Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative

1 Department of Mathematics, Jessore University of Science and Technology, Bangladesh
2 Department of Applied Mathematics, University of Rajshahi, Bangladesh

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

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable (G'/G,1/G)-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original (G'/G)-expansion method.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Exact solution; space time fractional modified BBM equation; space time fractional ZKBBM equation; conformable fractional derivative; solitary wave solution

Citation: M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque. New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Mathematics, 2019, 4(2): 199-214. doi: 10.3934/math.2019.2.199

References

  • 1. G. C. Wu, A fractional variational iteration method for solving fractional nonlinear differential equations, Comput. Math. Appl., 61 (2011), 2186-2190.
  • 2. J. Ji, J. B. Zhang, Y. J. Dong, The fractional variational iteration method improved with the Adomian series, Appl. Math. Lett., 25 (2012), 2223-2226.    
  • 3. M. T. Gencoglu, H. M. Baskonus, H. Bulut, Numerical simulations to the noninear model of interpersonal relationship with time fractional derivative, AIP Conf. Proc., 1798 (2017), 020103.
  • 4. S. Guo, L. Mei, The fractional variational iteration method using He's polynomial, Phys. Lett. A, 375 (2011), 309-313.    
  • 5. A. R. Seadawy, Approximation solutions to derivative nonlinear Schrodinger equation with computational applications by variational method, Eur. Phys. J. Plus, 130 (2015), 182.
  • 6. A. M. A. El-Sayed, S. H. Behiry, W. E. Raslan, Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation, Comput. Math. Appl., 59 (2010), 1759-1765.    
  • 7. A. M. A. El-Sayedand, M. Gaber, The Adomian's decomposition method for solving partial differential equation of fractional orderin finite domains, Phys. Lett. A, 359 (2006), 175-182.    
  • 8. S.S. Ray, A new approach for the application of Adomian's decomposition method for the solution to fractional space diffusion equation with insulated ends, Appl. Math. Comput., 202 (2008), 544-549.
  • 9. Z. Odibat, S. Momani, A Generalized Differential Transform Method for Linear Partian Differential Equations of fractional Order, Appl. Math. Lett., 21 (2008), 194-199.
  • 10. V. S. Erturk, S. Momani, Z. Odibat, Application of Generalized Transformation Method to Multi-order Fractional Differential Equations, Commun. Nonlinear. Sci., 13 (2008), 1642-1654.    
  • 11. M. Yavuz, N. Ozdemir, H. M. Baskonus, Solution of fractional partial differential equation using the operator involving non-singular kernal, Eur. Phys. J. Plus, 133 (2018), 1-12.    
  • 12. D. Kumar, J. Singh, H. M. Baskonus, et al., An effective computational approach for solving local fractional Telegraph equations, Nonlinear. Sci. Lett. A, 8 (2017), 200-206.
  • 13. M. Dehghan, J. Manafian, The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method, Z. Naturforsch., 64 (2009), 420-430.
  • 14. M. Dehghan, J. Manafian, A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses, Math. Meth. Appl. Sci., 33 (2010), 1384-1398.
  • 15. A. R. Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrodinger equation and its solutions, Optic, 139 (2017), 31-43.
  • 16. M. L. Wang, X. Z. Li, J. L. Zhang, The (G'/G)-expansion method and the traveling wave solutions to nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423.    
  • 17. B. Zhang, (G'/G)-expansion method for solving fractional partial differential equation in the theory of mathematical physics, Commun. Theor. Phys., 58 (2012), 623-630.
  • 18. M. A. Akbar, N. H. M. Ali, E. M. E. Zayed, A generalized and improved (G'/G)-expansion method for nonlinear evolution equation, Math. Probl. Eng., 20 (2012), 12-22.
  • 19. M. A. Akbar, N. H. M. Ali, E. M. E. Zayed, Abundant exact traveling wave solutions to the generalized Bretherton equation via the improved (G'/G)-expansion method, Commun. Theo. Phys., 57 (2012), 173-178.    
  • 20. H. M. Baskonusand H. Bulut, Regarding the prototype solutions for the nonlinear fractional order biological population model, AIP Conf. proc., 1738 (2016), 290004.
  • 21. H. Bulut, G. Yel, H. M. Baskonus, An application of improved Bernoulli sub-equation function method to the nonlinear time fractional Burgers equation, Tur. J. Math. Comput. Sci., 5 (2016), 1-17.
  • 22. M. Foroutan, I. Zamanpour, J. manafian, Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long range interactions, Eur. Phys. J. Plus, 132 (2017), 421.
  • 23. M. Foroutan, J. Manafian, A. Ranjbaran, Lump solution and its interaction to (3+1)-D potential-YTSF equation, Nonlinear. Dynam., 92 (2018), 2077-2092.    
  • 24. A. Esen, T. A. Sulaiman, H. Bulut, et al. Optical solutions to the space time fractional (1+1)-dimensional couple nonlinear Schrodinger equation, Optic, 167 (2018), 150-156.
  • 25. J. Manafian, On the complex structure of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities, Eur. Phys. J. Plus, 130 (2015), 1-20.    
  • 26. M. Dehghan, J. Manafian, A. Saadatmandi, Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method, Int. J. Mod. Phys. B, 25 (2011), 2965-2981.    
  • 27. M. A. Akbar, N. H. M. Ali, New solitary and periodic solutions to nonlinear evolution equation by Exp- function method, World Appl. Sci. J., 17 (2012),1603-1610.
  • 28. B. Lu, Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A, 376 (2012), 2045-2048.    
  • 29. S. M. Guo, L. Q. 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.    
  • 30. S. Zhang, H. Q. Zhang, Fractional sub-equation method and its application to the nonlinear fractional PDEs, Phys. Lett. A, 375 (2011), 1069-1073.    
  • 31. B. Lu, The first integral method for some time fractional differential equation, J. Math. Anal. Appl., 395 (2012), 684-693.    
  • 32. A. Bekir, O. Guner, O. Unsal, The First Integral Method for exact Solutions to nonlinear Fractional Differential Equation, J. Compt. Nonlinear. Dynam., 10 (2015).
  • 33. M. H. Uddin, M. A. Akbar, M. A. Khan, et al. Close Form Solutions to the Fractional Generalized Reaction Duffing Model and the Density Dependent Fractional Diffusion Reaction Equation, Appl. Comput. Math., 6 (2017), 177-184.    
  • 34. L. X. Li, E. Q. Li, M. L. Wang, The (G'/G)-expansion method and its application to travelling wave solutions to the Zakharovequation, Appl. Math. B., 25 (2010), 454-462.    
  • 35. E. M. E. Zayed, M. A. M. Abdelaziz, The two variable (G'/G)-expansion method for solving the nonlinear KdV-mkdV equation, Math. Probl. Eng., 2012 (2012), 725061.
  • 36. H. M. Baskonus, H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math., 13 (2015), 547-556.
  • 37. S. H. Seyedi, B. N. Saray, A. Ramazani, On the multiscale simulation of squeezing nanofluid flow by a highprecision scheme, Power Tech., 340 (2018), 264-273.    
  • 38. S. H. Seyedi, B. N. Saray, M. R. H. Nobari, Using interpolation scaling functions based on Galerkin method for solving non-Newtonian fluid flow between two vertical flat plates, Appl. Math. Comput., 269 (2015), 488-496.
  • 39. J. F. Alzaidy, Fractional sub-equation method and its application to the space time fractional differential equation in mathematical physics, British J. Math. Comput. Sci., 3 (2013), 153-163.    
  • 40. S. M. Ege, E. Misirli, The modified Kudryashov method for solving some fractional order nonlinear equations, Adv. Differ. Equations, 2014 (2014).
  • 41. A. Bekir, O. Guner, O. Unsal, The first integral method for exact solution to nonlinear fractional differential equations, J. Comput. Nonlinear Dynam., 10 (2015) 021020.
  • 42. M. Song, C. Yang, Exact traveling wave solutions to the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 216 (2010), 3234-3243.
  • 43. E. Aksoy, M. Kaplan, A. Bekir, Exponential rational function method for space time fractional differential equation, J. Waves Random. Complex Media, 26 (2016) 142-151.
  • 44. M. Ekici, E. M. E. Zayed, A. Sonmezoglu, A new fractional sub-equation for solving the space time fractional differential equation in mathematical physics, Comput. Methods Differ. Equations, 2 (2014), 153-170.
  • 45. R. Khalil, M. Al Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.    
  • 46. Y. Cenesiz, D. Baleanu, A. Kurt, et al. New exact solution to Burgers' type equations with conformable derivative, J. Waves Random. Complex Media, 27 (2016), 103-116.
  • 47. A. M. Wazwaz, Partial Differential Equations and Solitary Wave Theory, New York: Springer, 2009.
  • 48. S. T. Mohyud-Din, S. Bibi, Exact solutions for nonlinear fractional differential equations using (G'/G)-expansion method, Alexandria Eng. J., 57 (2018), 1003-1008.    

 

Reader Comments

your name: *   your email: *  

© 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

Copyright © AIMS Press All Rights Reserved