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Year 2018, Volume: 2 Issue: 2, 161 - 170, 29.12.2018

Ozkan Guner

https://doi.org/10.29002/asujse.422554
Cited By: 2

Abstract

References

  • [1] M. Antonova, A. Biswas, Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 734-748.
  • [2] L., Zhang, Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension. Appl. Math. Comput. 163 (2005) 343-355.
  • [3] A.M., Wazwaz, New solitary wave solutions to the modified Kawahara equation. Phys. Lett. A 360 (2007) 588-592.
  • [4] Z., Yang, W.P., Zhong, Analytical solutions to Sine-Gordon equation with variable coefficient. Romanian Reports in Physics 66 (2014) 262-273.
  • [5] D.M., Mothibi, C.M., Khalique, On the exact solutions of a modified Kortweg de Vries type equation and higher-order modified Boussinesq equation with damping term. Adv. Differ. Equ. 2013 (2013) 166.
  • [6] U.M., Abdelsalam, Traveling wave solutions for shallow water equations. Journal of Ocean Engineering and Science 2 (2017) 28-33.
  • [7] C., Cattani, T.A., Sulaiman, H.M., Baskonus, H., Bulut, On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel'd-Sokolov systems. Opt Quant Electron (2018) 50:138.
  • [8] C., Cattani, T.A., Sulaiman, H.M., Baskonus, H., Bulut, Solitons in an inhomogeneous Murnaghan's rod. Eur. Phys. J. Plus (2018) 133: 228.
  • [9] H.M., Baskonus, New acoustic wave behaviors to the Davey--Stewartson equation with power-law nonlinearity arising in fluid Dynamics. Nonlinear Dyn. 86 (2016) 177-183.
  • [10] H.M.,Baskonus, New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation. AIP Conference Proceedings 1798 (2017) 020018.
  • [11] H.M., Baskonus, H., Bulut New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics. Entropy 17 (2015) 4255-4270.
  • [12] H.M., Baskonus, H., Bulut An Effective Schema for Solving Some Nonlinear Partial Differential Equation Arising In Nonlinear Physics. Open Phys. 13 (2015) 280-289.
  • [13] J., Biazara, H., Ghazvini, He's variational iteration method for solving linear and non-linear systems of ordinary differential equations. Appl. Math. Comput. 191 (2007) 287-297.
  • [14] E., Babolian, A., Azizi, J., Saeidian, Some notes on using the homotopy perturbation method for solving time-dependent differential equations. Mathematical and Computer Modelling 50 (2009) 213-224.
  • [15] Z., Jin-Ming, Z., Yao-Ming, The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq Burgers equation. Chin. Phys. B 20 (2011) 010205.
  • [16] S.A., El-Wakil, M.A., Abdou, New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons & Fractals 31 (2007) 840-852.
  • [17] Guner, O., Bekir, A., Traveling wave solutions for time-dependent coefficient nonlinear evolution equations, Waves in Random and Complex Media, 25 3 (2015) 342-349
  • [18] M.S., Ismail, Numerical solution of complex modified Korteweg-de Vries equation by Petrov-Galerkin method. Appl. Math. Comput. 202 (2008) 520-531.
  • [19] S., Kutluay, A., Esen, Exp-function method for solving the general improved KdV equation. International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 717-725.
  • [20] A., Bekir, Application of the (G′/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372 (2008) 3400-3406.
  • [21] N., Taghizadeh, M., Mirzazadeh, The first integral method to some complex nonlinear partial differential equations. Journal of Computational and Applied Mathematics 235 (2011) 4871-4877.
  • [22] A., Bekir, O., Guner, Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev--Petviashvili equation and generalized Benjamin equation. Pramana - J. Phys. 81 (2013) 203-214.
  • [23] A., Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution. Phys. Lett. A 372 (2008a) 4601-4602.
  • [24] A., Biswas, 1-Soliton solution of (1+2) dimensional nonlinear Schrödinger's equation in dual-power law media. Phys. Lett. A 372 (2008b) 5941-5943.
  • [25] A., Biswas, M.D., Petkovic´, D., Milovic, Topological and non-topological exact soliton solution of the power law KdV equation. Commun Nonlinear Sci Numer Simulat. 15 (2010) 3263-3269.
  • [26] A., Biswas, D., Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger's equation. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1473-1484.
  • [27] M., Saha, A.K., Sarma, A., Biswas, Dark optical solitons in power law media with time-dependent coefficients. Phys. Lett. A 373 (2009) 4438-4441.
  • [28] K., Yun-Quan, Y., Jun, The First Integral Method to Study a Class of Reaction-Diffusion Equations. Commun. Theor. Phys. 43 (2005) 597-600.
  • [29] A., Bekir, O., Guner, Topological (dark) soliton solutions for the Camassa--Holm type equations. Ocean Engineering 74 (2013) 276-279.
  • [30] M., Abudiab, C.M., Khalique, Exact solutions and conservation laws of a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation. Advances in Difference Equations 2013 (2013):221.
  • [31] A.M., Wazwaz, Two forms of (3+1)-dimensional B-type Kadomtsev-Petviashvili equation: multiple-soliton solutions. Phys. Scr. 86 (2012) 035007.
  • [32] A.M., Wazwaz, Distinct kinds of multiple-soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys. Scr. 84 (2011) 055006.
  • [33] H.F., Shen, M.H., Tu, On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J. Math. Phys. 52 (2011) 032704.
  • [34] W.X., Ma, Y., Zhang, Y., Tang, J., Tu, Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218 (2012) 7174-7183.
  • [35] M.G., Asaad, W.X., Ma, Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev--Petviashvili equation and its modified counterpart. Appl. Math. Comput. 218 (2012) 5524-5542.
  • [36] W.X., Ma, A., Abdeljabbar, A bilinear Bäcklund transformation of a (3 + 1)-dimensional generalized KP equation. Appl. Math. Lett. 25 (2012) 1500-1504.
  • [37] W.X., Ma, A., Abdeljabbar, M.G., Asaad, Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl. Math. Comput. 217 (2011) 10016-10023.
  • [38] W.X., Ma, Z., Zhu, Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218 (2012) 11871-11879.
  • [39] A.M., Wazwaz, Variants of a (3+1)-dimensional generalized BKP equation:Multiple-front waves solutions. Computers & Fluids 97 (2014) 164-167.
  • [40] W.X., Ma, E., Fan, Linear superposition principle applying to Hirota bilinear equations, Computers and Mathematics with Applications. 61 (2011) 950-959.
  • [41] M., Boiti, J.J. -P., Leon, M., Manna, F., Pempinelli, On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Problems 2 (1986) 271-279.
  • [42] P.G., Estévez, S.B., Leble, A Wave Equation in 2+1: Painlevé Analysis and Solutions. Inverse Problems 11 (1995) 925-937.
  • [43] B., Tian, Y.T., Gao, Soliton-like solutions for a (2 + 1)-dimensional generalization of the shallow water wave equations. Chaos, Solitons & Fractals 7 (1996) 1497-1499.
  • [44] Y.T., Gao, B., Tian, Generalized Tanh Method with Symbolic Computation and Generalized Shallow Water Wave Equation. Computers Math. Applic. 33 (1997) 115-118.
  • [45] M.T., Darvishi, M., Najafi, L., Kavitha, M., Venkatesh, Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti--Leon--Manna--Pempinelli Equations. Commun. Theor. Phys. 58 (2012) 785-794.

New exact solution for (2+1) and (3+1) dimensional nonlinear partial differential equations

Year 2018, Volume: 2 Issue: 2, 161 - 170, 29.12.2018

Ozkan Guner

https://doi.org/10.29002/asujse.422554
Cited By: 2

Abstract

In this
paper, dark soliton solutions have been obtained for the (2+1)-dimensional
reaction-diffusion equation, the (3+1)-dimensional generalized B-type
Kadomtsev-Petviashvili (gBKP) equation and the (3+1)-dimensional
Boiti-Leon-Manna-Pempinelli (BLMP) equation using the solitary wave ansatz.
Ansatz approach is utilized to carry out this integration. The constraint
relations for each of the equations are given for the existence of dark soliton
solutions.

Keywords

Ansatz method dark soliton solutions exact solution

References

  • [1] M. Antonova, A. Biswas, Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 734-748.
  • [2] L., Zhang, Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension. Appl. Math. Comput. 163 (2005) 343-355.
  • [3] A.M., Wazwaz, New solitary wave solutions to the modified Kawahara equation. Phys. Lett. A 360 (2007) 588-592.
  • [4] Z., Yang, W.P., Zhong, Analytical solutions to Sine-Gordon equation with variable coefficient. Romanian Reports in Physics 66 (2014) 262-273.
  • [5] D.M., Mothibi, C.M., Khalique, On the exact solutions of a modified Kortweg de Vries type equation and higher-order modified Boussinesq equation with damping term. Adv. Differ. Equ. 2013 (2013) 166.
  • [6] U.M., Abdelsalam, Traveling wave solutions for shallow water equations. Journal of Ocean Engineering and Science 2 (2017) 28-33.
  • [7] C., Cattani, T.A., Sulaiman, H.M., Baskonus, H., Bulut, On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel'd-Sokolov systems. Opt Quant Electron (2018) 50:138.
  • [8] C., Cattani, T.A., Sulaiman, H.M., Baskonus, H., Bulut, Solitons in an inhomogeneous Murnaghan's rod. Eur. Phys. J. Plus (2018) 133: 228.
  • [9] H.M., Baskonus, New acoustic wave behaviors to the Davey--Stewartson equation with power-law nonlinearity arising in fluid Dynamics. Nonlinear Dyn. 86 (2016) 177-183.
  • [10] H.M.,Baskonus, New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation. AIP Conference Proceedings 1798 (2017) 020018.
  • [11] H.M., Baskonus, H., Bulut New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics. Entropy 17 (2015) 4255-4270.
  • [12] H.M., Baskonus, H., Bulut An Effective Schema for Solving Some Nonlinear Partial Differential Equation Arising In Nonlinear Physics. Open Phys. 13 (2015) 280-289.
  • [13] J., Biazara, H., Ghazvini, He's variational iteration method for solving linear and non-linear systems of ordinary differential equations. Appl. Math. Comput. 191 (2007) 287-297.
  • [14] E., Babolian, A., Azizi, J., Saeidian, Some notes on using the homotopy perturbation method for solving time-dependent differential equations. Mathematical and Computer Modelling 50 (2009) 213-224.
  • [15] Z., Jin-Ming, Z., Yao-Ming, The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq Burgers equation. Chin. Phys. B 20 (2011) 010205.
  • [16] S.A., El-Wakil, M.A., Abdou, New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons & Fractals 31 (2007) 840-852.
  • [17] Guner, O., Bekir, A., Traveling wave solutions for time-dependent coefficient nonlinear evolution equations, Waves in Random and Complex Media, 25 3 (2015) 342-349
  • [18] M.S., Ismail, Numerical solution of complex modified Korteweg-de Vries equation by Petrov-Galerkin method. Appl. Math. Comput. 202 (2008) 520-531.
  • [19] S., Kutluay, A., Esen, Exp-function method for solving the general improved KdV equation. International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 717-725.
  • [20] A., Bekir, Application of the (G′/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372 (2008) 3400-3406.
  • [21] N., Taghizadeh, M., Mirzazadeh, The first integral method to some complex nonlinear partial differential equations. Journal of Computational and Applied Mathematics 235 (2011) 4871-4877.
  • [22] A., Bekir, O., Guner, Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev--Petviashvili equation and generalized Benjamin equation. Pramana - J. Phys. 81 (2013) 203-214.
  • [23] A., Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution. Phys. Lett. A 372 (2008a) 4601-4602.
  • [24] A., Biswas, 1-Soliton solution of (1+2) dimensional nonlinear Schrödinger's equation in dual-power law media. Phys. Lett. A 372 (2008b) 5941-5943.
  • [25] A., Biswas, M.D., Petkovic´, D., Milovic, Topological and non-topological exact soliton solution of the power law KdV equation. Commun Nonlinear Sci Numer Simulat. 15 (2010) 3263-3269.
  • [26] A., Biswas, D., Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger's equation. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 1473-1484.
  • [27] M., Saha, A.K., Sarma, A., Biswas, Dark optical solitons in power law media with time-dependent coefficients. Phys. Lett. A 373 (2009) 4438-4441.
  • [28] K., Yun-Quan, Y., Jun, The First Integral Method to Study a Class of Reaction-Diffusion Equations. Commun. Theor. Phys. 43 (2005) 597-600.
  • [29] A., Bekir, O., Guner, Topological (dark) soliton solutions for the Camassa--Holm type equations. Ocean Engineering 74 (2013) 276-279.
  • [30] M., Abudiab, C.M., Khalique, Exact solutions and conservation laws of a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation. Advances in Difference Equations 2013 (2013):221.
  • [31] A.M., Wazwaz, Two forms of (3+1)-dimensional B-type Kadomtsev-Petviashvili equation: multiple-soliton solutions. Phys. Scr. 86 (2012) 035007.
  • [32] A.M., Wazwaz, Distinct kinds of multiple-soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys. Scr. 84 (2011) 055006.
  • [33] H.F., Shen, M.H., Tu, On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J. Math. Phys. 52 (2011) 032704.
  • [34] W.X., Ma, Y., Zhang, Y., Tang, J., Tu, Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218 (2012) 7174-7183.
  • [35] M.G., Asaad, W.X., Ma, Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev--Petviashvili equation and its modified counterpart. Appl. Math. Comput. 218 (2012) 5524-5542.
  • [36] W.X., Ma, A., Abdeljabbar, A bilinear Bäcklund transformation of a (3 + 1)-dimensional generalized KP equation. Appl. Math. Lett. 25 (2012) 1500-1504.
  • [37] W.X., Ma, A., Abdeljabbar, M.G., Asaad, Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl. Math. Comput. 217 (2011) 10016-10023.
  • [38] W.X., Ma, Z., Zhu, Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218 (2012) 11871-11879.
  • [39] A.M., Wazwaz, Variants of a (3+1)-dimensional generalized BKP equation:Multiple-front waves solutions. Computers & Fluids 97 (2014) 164-167.
  • [40] W.X., Ma, E., Fan, Linear superposition principle applying to Hirota bilinear equations, Computers and Mathematics with Applications. 61 (2011) 950-959.
  • [41] M., Boiti, J.J. -P., Leon, M., Manna, F., Pempinelli, On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Problems 2 (1986) 271-279.
  • [42] P.G., Estévez, S.B., Leble, A Wave Equation in 2+1: Painlevé Analysis and Solutions. Inverse Problems 11 (1995) 925-937.
  • [43] B., Tian, Y.T., Gao, Soliton-like solutions for a (2 + 1)-dimensional generalization of the shallow water wave equations. Chaos, Solitons & Fractals 7 (1996) 1497-1499.
  • [44] Y.T., Gao, B., Tian, Generalized Tanh Method with Symbolic Computation and Generalized Shallow Water Wave Equation. Computers Math. Applic. 33 (1997) 115-118.
  • [45] M.T., Darvishi, M., Najafi, L., Kavitha, M., Venkatesh, Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti--Leon--Manna--Pempinelli Equations. Commun. Theor. Phys. 58 (2012) 785-794.
There are 45 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

Ozkan Guner 0000-0002-9900-2844

Publication Date December 29, 2018
Submission Date May 10, 2018
Acceptance Date October 17, 2018
Published in Issue Year 2018Volume: 2 Issue: 2

Cite

APA Guner, O. (2018). New exact solution for (2+1) and (3+1) dimensional nonlinear partial differential equations. Aksaray University Journal of Science and Engineering, 2(2), 161-170. https://doi.org/10.29002/asujse.422554

Cited By

Soliton solutions of (2+1)-dimensional non-linear reaction-diffusion model via Riccati-Bernoulli approach
Thermal Science
https://doi.org/10.2298/TSCI22S2811A
A New Approach to (3+1) Dimensional Boiti–Leon–Manna–Pempinelli Equation
Applied Mathematics and Nonlinear Sciences
Gülnur Yel
https://doi.org/10.2478/amns.2020.1.00029
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