The Riccati-Bernoulli Sub-ODE Technique for Solving the Deterministic (Stochastic) Generalized-Zakharov System

Mahmoud A.E. Abdelrahman, M. A. Sohaly

Article ID: 810
Vol 4, Issue 1, 2021

VIEWS - 1129 (Abstract) 451 (PDF)

Abstract


This article concerns  with  the  construction of the  analytical traveling  wave so- lutions  for the Generalized-Zakharov System  by the Riccati-Bernoulli Sub- ODE technique. Also, we will discuss this  technique  in random  case by using random  traveling  wave trans- formation  in order  to  find what  is the  effect of the  randomness  input  for this  technique. We presented the Generalized-Zakharov System as an example to show the difference effect between the deterministic and stochastic Riccati-Bernoulli Sub-ODE technique.  The first moment of random solution is computed for different statistical probability distributions.


Keywords


Riccati-Bernoulli Sub-ODE method; generalized-Zakharov system; (stochastic) traveling wave solutions; (stochastic) solitary wave solutions; random variable

Full Text:

PDF


References


1. . Abdelrahman MAE, Kunik M. The interaction of waves for the ultra-relativistic Euler equations, J. Math. Anal. Appl. 2014; 409: 1140-1158.

2. . Abdelrahman MAE, Kunik M. The Ultra-Relativistic Euler Equations, Math. Meth. Appl. Sci.38 2015; 1247-1264.

3. . Abdelrahman MAE. Global Solutions for the Ultra-Relativistic Euler Equations, Nonlinear Analysis 2017; 155: 140-162.

4. . Abdelrahman MAE. On the Shallow Water Equations, Z. Naturforsch., 72(9)a 2017; 873-879.

5. . Abdelrahman MAE. Numerical investigation of the wave-front tracking algorithm for the full ultra-relativistic Euler equations, International Journal of Nonlinear Sciences and Nu- merical Simulation DOI: https://doi.org/10.1515/ijnsns-2017-0121.

6. . Razborova P, Ahmed B, Biswas A. Solitons, shock waves and conservation laws of Rosenau-KdV-RLW equation with power law nonlinearity. Appl. Math. Inf. Sci. 2014; 8(2): 485-491.

7. . Biswas AQ, Mirzazadeh M. Dark optical solitons with power law nonlinearity using (G'/G)-expansion. Optik 2014; 125: 4603-4608.

8. . Younis M, Ali S, Mahmood SA. Solitons for compound KdV Burgers equation with variable coefcients and power law nonlinearity. Nonlinear Dyn. 2015; 81: 1191-1196.

9. . Bhrawy AH. An efficient Jacobi pseudo spectral approximation for nonlinear complex gen- eralized Zakharov system. Appl. Math. Comput.2014; 247: 30-46.

10. . Abdelrahman MAE, Sohaly MA. Solitary Waves for the Modified Korteweg-De Vries Equation in Deterministic Case and Random Case. J Phys Math. 2017; 8(1), [DOI:10.4172/2090-0902.1000214].

11. . Abdelrahman MAE, Sohaly MA. Solitary waves for the nonlinear Schro¨dinger problem with the probability distribution function in stochastic input case. Eur. Phys. J. Plus 2017.

12. . Malfliet W. Solitary wave solutions of nonlinear wave equation, Am. J. Phys.1992; 60: 650-654.

13. . Malfliet W. Hereman W, The tanh method: Exact solutions of nonlinear evolution and wave equations, Phys.Scr. 1996; 54: 563-568.

14. . Wazwaz AM. The tanh method for travelling wave solutions of nonlinear equations, Appl.Math. Comput., 2004; 154: 714-723.

15. . Dai CQ, Zhang JF. Jacobian elliptic function method for nonlinear differential difference equations, Chaos Solutions Fractals, 2006; 27: 1042-1049.

16. . Fan E, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 2002; 305: 383-392.

17. . Liu S, Fu Z, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 2001; 289: 69-74.

18. . He JH, Wu XH. Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 2006; 30: 700-708.

19. . Aminikhad H, Moosaei H, Hajipour M. Exact solutions for nonlinear partial differential equations via Exp-function method, Numer. Methods Partial Differ. Equations 2009; 26: 1427-1433.

20. . Wazwaz AM. Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE. Method, Comput. Math. Appl., 2005; 50: 1685-1696.

21. . Wazwaz AM. A sine-cosine method for handling nonlinear wave equations, Math. Comput.Modelling 2004; 40: 499-508.

22. . Yan C. A simple transformation for nonlinear waves, Phys. Lett. A 1996; 224: 77-84.

23. . Fan E, Zhang H. A note on the homogeneous balance method, Phys. Lett. A 1998; 246: 403-406.

24. . Wang ML. Exct solutions for a compound KdV-Burgers equation, Phys. Lett. A 1996; 213: 279-287.

25. . Abdou MA. The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos Solitons Fractals 2007; 31: 95-104.

26. . Ren YJ, Zhang HQ. A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Chaos Solitons Fractals 2006; 27: 959-979.

27. . Zhang JL, Wang ML, Wang YM, et al. The improved F-expansion method and its applications, Phys.Lett.A 2006; 350: 103-109.

28. . EL-Wakil SA, Abdou MA. New exact travelling wave solutions using modified extented tanh-function method, Chaos Solitons Fractals 2007; 31: 840-852.

29. . Fan E. Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 2000; 277: 212-218.

30. . Wazwaz AM. The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput. 2007; 187: 1131-1142.

31. . Wang ML, Zhang JL, XZ LI. (G'/G)-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A 2008; 372: 417-423.

32. . Zhang S, Tong JL, Wang W. Ageneralized (G'/G)-expansion method for the mKdv equation with variable coefficients, Phys. Lett. A 2008; 372: 2254-2257.

33. . El-Wakil SA, Abdou MA, Elhanbaly A. Phys. Lett. A 2006; 353: 40.

34. . Abdelrahman MAE. A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations, Nonlinear Engineering Modeling and Application (to appear).

35. . Sohaly M. Mean Square Heuns Method Convergent for Solving Random Differential Initial Value Problems of First Order. American Journal of Computational Mathematics 2014; 4: 474-481. doi: 10.4236/ajcm.2014.45040.

36. . Yassen M, Sohaly M, Elbaz I. Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense. American Journal of Computational Mathematics 2016; 6: 66-73. doi: 10.4236/ajcm.2016.62008

37. . Sohaly MA, Yassen MT,Elbaz IM. The Variational Methods for Solving Random Models, International Journal of Innovative Research in Computer Science & Technology (IJIRCST) 2017; 5(2): 2347-5552.

38. . Yang XF, Deng ZC, Wei Y. A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Diff. Equa. 2015; 1: 117-133.




DOI: https://doi.org/10.24294/ijmss.v1i3.810

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Creative Commons License

This site is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.