
AIMS Mathematics, 2019, 4(3): 397411. doi: 10.3934/math.2019.3.397
Research article Special Issues
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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
Received: , Accepted: , Published:
Special Issues: New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators
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 NorthHolland 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), 7681.
7. Z. Bin, $(G'/G)$expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys., 58 (2012), 623630.
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), 595603.
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 KolmogorovPetrovskiiPiskunovequation, JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 13 (2018), 1733.
10. M. T. Islam, M. A. Akbar and M. A. K. Azad, The exact traveling wave solutions to the nonlinear spacetime fractional modified BenjaminBonaMahony equation, JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 13 (2018), 5671.
11. J. F. Alzaidy, The fractional subequation method and exact analytical solutions for some fractional PDEs, American Journal of Mathematical Analysis, 1 (2013), 1419.
12. S. Guo, L. Mei, Y. Li, et al. The improved fractional subequation method and its applications to the spacetime fractional differential equations in fluid mechanics, Phys. Lett. A, 376 (2012), 407411.
13. B. Zheng, Expfunction 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 Expfunction method combined with complex transform method applied to fractional differential equations, Adv. Nonlinear Anal, 4 (2015), 201208.
15. B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684693.
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), 177184.
17. W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time fractional differential equations, Pramana, 81 (2013), 377384.
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), 18.
19. Y. Pandir and Y. Gurefe, New exact solutions of the generalized fractional ZakharovKuznetsov equations, Life Sci. J., 10 (2013), 27012705.
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), 897902.
21. C. Chen and Y. L. Jiang, Lie group analysis method for two classes of fractional partial differential equations, Commun. Nonlinear Sci., 26 (2015), 2435.
22. G. C. Wu, A fractional characteristic method for solving fractional partial differential equations, Appl. Math. Lett., 24 (2011), 10461050.
23. A. R. Seadawy, Travellingwave solutions of a weakly nonlinear twodimensional higherorder KadomtsevPetviashvili dynamical equation for dispersive shallowwater 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 RiemannLiouville derivative, Int. J. Nonlin. Sci. Num., 17 (2016), 413420.
25. G. H. Gao, Z. Z. Sun and Y. N. Zhang, A finite difference scheme for fractional subdiffusion equations on an unbounded domain using artificial boundary conditions, J. Comput. Phys., \textbf{231 (2012), 28652879.
26. W. Deng, Finite element method for the space and time fractional FokkerPlanck equation, SIAM J. Numer. Anal., 47 (2008), 204226.
27. S. Momani, Z. Odibat and V. S. Erturk, Generalized differential transform method for solving a space and timefractional diffusionwave equation, Phys. Lett. A, 370 (2007), 379387.
28. K. A. Gepreel, The homotopy perturbation method applied to nonlinear fractional KadomtsevPetviashviliPiskkunov equations, Appl. Math. Lett., 24 (2011), 14581434.
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), 220229.
30. A. M. A. ElSayed, S. H. Behiry and W. E. Raslan, Adomian's decomposition method for solving an intermediate fractional advectiondispersion equation, Comput. Math. Appl., 59 (2010), 17591765.
31. M. Inc, The approximate and exact solutions of the space and timefractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., \textbf{345 (2008), 476484.
32. R. Khalil, M. A. Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 6570.
33. M. T. Islam, M. A. Akbar and A. K. Azad, A Rational $(G'/G)$expansion method and its application to the modified KdVBurgers equation and the (2+1)dimensional Boussinesq equation, Nonlinear Studies, 22 (2015), 635645.
34.Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Mathematical & Computational Applications, 15 (2010), 970973.
35. R. L. Mace and M. A. Hellberg, The Kortewegde VriesZakharovKuznetsov equation for electronacoustic waves, Phys. Plasmas, 8 (2001), 26492656.
36. O. Guner, E. Aksoy, A. Bekir, et al. Different methods for (3+1)dimensional spacetime fractional modified KdVZakharovKuznetsov equation, Comput. Math. Appl., 71 (2016), 12591269.
37. E. A. B. AbdelSalam and E. A. E. Gumma, Analytical solution of nonlinear spacetime fractional differential equations using the improved fractional Riccati expansion method, Ain Shams Engineering Journal, 6 (2015), 613620.
38. A. K. Khalifaa, K. R. Raslana and H. M. Alzubaidi, A collocation method with cubic Bsplines for solving the MRLW equation, J. Comput. Appl. Math., 212 (2008), 406418.
39. K. R. Raslan, Numerical study of the Modified Regularized Long Wave (MRLW) equation, Chaos, Solitons & Fractals, 42 (2009), 18451853.
40. K. R. Raslan and S. M. Hassan, Solitary waves for the MRLW equation, Applied Mathematics Letters, 22 (2009), 984989.
41.M. Kaplan, A. Bekir, A. Akbulut, et al. The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015), 13741383.
42. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York NY, USA, 2012.
43. G. Jumarie, Modified RiemannLiouville Derivative and Fractional Taylor Series of NonDifferentiable Functions Further Results, Comput. Math. Appl., 51 (2006), 13671376.
44. J.H. He, A Tutorial Review on Fractal Space time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), 36983718.
45. S. Aman, Q. AlMdallal 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. AlMdallal, On fractionalLegendre spectral Galerkin method for fractional Sturm Liouville problems, Chaos, Solitons and Fractals, 116 (2018), 261267.
47. T. Abdeljawad, Q. AlMdallal and F. Jarad, Fractional logistic models in the frame of fractional operators generated by conformable derivatives, Chaos, Solitons and Fractals, 119 (2019), 94101.
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), 411.
49 T. Abdeljawad, Q. AlMdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), 18.
50. Q. AlMdallal, K. A. Abro and I. Khan, Analytical solutions fractional Walter's B fluid with applications, Complexity, 2018 (2018), 110.
51. P. Agarwal, Q. AlMdallal, Y. J. Cho, et al. Fractional differential equations for the generalized MittagLeffler function, Adv. Differ. EquNY, 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), 7291.
53. J. V. da C. Sousa and E. C. de Oliveira, MittagLeffler functions and the truncated $\upsilon$fractional derivative, Mediterr. J. Math., 14 (2017), 244.
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