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On the new wave solutions to the MCH equation

  • Mahmoud A. E. Abdelrahman
  • M. A. Sohaly
Original Paper
  • 13 Downloads

Abstract

We will apply two mathematical methods, namely the exp\((-\varphi (\xi ))\)-expansion and sine–cosine in deterministic and stochastic ways for solving the deterministic and stochastic cases of the simplified MCH equation. This nonlinear equation can be turned into another nonlinear ordinary differential equation by appropriate transformation. The methods that we use are efficient and powerful in solving wide classes of nonlinear equations. The solutions obtained are in the form of trigonometric, hyperbolic, exponential and rational functions and a variety of special solutions like kink-shaped, bell-type soliton solutions. Moreover, these solutions reflect some interesting physical interpretation for nonlinear problems. We will also study the stochastic case from the point random wave transformation or some disturbances in the equation itself. The convergence conditions will be discussed.

Keywords

exp\((-\varphi (\xi ))\)-expansion technique Sine–cosine technique Deterministic (random) MCH equation Deterministic (stochastic) solitary wave solutions Convergence 

PACS Nos.

02.30.Jr 02.60.Cb 04.20.Jb 

References

  1. [1]
    A Wazwaz Appl. Math. Comput. 163 1165 (2005)MathSciNetGoogle Scholar
  2. [2]
    R Camassa and D Holm Phys. Rev. Lett. 71 1661 (1993)ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    Tian and X Song Chaos Solitons and Fractals. 19 621 (2004)Google Scholar
  4. [4]
    J P Boyd Appl. Math. Comput. 81 173 (1997)MathSciNetGoogle Scholar
  5. [5]
    H Bulut, T A Sulaiman, F Erdogan and H M Baskonus Eur. Phys. J. Plus 132 350 (2017)CrossRefGoogle Scholar
  6. [6]
    Md N Alam and M A Akbar J. Assoc. Arab Univ. Basic Appl. Sci. 17 6 (2015)Google Scholar
  7. [7]
    M A E Abdelrahman and M Kunik Math. Meth. Appl. Sci. 38 1247 (2015)CrossRefGoogle Scholar
  8. [8]
    M A E Abdelrahman Nonlinear Anal . 155 140 (2017)MathSciNetCrossRefGoogle Scholar
  9. [9]
    M A E Abdelrahman Z. Naturforsch. 72 873 (2017)ADSCrossRefGoogle Scholar
  10. [10]
    M A E Abdelrahman Int. J. Nonlinear Sci. Numer. Simul. (2014).  https://doi.org/10.1515/ijnsns-0121
  11. [11]
    P Razborova, B Ahmed and A Biswas Appl. Math. Inf. Sci. 8 485 (2014)MathSciNetCrossRefGoogle Scholar
  12. [12]
    A Biswas and M Mirzazadeh Optik 125 4603 (2014)ADSCrossRefGoogle Scholar
  13. [13]
    M Younis, S Ali and S A Mahmood Nonlinear Dyn. 81 1191 (2015)CrossRefGoogle Scholar
  14. [14]
    A H Bhrawy Appl. Math. Comput. 247 30 (2014)MathSciNetGoogle Scholar
  15. [15]
    M A E Abdelrahman and M A Sohaly J. Phys. Math. 8 (2017).  https://doi.org/10.4172/2090-0902.1000214
  16. [16]
    M A E Abdelrahman and M A Sohaly Eur. Phys. J. Plus (2017)Google Scholar
  17. [17]
    W Malfliet Am. J. Phys. 60 650 (1992)ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    W Malfliet and W Hereman Phys. Scr. 54 563 (1996)ADSCrossRefGoogle Scholar
  19. [19]
    A M Wazwaz Appl. Math. Comput. 154 714 (2004)MathSciNetGoogle Scholar
  20. [20]
    C Q Dai and J F Zhang Chaos Solitons Fractals 27 1042 (2006)ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    E Fan and J Zhang Phys. Lett. A 305 383 (2002)ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    J H He and X H Wu Chaos Solitons Fractals 30 700 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    H Aminikhad, H Moosaei and M Hajipour Numer. Methods Partial Differ. Equ. 26 1427 (2009)Google Scholar
  24. [24]
    M A E Abdelrahman Nonlinear Eng. Model. Appl. (2018).  https://doi.org/10.1515/nleng-2017-0145
  25. [25]
    A M Wazwaz Comput. Math. Appl. 50 1685 (2005)MathSciNetCrossRefGoogle Scholar
  26. [26]
    A M Wazwaz Math. Comput. Modell. 40 499 (2004)MathSciNetCrossRefGoogle Scholar
  27. [27]
    C Yan Phys. Lett. A 224 77 (1996)ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    E Fan and H Zhang Phys. Lett. A 246 403 (1998)ADSCrossRefGoogle Scholar
  29. [29]
    M L Wang Phys. Lett. A 213 279 (1996)ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    Y J Ren and H Q Zhang Chaos Solitons Fractals 27 959 (2006)ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    J L Zhang, M L Wang, Y M Wang and Z D Fang Phys. Lett. A 350 103 (2006)ADSCrossRefGoogle Scholar
  32. [32]
    E Fan Phys. Lett. A 277 212 (2000)ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    A M Wazwaz Appl. Math. Comput. 187 1131 (2007)MathSciNetGoogle Scholar
  34. [34]
    M L Wang, J L Zhang and X Z Li Phys. Lett. A 372 417 (2008)ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    S Zhang, J L Tong and W Wang Phys. Lett. A 372 2254 (2008)ADSCrossRefGoogle Scholar
  36. [36]
    J Singh D Kumar, D Baleanu, S Rathore Appl. Math. Comput. 335 12 (2018)MathSciNetGoogle Scholar
  37. [37]
    J Choi, D Kumar, J Singh and R Swroop J. Korean Math. Soc. 54 1209 (2017)MathSciNetGoogle Scholar
  38. [38]
    D Kumar, J Singh, and D Baleanu Math. Methods Appl. Sci. 40 5642 (2017)ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    D Kumar, J Singh, D Baleanu and Sushila Phys. A 372 155 (2018)Google Scholar
  40. [40]
    J Singh, M M Rashidi, Sushila and D Kumar Neural Comput. Appl. (2017).  https://doi.org/10.1007/s00521-017-3198-y.
  41. [41]
    M A El-Tawil and M A Sohaly Open J. Discret Math. 66 (2011)Google Scholar
  42. [42]
    M A El-Tawil and M A Sohaly Int. Conf. Math. Trends Dev. ICMTD12 Google Scholar
  43. [43]
    A Navarro-Quiles, J V Romero, M D Rosell and M A Sohaly Abstr. Appl. Anal. 2016 7 (2016) Article ID 5391368.  https://doi.org/10.1155/2016/5391368
  44. [44]
    M Sohaly Electronic J. Math. Anal. Appl. 2 164 (2014)Google Scholar
  45. [45]
    A M Wazwaz Appl. Math. Comput. 159 559 (2004)MathSciNetGoogle Scholar
  46. [46]
    F Tascan and A Bekir Appl. Math. Comput. 215 3134 (2009)MathSciNetGoogle Scholar
  47. [47]
    K Hosseini, A Bekir and R Ansari Opt. Quant. Electron. 49 131 (2017)CrossRefGoogle Scholar
  48. [48]
    S T Mohyud-Din and A Ali Fundam. Inform. 151 173 (2017)CrossRefGoogle Scholar
  49. [49]
    K Hosseini, K Mayeli, A Bekir and O Guner Commun. Theor. Phys. 69 1 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of ScienceTaibah UniversityAl-Madinah Al-MunawarahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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