Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative

M. Tarikul Islam; M. Ali Akbar; M. Abul Kalam Azad

doi:10.3934/Math.2018.4.625

AIMS Mathematics, 2018, 3(4): 625-646. doi: 10.3934/Math.2018.4.625. Research article

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Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative

M. Tarikul Islam, M. Ali Akbar, M. Abul Kalam Azad

1 Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh
2 Department of Applied Mathematics, Rajshahi University, Rajshahi, Bangladesh

Received: , Accepted: , Published:

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Fractional order nonlinear evolution equations involving conformable fractional derivative are formulated and revealed for attractive solutions to depict the physical phenomena of nonlinear mechanisms in the real world. The core aim of this article is to explore further new general exact traveling wave solutions of nonlinear fractional evolution equations, namely, the space time fractional (2+1)-dimensional dispersive long wave equations, the (3+1)-dimensional space time fractional mKdV-ZK equation and the space time fractional modified regularized long-wave equation. The mentioned equations are firstly turned into the fractional order ordinary differential equations with the aid of a suitable composite transformation and then hunted their solutions by means of recently established fractional generalized ($D_\xi^\alpha G/G$)-expansion method. This productive method successfully generates many new and general closed form traveling wave solutions in accurate, reliable and efficient way in terms of hyperbolic, trigonometric and rational. The obtained results might play important roles for describing the complex phenomena related to science and engineering and also be newly recorded in the literature for their high acceptance. The suggested method will draw the attention to the researchers to establish further new solutions to any other nonlinear evolution equations.

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Keywords: The fractional generalized ($D_\xi^\alpha G/G$)-expansion method; conformable fractional derivative; composite transformation; fractional order nonlinear evolution equations; closed form solutions

Citation: M. Tarikul Islam, M. Ali Akbar, M. Abul Kalam Azad. Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative. AIMS Mathematics, 2018, 3(4): 625-646. doi: 10.3934/Math.2018.4.625

References:

1. K.B. Oldham, J. Spanier, The Fractional Calculus, New York: Academic Press, 1974.
2. I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., 198 (1999) 1-340.
3. A. Coronel-Escamilla, J. F. Gomez-Aguilar, L. Torres, et al. A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A., 491 (2018), 406-424.
4. A. Atangana, J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Meth. Part. D. E., (2017).
5. D. Baleanu, M. Inc, A. Yusuf et al. Time fractional third-order evolution equation: Symmetry analysis, explicit solutions and conservation laws, J. Compt. Nonliner Dynam., 13 (2018), 021011.
6. A. Akgul, D. Baleanu, M. Inc, et al. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method, Open. Phys., 14 (2016), 685-689.
7. E. C. Aslan, M. Inc, Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis, Wave. Random. Complex., 27 (2017), 594-601.
8. M. Inc, E. Ulutas, A. Biswas, Singular solitons and other solutions to a couple of nonlinear wave equations, Chin. Phys. B., 22 (2013), 060204.
9. I. E. Inan, Y. Ugurlu, M. Inc, New applications of the (G'/G,1/G)-expansion method, Acta. Phys. Pol. A., 128 (2015), 245-252.
10. M. N. Alam, M. A. Akbar, The new approach of the generalized (G'/G,1/G)-expansion method for nonlinear evolution equations, Ain Shams Eng. J., 5 (2014), 595-603.
11. D. Baleanu, Y. Ugurlu, M. Inc and B. Kilic, Improved (G'/G,1/G)-expansion method for the time fractional Biological population model and Cahn-Hilliard equation, J. Comput. Nonliner. Dynam., 10 (2015), 051016.
12. M. T. Islam, M. A. Akbar, M. A. K. Azad, The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation, J. Mech. Cont. Math. Sci., 13 (2018), 56-71.
13. B. Zheng, Exp-function method for solving fractional partial differential equations, Sci. World J., 2013 (2013), 465723.
14. O. Guner, A. Bekir, 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. J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some fractional PDEs, Am. J. Math. Anal., 1 (2013), 14-19.
16. H. Yépez-Martinez, J. F. Gómez-Aguilar, D. Baleanu, Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion, Optik, 155 (2018), 357-365.
17. H. Yépez-Martinez, J. F. Gómez-Aguilar, A. Atangana, First integral method for nonlinear differential equations with conformable derivative, Math. Modell. Nat. Phenom., 13 (2018), 14.
18. H. Yépez-Martinez, J. F. Gómez-Aguilar, I. O. Sosa, et al. The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mex. Fis., 62 (2016), 310-316.
19. M. Inc, I. E. Inan, Y. Ugurlu, New applications of the functional variable method, Optik, 136 (2017), 374-381.
20. H. Bulut, H. M. Baskonus, Y. Pandir, The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstr. Appl. Anal., 2013 (2013), 636802.
21. Y. Pandir, Y. Gurefe, New exact solutions of the generalized fractional Zakharov-Kuznetsov equations, Life Sci. J., 10 (2013), 2701-2705.
22. N. Taghizadeh, M. Mirzazadeh, M. Rahimian et al. Application of the simplest equation method to some time fractional partial differential equations, Ain Shams Eng. J., 4 (2013), 897-902.
23. C. Chen, Y. L. Jiang, Lie group analysis method for two classes of fractional partial differential equations, Commun. Nonlinear Sci., 26 (2015), 24-35.
24. G. C. Wu, A fractional characteristic method for solving fractional partial differential equations, Appl. Math. Lett., 24 (2011), 1046-1050.
25. 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.
26. A. Akbulut, M. Kaplan, A. Bekir, Auxiliary equation method for fractional differential equations with modified Riemann-Liouville derivative, Int. J. Nonlin. Sci. Num., 17 (2016).
27. W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), 204-226.
28. S. Momani, Z. Odibat, V. S. Erturk, Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. Lett. A., 370 (2007), 379-387.
29. Y. Hu, Y. Luo, 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, 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., 345 (2008), 476-484.
32. G. H. Gao, Z. Z. Sun, Y. N. Zhang, A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions, J. Comput. Phys., 231 (2012), 2865-2879.
33. K. A. Gepreel, The homotopy perturbation method applied to nonlinear fractional Kadomtsev-Petviashvili-Piskkunov equations, Appl. Math. Lett., 24 (2011), 1428-1434.
34. J. F. Gómez-Aguilar, H. Yépez-Martinez, J. Torres-Jimenez et al. Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv. Differ. Equations, 2017 (2017), 68.
35. V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martinez, et al. Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equations, 2016 (2016), 164.
36. H. Yépez-Martinez, J. F. Gómez-Aguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Math. Model. Nat. Phenom., 13 (2018), 13.
37. E. C. Aslan, M. Inc, Numerical solutions and comparisons for nonlinear time fractional Ito coupled system, J. Comput. Theor. Nanos., 13 (2016), 5426-5431.
38. M. Inc, Some special structures for the generalized nonlinear Schrodinger equation with nonlinear dispersion, Wave. Random. Complex., 23 (2013), 77-88.
39. R. Khalil, M. Al Horani, A. Yousef et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
40. A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
41. M. Eslami, H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53 (2016), 475-485.
42. Y. Cenesiz, A. Kurt, The new solution of time fractional wave equation with conformable fractional derivative definition, J. New Theory, 7 (2015), 79-85.
43. M. Boiti, J. Leon and P. Pempinelli, Spectral transform for a two spatial dimension extension of the dispersive long wave equation, Inverse Probl., 3 (1987), 371-387.
44. R. L. Mace, M. A. Hellberg, The Korteweg-de Vries-Zakharov-Kuznetsov equation for electron-acoustic waves, Phys. Plasmas, 8 (2001), 2649-2656.
45. 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.
46. A. K. Khalifaa, K. R. Raslana, H. M. Alzubaidi, A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212 (2008), 406-418.
47. K. R. Raslan, Numerical study of the Modified Regularized Long Wave (MRLW) equation, Chaos, Solitons Fractals, 42 (2009), 1845-1853.
48. K. R. Raslan, S. M. Hassan, Solitary waves for the MRLW equation, Appl. Math. Lett., 22 (2009), 984-989.
49. M. Kaplan, A. Bekir, A. Akbulut, et al. The modified simple equation method for nonlinear fractional differential equations, Rom. Journ. Phys., 60 (2015), 1374-1383.
50. A. B. E. Abdel Salam, E. A. E. Gumma, Analytical solution of nonlinear space-time fractional differential equations using the improved fractional Riccati expansion method, Ain Shams Eng. J., 6 (2015), 613-620.
51. E. Fan, H. Zhang, A note on the homogeneous balance method, Phys. Lett. A., 246 (1998), 403-406.

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