Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 13, 2019

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

Esin Aksoy, Ahmet Bekir and Adem C Çevikel


In this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.


[1] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.Search in Google Scholar

[2] I. Podlubny, Fractional differential erquations, Academic Press, California, 1999.Search in Google Scholar

[3] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.Search in Google Scholar

[4] S. Zhang, Q-A. Zong, D. Liu and Q. Gao, A generalized exp-function method for fractional riccati differential equations, Commun. Fraction. Calc. 1(1) (2010), 48–51.10.1155/2010/764738Search in Google Scholar

[5] A. Bekir, Ö. Güner and A. C. Cevikel, Fractional complex transform and exp-function methods for fractional differential equations, Abstr. Appl. Anal. 2013 (2013), 426462.10.1155/2013/426462Search in Google Scholar

[6] S. Zhang and H-Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A. 375 (2011), 1069–1073.10.1016/j.physleta.2011.01.029Search in Google Scholar

[7] J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs, Am. J. Math. Anal. 1(1) (2013), 14–19.Search in Google Scholar

[8] A. Bekir and E. Aksoy. Application of the sub-equation method to some differential equations of time-fractional order, J. Comput. Nonlinear Dyn. 10 (2015), 054503–1.10.1115/1.4028826Search in Google Scholar

[9] H. Y. Mart’ınez, J. M. Reyes and I. O. Sosa, Fractional sub-equation method and analytical solutions to the Hirota–Satsuma coupled KdV equation and coupled mKdV equation, Br. J. Math. Comput. Sci. 4(4) (2014), 572–589.10.9734/BJMCS/2014/7059Search in Google Scholar

[10] B. Zheng, (G'/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys. 58 (2012), 623–630.10.1088/0253-6102/58/5/02Search in Google Scholar

[11] K. A. Gepreel and S. Omran, Exact solutions for nonlinear partial fractional differential equations, Chin. Phys. B. 21(11) (2012), 110204.10.1088/1674-1056/21/11/110204Search in Google Scholar

[12] A. Akgül, A. Kiliçman and M. Inc, Improved (G'/G)-expansion method for the space and time fractional Foam Drainage and KdV equations, Abstr. Appl. Anal. 2013(2013), 414353.10.1155/2013/414353Search in Google Scholar

[13] B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl. 395 (2012), 684–693.10.1016/j.jmaa.2012.05.066Search in Google Scholar

[14] M. Younis, A new approach for the exact solutions of nonlinear equations of fractional order via modified simple equation method, Appl. Math. 5 (2014), 1927–1932.10.4236/am.2014.513186Search in Google Scholar

[15] W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana J. Phys. 81(3) (2013), 377–384.10.1007/s12043-013-0583-7Search in Google Scholar

[16] A. Bekir, Ö. Güner, E. Aksoy and Y. Pandır, Functional variable method for the nonlinear fractional differential equations, AIP Conf. Proc. 1648 (2015), 730001.10.1063/1.4912955Search in Google Scholar

[17] H. Bulut, H. M. Baskonus and Y. Pandır, The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstr. Appl. Anal. 2013 (2013), 636802.10.1155/2013/636802Search in Google Scholar

[18] E. A-B. Abde-Salam and E. A. Yousif, Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method, Math. Prob. Eng. 2013 (2013), 846283.Search in Google Scholar

[19] N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (11) (2012), 2248–2253.10.1016/j.cnsns.2011.10.016Search in Google Scholar

[20] S. T. Demiray, Y. Pandir and H. Bulut, Generalized Kudryashov method for time-fractional differential equations, Abstr. Appl. Anal. 2014 (2014), 901540.Search in Google Scholar

[21] M. Mirzazadeh, M. Ekici, A. Sonmezoglu, S. Ortakaya, M. Eslami and A. Biswas, Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics, Eur. Phys. J. Plus. 131 (2016), 166.10.1140/epjp/i2016-16166-7Search in Google Scholar

[22] Q. Zhou and M. Mirzazadeh, Analytical solitons for Langmuir waves in plasma physics with cubic nonlinearity and perturbations, Z. Naturforsch. A. 71 (2016), 807–815.10.1515/zna-2016-0201Search in Google Scholar

[23] M. Eslami, B. Fathi Vajargah, M. Mirzazadeh and A. Biswas, Application of first integral method to fractional partial differential equations, Indian J. Phys. 88(2) (2014), 177–184.10.1007/s12648-013-0401-6Search in Google Scholar

[24] M. Mirzazadeh, Analytical study of solitons to nonlinear time fractional parabolic equations, Nonlinear Dyn. 85(4) (2016), 2569–2576.10.1007/s11071-016-2845-7Search in Google Scholar

[25] G. Jumarie, Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (2006), 1367–1376.10.1016/j.camwa.2006.02.001Search in Google Scholar

[26] G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouvillie derivative for nondifferentiable functions. Appl. Maths. Lett. 22 (2009), 378–385.10.1016/j.aml.2008.06.003Search in Google Scholar

[27] Z. B. Li and J. H. He, Fractional complex transform for fractional differential equations, Math. Comput. Appl. 15 (2010), 970–973.10.3390/mca15050970Search in Google Scholar

[28] J-H. He and Z. B. Li, Converting fractional differential equations into partial differential equations. Therm. Sci. 16(2) (2012), 331–334.10.2298/TSCI110503068HSearch in Google Scholar

[29] H. Y. Mart’ınez, J. M. Reyes and I. O. Sosa, Fractional sub-equation method and analytical solutions to the Hirota–Satsuma coupled KdV equation and coupled mKdV equation, Br. J. Math. Comput. Sci. 4(4) (2014), 572–589.10.9734/BJMCS/2014/7059Search in Google Scholar

Received: 2015-10-21
Accepted: 2019-05-28
Published Online: 2019-06-13
Published in Print: 2019-08-27

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow