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Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model

Year 2018, Volume: 1 Issue: 2, 184 - 190, 25.12.2018
https://doi.org/10.33401/fujma.486881

Abstract

In this article, we employ Lie group analysis to obtain symmetry reduction of a class of (3+1)-dimensional nonlinear model. This nonlinear model plays a critical role in the study of nonlinear sciences. By the exp$(-\varphi(z))$-expansion method, we construct explicit solutions for the proposed equation. Four types of explicit solutions are obtained, which are hyperbolic, exponential, trigonometric and rational function solutions.

References

  • [1] J. Q. Mei, H. Q. Zhang, New soliton-like and periodic-like solutions for the KdV equation, Appl. Math. Comput., 169 (2005), 589-599.
  • [2] A. G. Cui, H. Y. Li, C. Y. Zhang, A splitting method for shifted skew-Hermitian linear system, J. Inequal. Appl., 2016, 160 (2016); doi: 10.1186/s13660- 016-1105-1.
  • [3] V. E. Zakharov, E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP 39 (1974), 285-288.
  • [4] D. J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent Sources, J. Phys. Soc. Japan, 71(11) (2002), 2649-2656.
  • [5] A. M. Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 12 (2007), 1395-1404.
  • [6] F. Tascan, A. Bekir, M. Koparan, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009), 1810-1815.
  • [7] S. Tang, Y. Xiao, Z. Wang, Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation, Appl. Math. Comput., 210(1) (2009), 39-47.
  • [8] A. J. M. Jawad, M. D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217(2) (2010), 869-877.
  • [9] D. Feng, T. He, J. L¨u, Bifurcations of travelling wave solutions for (2+1)-dimensional Boussinesq type equation, Appl. Math. Comput., 185(1) (2007), 402-414.
  • [10] K. Khan, M. A. Akbar, Study of analytical method to seek for exact solutions of variant Boussinesq equations, Springerplus, 3 (2014), 17 pages.
  • [11] W. J. Yuan, Y. Z. Li, J. M. Lin, Meromorphic solutions of an auxiliary ordinary differential equation using complex method, Math. Meth. Appl. Sci., 36 (2013), 1776-1782.
  • [12] W. J. Yuan, F. N. Meng, Y. Huang, Y. H. Wu, All traveling wave exact solutions of the variant Boussinesq equations, Appl. Math. Comput., 268 (2015), 865-872.
  • [13] Y. Y. Gu, W. J. Yuan, N. Aminakbari, Q. H. Jiang, Exact solutions of the Vakhnenko-Parkes equation with complex method, J. Funct. Spaces, (2017), Article ID 6521357, 6 pages.
  • [14] Y. Y. Gu, N. Aminakbari, W. J. Yuan, Y. H. Wu, Meromorphic solutions of a class of algebraic differential equations related to Painleve equation III, Houston J. Math., 43(4) (2017), 1045-1055.
  • [15] Y. Y. Gu, W. J. Yuan, N. Aminakbari, J. M. Lin, Meromorphic solutions of some algebraic differential equations related Painleve equation IV and its applications, Math. Meth. Appl. Sci., 41(10) (2018), 3832-3840.
  • [16] K. Khan, M. A. Akbar, The exp$(-\phi(\xi))$-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation, Int. J. Dyn. Syst. Diff K. er. Equ., 5(1) (2014), 72-83.
  • [17] S. M. R. Islam, K. Khan, M. A. Akbar, Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations, Springerplus, 4 (2015), 11 pages.
  • [18] N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov-Ivanov equation by using exp$(-\varphi(\xi))$-expansion method, Optik, 139 (2017), 72-76.
  • [19] C. Tian, Lie group and its applications in partial differential equations, Higher Education Press, Beijing, 2001.
  • [20] H. Liu, J. Li, Q. Zhang, Lie symmetry analysis and exact explicit solutions for gener Burgers’ equation, J. Comput. Appl. Math., 228(1) (2009), 1-9.
  • [21] B. Ren, Symmetry reduction related with nonlocal symmetry for Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 456-463.
  • [22] H. Djourdem, S. Benaicha, Solvability for a nonlinear third-order three-point boundary value problem, Univers. J. Math. Appl., 1(2) (2018), 125-131.
  • [23] İ. Çelik, Chebyshev Wavelet collocation method for solving a class of linear and nonlinear nonlocal boundary value problems, Fundam. J. Math. Appl., 1(1) (2018), 25-35.
  • [24] B. Noureddine, S. Benaicha, Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion, J. Math. Sci. Model. 1(1) (2018), 45-55.
  • [25] B. Noureddine, S. Benaicha, H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Univers. J. Math. Appl. 1(1) (2018), 39-45.
Year 2018, Volume: 1 Issue: 2, 184 - 190, 25.12.2018
https://doi.org/10.33401/fujma.486881

Abstract

References

  • [1] J. Q. Mei, H. Q. Zhang, New soliton-like and periodic-like solutions for the KdV equation, Appl. Math. Comput., 169 (2005), 589-599.
  • [2] A. G. Cui, H. Y. Li, C. Y. Zhang, A splitting method for shifted skew-Hermitian linear system, J. Inequal. Appl., 2016, 160 (2016); doi: 10.1186/s13660- 016-1105-1.
  • [3] V. E. Zakharov, E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP 39 (1974), 285-288.
  • [4] D. J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent Sources, J. Phys. Soc. Japan, 71(11) (2002), 2649-2656.
  • [5] A. M. Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 12 (2007), 1395-1404.
  • [6] F. Tascan, A. Bekir, M. Koparan, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009), 1810-1815.
  • [7] S. Tang, Y. Xiao, Z. Wang, Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation, Appl. Math. Comput., 210(1) (2009), 39-47.
  • [8] A. J. M. Jawad, M. D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217(2) (2010), 869-877.
  • [9] D. Feng, T. He, J. L¨u, Bifurcations of travelling wave solutions for (2+1)-dimensional Boussinesq type equation, Appl. Math. Comput., 185(1) (2007), 402-414.
  • [10] K. Khan, M. A. Akbar, Study of analytical method to seek for exact solutions of variant Boussinesq equations, Springerplus, 3 (2014), 17 pages.
  • [11] W. J. Yuan, Y. Z. Li, J. M. Lin, Meromorphic solutions of an auxiliary ordinary differential equation using complex method, Math. Meth. Appl. Sci., 36 (2013), 1776-1782.
  • [12] W. J. Yuan, F. N. Meng, Y. Huang, Y. H. Wu, All traveling wave exact solutions of the variant Boussinesq equations, Appl. Math. Comput., 268 (2015), 865-872.
  • [13] Y. Y. Gu, W. J. Yuan, N. Aminakbari, Q. H. Jiang, Exact solutions of the Vakhnenko-Parkes equation with complex method, J. Funct. Spaces, (2017), Article ID 6521357, 6 pages.
  • [14] Y. Y. Gu, N. Aminakbari, W. J. Yuan, Y. H. Wu, Meromorphic solutions of a class of algebraic differential equations related to Painleve equation III, Houston J. Math., 43(4) (2017), 1045-1055.
  • [15] Y. Y. Gu, W. J. Yuan, N. Aminakbari, J. M. Lin, Meromorphic solutions of some algebraic differential equations related Painleve equation IV and its applications, Math. Meth. Appl. Sci., 41(10) (2018), 3832-3840.
  • [16] K. Khan, M. A. Akbar, The exp$(-\phi(\xi))$-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation, Int. J. Dyn. Syst. Diff K. er. Equ., 5(1) (2014), 72-83.
  • [17] S. M. R. Islam, K. Khan, M. A. Akbar, Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations, Springerplus, 4 (2015), 11 pages.
  • [18] N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov-Ivanov equation by using exp$(-\varphi(\xi))$-expansion method, Optik, 139 (2017), 72-76.
  • [19] C. Tian, Lie group and its applications in partial differential equations, Higher Education Press, Beijing, 2001.
  • [20] H. Liu, J. Li, Q. Zhang, Lie symmetry analysis and exact explicit solutions for gener Burgers’ equation, J. Comput. Appl. Math., 228(1) (2009), 1-9.
  • [21] B. Ren, Symmetry reduction related with nonlocal symmetry for Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 456-463.
  • [22] H. Djourdem, S. Benaicha, Solvability for a nonlinear third-order three-point boundary value problem, Univers. J. Math. Appl., 1(2) (2018), 125-131.
  • [23] İ. Çelik, Chebyshev Wavelet collocation method for solving a class of linear and nonlinear nonlocal boundary value problems, Fundam. J. Math. Appl., 1(1) (2018), 25-35.
  • [24] B. Noureddine, S. Benaicha, Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion, J. Math. Sci. Model. 1(1) (2018), 45-55.
  • [25] B. Noureddine, S. Benaicha, H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Univers. J. Math. Appl. 1(1) (2018), 39-45.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Yongyi Gu 0000-0002-6651-1714

Publication Date December 25, 2018
Submission Date November 23, 2018
Acceptance Date December 19, 2018
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Gu, Y. (2018). Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model. Fundamental Journal of Mathematics and Applications, 1(2), 184-190. https://doi.org/10.33401/fujma.486881
AMA Gu Y. Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model. Fundam. J. Math. Appl. December 2018;1(2):184-190. doi:10.33401/fujma.486881
Chicago Gu, Yongyi. “Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model”. Fundamental Journal of Mathematics and Applications 1, no. 2 (December 2018): 184-90. https://doi.org/10.33401/fujma.486881.
EndNote Gu Y (December 1, 2018) Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model. Fundamental Journal of Mathematics and Applications 1 2 184–190.
IEEE Y. Gu, “Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model”, Fundam. J. Math. Appl., vol. 1, no. 2, pp. 184–190, 2018, doi: 10.33401/fujma.486881.
ISNAD Gu, Yongyi. “Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model”. Fundamental Journal of Mathematics and Applications 1/2 (December 2018), 184-190. https://doi.org/10.33401/fujma.486881.
JAMA Gu Y. Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model. Fundam. J. Math. Appl. 2018;1:184–190.
MLA Gu, Yongyi. “Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model”. Fundamental Journal of Mathematics and Applications, vol. 1, no. 2, 2018, pp. 184-90, doi:10.33401/fujma.486881.
Vancouver Gu Y. Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model. Fundam. J. Math. Appl. 2018;1(2):184-90.

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