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New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the (2 + 1)‐Dimensional Evolution Equation

Author

Listed:
  • Hasibun Naher
  • Farah Aini Abdullah

Abstract

The generalized Riccati equation mapping is extended with the basic (G′/G)‐expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty‐seven traveling wave solutions for the (2+1)‐dimensional modified Zakharov‐Kuznetsov equation by applying this method. Further, the auxiliary equation G′(η) = w + uG(η) + vG2(η) is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

Suggested Citation

  • Hasibun Naher & Farah Aini Abdullah, 2012. "New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the (2 + 1)‐Dimensional Evolution Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:486458
    DOI: 10.1155/2012/486458
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    References listed on IDEAS

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    1. Qiming Liu & Rui Xu, 2012. "Periodic Solutions of a Cohen-Grossberg-Type BAM Neural Networks with Distributed Delays and Impulses," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-17, April.
    2. Khaled A. Gepreel & Taher A. Nofal & Ali A. Al-Thobaiti, 2012. "The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Hasibun Naher & Farah Aini Abdullah, 2012. "The Improved ( ð º â€² / ð º ) -Expansion Method for the (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-20, August.
    4. Ming Song & Zhengrong Liu, 2012. "Periodic Wave Solutions and Their Limits for the Generalized KP‐BBM Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    5. F. A. Godínez & M. A. Escobedo & M. Navarrete, 2012. "Homotopy Analysis Method for the Rayleigh Equation Governing the Radial Dynamics of a Multielectron Bubble," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-20, September.
    6. Khaled A. Gepreel & A. R. Shehata, 2012. "Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-15, March.
    7. Ming Song & Zhengrong Liu, 2012. "Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-25, July.
    8. Yongkun Li & Chao Wang, 2012. "Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    9. Yang Liu & Hong Li & Jinfeng Wang & Wei Gao, 2012. "A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-24, June.
    10. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2012. "Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K(m, n) Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    11. Khaled A. Gepreel & Taher A. Nofal & Ali A. Al-Thobaiti, 2012. "The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-30, October.
    12. Xianbin Wu & Weiguo Rui & Xiaochun Hong, 2012. "Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K ( m , n ) Equation," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-23, March.
    13. Qiming Liu & Rui Xu, 2012. "Periodic Solutions of a Cohen‐Grossberg‐Type BAM Neural Networks with Distributed Delays and Impulses," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    14. J. F. Alzaidy, 2012. "Extended Mapping Method and Its Applications to Nonlinear Evolution Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-14, September.
    15. Hasibun Naher & Farah Aini Abdullah & M. Ali Akbar, 2012. "New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-14, January.
    16. Hasibun Naher & Farah Aini Abdullah & M. Ali Akbar, 2012. "New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp‐Function Method," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    17. F. A. Godínez & M. A. Escobedo & M. Navarrete, 2012. "Homotopy Analysis Method for the Rayleigh Equation Governing the Radial Dynamics of a Multielectron Bubble," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    18. Khaled A. Gepreel & A. R. Shehata, 2012. "Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    19. J. F. Alzaidy, 2012. "Extended Mapping Method and Its Applications to Nonlinear Evolution Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    20. Yongkun Li & Chao Wang, 2012. "Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-28, March.
    21. Yang Liu & Hong Li & Jinfeng Wang & Wei Gao, 2012. "A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    22. Hasibun Naher & Farah Aini Abdullah, 2012. "The Improved (G’/G)‐Expansion Method for the (2+1)‐Dimensional Modified Zakharov‐Kuznetsov Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    23. Ni Hua & Tian Li-Xin & Yao Hong-Xing, 2012. "On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka-Volterra Type System with Feedback Control," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-15, March.
    24. Ni Hua & Tian Li-Xin & Yao Hong-Xing, 2012. "On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka‐Volterra Type System with Feedback Control," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    25. Zhu, Shun-dong, 2008. "The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2+1)-dimensional Boiti–Leon–Pempinelle equation," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1335-1342.
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