Citation: M. Ali Akbar, Norhashidah Hj. Mohd. Ali, M. Tarikul Islam. Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics[J]. AIMS Mathematics, 2019, 4(3): 397-411. doi: 10.3934/math.2019.3.397
[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. doi: 10.1016/j.joes.2017.12.003 |
[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. doi: 10.1088/0253-6102/58/5/02 |
[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. doi: 10.1016/j.asej.2013.12.008 |
[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. doi: 10.26782/jmcms.2018.04.00002 |
[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. doi: 10.26782/jmcms.2018.06.00004 |
[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. doi: 10.1016/j.physleta.2011.10.056 |
[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. doi: 10.1016/j.jmaa.2012.05.066 |
[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. doi: 10.1007/s12648-013-0401-6 |
[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. doi: 10.1007/s12043-013-0583-7 |
[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. doi: 10.1016/j.asej.2013.01.006 |
[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. doi: 10.1016/j.cnsns.2015.01.018 |
[22] | G. C. Wu, A fractional characteristic method for solving fractional partial differential equations, Appl. Math. Lett., 24 (2011), 1046-1050. doi: 10.1016/j.aml.2011.01.020 |
[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. doi: 10.1016/j.physleta.2007.05.083 |
[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. doi: 10.1016/j.cam.2007.04.005 |
[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. doi: 10.1016/j.camwa.2009.08.065 |
[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. doi: 10.1016/j.cam.2014.01.002 |
[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. doi: 10.1063/1.1363665 |
[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. doi: 10.1016/j.camwa.2016.02.004 |
[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. doi: 10.1016/j.asej.2014.10.014 |
[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. doi: 10.1016/j.cam.2006.12.029 |
[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. doi: 10.1016/j.aml.2009.01.020 |
[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. doi: 10.1016/j.camwa.2006.02.001 |
[44] | J.-H. He, A Tutorial Review on Fractal Space time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), 3698-3718. doi: 10.1007/s10773-014-2123-8 |
[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. doi: 10.1016/j.chaos.2018.09.032 |
[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. doi: 10.1016/j.chaos.2018.12.015 |
[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. doi: 10.1016/j.cnsns.2018.01.005 |
[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. |