5) (2012) 743-752.
  • O. F. Gozukizil and T. Aydemir, Exact traveling wave solutions for some nonlinear parrrtial differential equations by using the (G′/G)-expansion method.
  • M. Alquran and A. Qawasmeh, Soliton sulutions of shallow water wave equations by means of (G′/G)-expansion method, J.Appl. Ana.Comput., 4 (2014) 221-229
Toplam 49 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Articles
Yazarlar

Jesmin Akter Bu kişi benim

M. Ali Akbar Bu kişi benim

Yayımlanma Tarihi 31 Aralık 2016
Yayımlandığı Sayı Yıl 2016 Cilt: 4 Sayı: 4

Kaynak Göster

APA Akter, J., & Akbar, M. A. (2016). Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences, 4(4), 12-26.
AMA Akter J, Akbar MA. Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences. Aralık 2016;4(4):12-26.
Chicago Akter, Jesmin, ve M. Ali Akbar. “Solitary Wave Solutions to Two Nonlinear Evolution Equations via the Modified Simple Equation Method”. New Trends in Mathematical Sciences 4, sy. 4 (Aralık 2016): 12-26.
EndNote Akter J, Akbar MA (01 Aralık 2016) Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences 4 4 12–26.
IEEE J. Akter ve M. A. Akbar, “Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method”, New Trends in Mathematical Sciences, c. 4, sy. 4, ss. 12–26, 2016.
ISNAD Akter, Jesmin - Akbar, M. Ali. “Solitary Wave Solutions to Two Nonlinear Evolution Equations via the Modified Simple Equation Method”. New Trends in Mathematical Sciences 4/4 (Aralık 2016), 12-26.
JAMA Akter J, Akbar MA. Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences. 2016;4:12–26.
MLA Akter, Jesmin ve M. Ali Akbar. “Solitary Wave Solutions to Two Nonlinear Evolution Equations via the Modified Simple Equation Method”. New Trends in Mathematical Sciences, c. 4, sy. 4, 2016, ss. 12-26.
Vancouver Akter J, Akbar MA. Solitary wave solutions to two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences. 2016;4(4):12-26.