Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics

1 Department of Applied Mathematics, University of Rajshahi, Bangladesh
2 School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia
3 Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh

Special Issues: New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators

Nonlinear evolution equations (NLEEs) of fractional order play important role to explain the inner mechanisms of complex phenomena in various fields of the real world. In this article, nonlinear evolution equations of fractional order; namely, the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation, the time fractional biological population model and the space-time fractional modified regularized long-wave equation are revealed for seeking closed form analytic solutions. The offered equations are first transformed into ordinary differential equations of integer order with the help of a suitable composite transformation and the conformable fractional derivative. Then the rational $(G'/G)$-expansion method, which is reliable, efficient and computationally attractive, is employed to construct the traveling wave solutions successfully. The obtained solutions are appeared to be exact, much more new and general than the existing results in the literature.
  Figure/Table
  Supplementary
  Article Metrics

References

1. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.

2. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.

3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, 1999.

4.R. Hilfer, Applications of fractional Calculus in Physics, World Scientific, 2000.

5. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

6. T. Islam, M. A. Akbar and A. K. Azad, Traveling wave solutions to some nonlinear fractional partial differential equations through the rational $(G'/G)$-expansion method, Journal of Ocean Engineering and Science, 3 (2018), 76-81.    

7. Z. Bin, $(G'/G)$-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys., 58 (2012), 623-630.    

8. M. N. Alam and M. A. Akbar, The new approach of the generalized $(G'/G)$-expansion method for nonlinear evolution equations, Ain Shams Engineering Journal, 5 (2014), 595-603.    

9. M. H. Uddin, M. A. Akbar, M. A. Khan, et al. Families of exact traveling wave solutions to the space time fractional modified KdV equation and the fractional Kolmogorov-Petrovskii-Piskunovequation, JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 13 (2018), 17-33.    

10. M. T. Islam, M. A. Akbar and M. A. K. Azad, The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation, JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 13 (2018), 56-71.    

11. J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some fractional PDEs, American Journal of Mathematical Analysis, 1 (2013), 14-19.

12. S. Guo, L. Mei, Y. Li, et al. The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A, 376 (2012), 407-411.    

13. B. Zheng, Exp-function method for solving fractional partial differential equations, The Scientific World Journal, 2013 (2013), 465723.

14. O. Guner, A. Bekir and H. Bilgil, A note on Exp-function method combined with complex transform method applied to fractional differential equations, Adv. Nonlinear Anal, 4 (2015), 201-208.

15. B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684-693.    

16. M. Eslami, B. F. Vajargah, M. Mirzazadeh, et al. Application of first integral method to fractional partial differential equations, Indian J. Phys., 88 (2014), 177-184.    

17. W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time fractional differential equations, Pramana, 81 (2013), 377-384.    

18. H. Bulut, H. M. Baskonus and Y. Pandir, The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstr. Appl. Anal., 2013 (2013), 1-8.

19. Y. Pandir and Y. Gurefe, New exact solutions of the generalized fractional Zakharov-Kuznetsov equations, Life Sci. J., 10 (2013), 2701-2705.

20. N. Taghizadeh, M. Mirzazadeh, M. Rahimian, et al. Application of the simplest equation method to some time fractional partial differential equations, Ain Shams Engineering Journal, 4 (2013), 897-902.    

21. C. Chen and Y. L. Jiang, Lie group analysis method for two classes of fractional partial differential equations, Commun. Nonlinear Sci., 26 (2015), 24-35.    

22. G. C. Wu, A fractional characteristic method for solving fractional partial differential equations, Appl. Math. Lett., 24 (2011), 1046-1050.    

23. 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), 29.

24. A. Akbulut, M. Kaplan and A. Bekir, Auxiliary equation method for fractional differential equations with modified Riemann-Liouville derivative, Int. J. Nonlin. Sci. Num., 17 (2016), 413-420.

25. G. H. Gao, Z. Z. Sun and Y. N. Zhang, A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions, J. Comput. Phys., \textbf{231 (2012), 2865-2879.

26. W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), 204-226.

27. S. Momani, Z. Odibat and V. S. Erturk, Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. Lett. A, 370 (2007), 379-387.    

28. K. A. Gepreel, The homotopy perturbation method applied to nonlinear fractional Kadomtsev-Petviashvili-Piskkunov equations, Appl. Math. Lett., 24 (2011), 1458-1434.

29. Y. Hu, Y. Luo and Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method, J. Comput. Appl. Math., 215 (2008), 220-229.    

30. A. M. A. El-Sayed, S. H. Behiry and W. E. Raslan, Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation, Comput. Math. Appl., 59 (2010), 1759-1765.    

31. M. Inc, The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., \textbf{345 (2008), 476-484.

32. R. Khalil, M. A. Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.    

33. M. T. Islam, M. A. Akbar and A. K. Azad, A Rational $(G'/G)$-expansion method and its application to the modified KdV-Burgers equation and the (2+1)-dimensional Boussinesq equation, Nonlinear Studies, 22 (2015), 635-645.

34.Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical & Computational Applications, 15 (2010), 970-973.

35. R. L. Mace and M. A. Hellberg, The Korteweg-de Vries-Zakharov-Kuznetsov equation for electron-acoustic waves, Phys. Plasmas, 8 (2001), 2649-2656.    

36. O. Guner, E. Aksoy, A. Bekir, et al. Different methods for (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation, Comput. Math. Appl., 71 (2016), 1259-1269.    

37. E. A. B. Abdel-Salam and E. A. E. Gumma, Analytical solution of nonlinear space-time fractional differential equations using the improved fractional Riccati expansion method, Ain Shams Engineering Journal, 6 (2015), 613-620.    

38. A. K. Khalifaa, K. R. Raslana and H. M. Alzubaidi, A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212 (2008), 406-418.    

39. K. R. Raslan, Numerical study of the Modified Regularized Long Wave (MRLW) equation, Chaos, Solitons & Fractals, 42 (2009), 1845-1853.

40. K. R. Raslan and S. M. Hassan, Solitary waves for the MRLW equation, Applied Mathematics Letters, 22 (2009), 984-989.    

41.M. Kaplan, A. Bekir, A. Akbulut, et al. The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015), 1374-1383.

42. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York NY, USA, 2012.

43. G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results, Comput. Math. Appl., 51 (2006), 1367-1376.    

44. J.-H. He, A Tutorial Review on Fractal Space time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), 3698-3718.    

45. S. Aman, Q. Al-Mdallal and I. Khan, Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium, Journal of King Saud University - Science, 2018.

46. Q. M. Al-Mdallal, On fractional-Legendre spectral Galerkin method for fractional Sturm- Liouville problems, Chaos, Solitons and Fractals, 116 (2018), 261-267.    

47. T. Abdeljawad, Q. Al-Mdallal and F. Jarad, Fractional logistic models in the frame of fractional operators generated by conformable derivatives, Chaos, Solitons and Fractals, 119 (2019), 94-101.    

48. R. Almeida, N. R. O. Bastos and M. T. T. Monteiro, A fractional Malthusian growth model with variable order using an optimization approach, Statistics, Optimization and Information Computing, 6 (2018), 4-11.

49 T. Abdeljawad, Q. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), 1-8.

50. Q. Al-Mdallal, K. A. Abro and I. Khan, Analytical solutions fractional Walter's B fluid with applications, Complexity, 2018 (2018), 1-10.

51. P. Agarwal, Q. Al-Mdallal, Y. J. Cho, et al. Fractional differential equations for the generalized Mittag-Leffler function, Adv. Differ. Equ-NY, 2018 (2018), 58.

52. J. V. da C. Sousa and E. C. de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci., 60 (2018), 72-91.    

53. J. V. da C. Sousa and E. C. de Oliveira, Mittag-Leffler functions and the truncated $\upsilon$-fractional derivative, Mediterr. J. Math., 14 (2017), 244.

© 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