Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Approximate solutions to nonlinear fractional order partial differential equations arising in ion-acoustic waves

1 Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan
2 Department of Mathematics, Shaheed Benazir Bhutto University Sheringal Dir(U), Khyber Pakhtunkhwa, Pakistan
3 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan
4 Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan

Special Issues: Initial and Boundary Value Problems for Differential Equations

An approximative procedure named asymptotic homotopy perturbation method (AHPM) is introduced to obtain solutions of the non-linear fractional order models. The two special cases, FZK(3; 3; 3) and FZK(2; 2; 2) of fractional Zakharov-Kuznetsov equations are chosen for the illustrative purpose of our method. AHPM is a very recent new procedure as compare with other existing homotopy perturbation procedures. A new auxiliary function has been introduced in AHPM. The AHPM solutions are compared with solutions of fractional complex transform FCT using variational iteration method VIM and exact solutions. Further, the surface graph of AHPM solutions are compared with surface graph of solutions of homotopy perturbation transform method (HPTM) solutions. In comparison, the solutions computed by AHPM are in agreement with exact solutions of the problems. The simulation section reveals that our new developed procedure is effective and explicit.
  Article Metrics


1. A. R. Seadawy, The solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method, Appl. Math. Sci., 6 (2012), 4081-4090.

2. A. R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas, Phys. Plasmas,21 (2014), Article ID: 052107.

3. A. R. Seadawy, Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations, Appl. Math. Inf. Sci., 10 (2016), 209-214.    

4. A. R. Seadawy, Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma, Comput. Math. Appl., 71 (2016), 201-212.    

5. A. R. Seadawy and D. Lu, Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma, Results Phys., 6 (2016), 590-593.    

6. A. R. Seadawy, Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma, Math. Methods Appl. Sci., 40 (2017), 1598-1607.    

7. A. R. Seadawy, Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves, Eur. Phys. J. Plus, 132 (2017), Article: 29.

8. A. R. Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions, Optik Int. J. Light Electron Opt., 139 (2017), 31-43.    

9. A. R. Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, J. Electromagn. Waves Appl., 31 (2017), 1353-1362.    

10. A. R. Seadawy, Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas, Pramana,89 (2017), Article: 49.

11. Y. Rangkuti, B. M. Batiha and M. T. Shatnawi, Solutions of fractional Zakharov–Kuznetsov equations by fractional complex transform, Int. J. Appl. Math. Res., 5 (2016), 24-28.    

12. D. Kumar, J. Singh and S. Kumar, Numerical computation of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves, J. Egypt. Math. Soc., 22 (2014), 373-378.    

13. I. Podlubny, Fractional Differential Equations, Academic Press, Academic Press, New York, 1999.

14. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386.

15. J. H. He, Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineering'98, Dalian, China,1998 (1998), 288-291.

16. J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86-90.

17. A. Luchko and R. Groneflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.

18. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc. New York, 1993.

19. K. B. Oldham and J. Spanier, The fractional calculus, New York: Academic Press, 1974.

20. M. Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, Geophys. J. Int., 13 (1967), 529-539.    

21. S. Ali, S. Bushnaq, K. Shah, et al. Numerical treatment of fractional order Cauchy reaction diffusion equations, Chaos Solitons Fractals, 103 (2017), 578-587.    

22. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80-90.    

23. C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813-823.    

24. V. E. Lynch, B. A. Carrerasand, D. del-Castillo-Negrete, et al. Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192 (2003), 406-421.    

25. S. Momani and K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), 1351-1365.

26. S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul., 70 (2005), 110-118.    

27. S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Solitons Fractals, 28 (2006), 930-937.    

28. S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494.

29. Z. Odibat and S. Momani, Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput., 181 (2006), 1351-1358.

30. Z. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27-34.

31. S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 31 (2007), 1248-1255.    

32. S. Momani and Z. Odibat, Numerical approach to differential equations of fractional order, J. Comput. Appl. Math., 207 (2007), 96-110.    

33. Z. Odibat and S. Momani, Numerical methods for solving nonlinear partial differential equations of fractional order, Appl. Math. Modell., 32 (2008), 28-39.    

34. Z. Odibat and S. Momani, Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36 (2008), 167-174.    

35. S. Momani and Z. Odibat, Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., 54 (2007), 910-919.    

36. S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365 (2007), 345-350.    

37. A. Yıldırım and Y. Gülkanat, Analytical approach to fractional Zakharov-Kuznetsov equations by He's homotopy perturbation method, Commun. Theor. Phys., 53 (2010), 1005-1010.    

38. Y. Khan, N. Faraz and A. Yildirim, New soliton solutions of the generalized Zakharov equations using He's variational approach, Appl. Math. Lett., 24 (2011), 965-968.    

39. S. Momani and A. Yıldırım, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J. Comput. Math., 87 (2010), 1057-1065.    

40. V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys., 39 (1974), 285-288.

41. S. Monro and E. J. Parkes, The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions, J. Plasma Phys., 62 (1999), 305-317.    

42. S. Monro and E. J. Parkes, Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation, J. Plasma Phys., 64 (2000), 411-426.    

43. S. J. Liao, On the Proposed Analysis Technique for Nonlinear Problems and its Applications , In: Ph.D. Dissertation, Shanghai Jiao Tong University, Shanghai, China, 1992.

44. J. H. He, An approximation sol. technique depending upon an articial parameter, Commun. Nonlinear Sci. Numer. Simul., 3 (1998), 92-97.    

45. V. Marinca and N. Herisanu, Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass Transfer, 35 (2008), 710-715.    

46. N. Herisanu and V. Marinca, Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine, Z. Naturforsch. A, 67 (2012), 509-516.    

47. N. Herisanu, V. Marinca, G. Madescu, et al. Dynamic response of a permanent magnet synchronous generator to a wind gust, Energies, 12 (2019), Article: 915.

48. V. Marinca and N. Herisanu, On the flow of a Walters-type B' viscoelastic fluid in a vertical channel with porous wall, Int. J. Heat Mass Transfer, 79 (2014), 146-165.    

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved