A new method for solving nonlinear fractional differential equations

Abstract

In this paper, a new extended Kudryashov method for solving fractional nonlinear differential equations is proposed. The fractional derivative in this paper is considered in the sense of modified Riemann-Liouville. We also handle the time-fractional fifthorder SawadaKotera equation and the time-fractional generalized HirotaSatsuma coupled KdV equation to illustrate the the simplicity and the effectiveness of this method. Solutions of these equations are obtained in analytical traveling wave solution form including hyperbolic and trigonometric functions.

Keywords

• Time-fractional fifth-order Sawada-Kotera equation,time-fractional generalized Hirota-Satsuma coupled KdV equation

References

1. Aminikhah, H., Sheikhani, A. R., Rezazadeh H., 2008, Exact solutions for the fractional differential equations by using the first integral method, Nonlinear Engineering 4 (1) , 15–22p.
2. Boudjehem, B., Boudjehem, D., 2011, Parameter tuning of a fractional-order PI Controller using the ITAE Criteria, Fractional Dynamics and Control, 2011, 49–57p.
3. Bekir, A. and Guner, O., 2013, Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method, Chinese Physics B , 2013, 22p.
4. Ege, S. M. and Misirli, E., 2014, The modified Kudryashov method for solving some fractional-order nonlinear equations, Advances in Difference Equations, 135, 1–13p.
5. Ege, S. M. and Misirli, E., 2014, Solutions of the space-time fractional foam-drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method,International Journal of Research in Advent Technology, 2321(9637), 384–388p.
6. Ege, S. M., 2015, On semianalytical solutions of some nonlinear physical evolution equations with polynomial type auxilary equation,PhD Thesis, Ege University, January, 2015
7. Fan, E., 2000, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A , 246, 403p.
8. Feng, Z. S. and Wang, X. H., 2003, The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation, Phys. Lett. A , Analytic solutions for nonlinear partial fractional differential equations 308, 173–178p.

9. Fu, Z. T., Liu, S. K., Liu, S. D. and Zhao, Q., 2001, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A, 290, 72–76p.
10. Ganjia, Z.Z., Ganjia, D.D. Rostamiyan, Y., 2009, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique, Applied Mathematical Modelling 33, 3107–3113p.
11. Guan J., 2014, Fractional Form Jacobi Elliptic Function Solutions For the second order Benjamin Ono equation, Sch. J. Eng. Tech., 2(3C), 456–458p.
12. Guner, O., Bekir, A. and Bilgil, H., 2015, A note on exp-function method combined with complex transform method applied to fractional differential equations, Advances in Nonlinear Analysis, 4(3), 201–208p.
13. Guoa, S., Meia, Y., Lia, Y. and Sunb, Y. , 2012, The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,Physics Letters A, 376, 407-411p.
14. Jumarie, G. , 2006, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Compt. Math. Appl., 51, 1367-1376p.
15. Jumarie, G. , 2007, Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Compt., 24, 31-48p.
16. J. H. He, "A Tutorial Review on Fractal Spacetime and Fractional Calculus," Int J. Theor. Phys.,53, pp.3698-3718, 2014.
17. Kabir, M. M., Khajeh, A., Aghdam, E. A. and Koma, A. Y., 2011, Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations, Math. Methods Appl. Sci., 35, 213–219p.
18. Kaplan, M., Bekir, A., Akbulut A. and Aksoy, E., 2015, The Modified Simple Equation Method for Nonlinear Fractional Differential Equations, Romanian J. Phys., ISSN 1221-146, 1–10p.
19. Kilic, B., Inc, M., 2015, The First Integral Method for the time fractional Kaup-Boussinesq System with time dependent coefficient, Journal Applied Mathematics and Computation, 254, 70-74p.
20. Kudryashov, N. A. , 2012, One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci., 17, 2248–2253p.
21. Liu, W., Chen, K., 2013, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana J. Phys., 81, 377-384p.
22. Lu, B., 2012, The first integral method for some time fractional differential equations, Journal of Mathematical Analysis and Applications, 395, 684–693p.
23. Malinowska, A. B., 2011, Fractional variational calculus for nondifferentiable functions, Fractional Dynamics and Control, 2011, 77–108p.
24. Meng, F., Feng, Q., 2012, A new fractional subequation method and its applications for space-time fractional partial differential equations, Journal of Apllied Mathematics, 2012, 1–10p.
25. Miller, K. S. and Ross, B, 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York.
26. Nofal, T. A. , Gepreel K. A. and Farag, M. H., 2014, Analytic solutions for nonlinear partial fractional differential equations, International Journal of Scientific and Engineering Research 5(12), 1519p.
27. Podlubny, I., 1999, Fractional Differential Equations, Academic Press, California.
28. Shang, N., Zheng, B., 2013, Exact Solutions for Three Fractional Partial Differential Equations by the (G/G) Method, Int. J. of Appl. Math., 43:3, 1–6p.
29. Zheng, B., 2013, Exp-function method for solving fractional partial differential equations, Scientific World Journal, 2013, 1–8p.
30. Zheng, B., 2012,(G’/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics, Commun. Theor. Phys., 58, 623–630p.
31. Zheng, B., 2014, A new fractional Jacobi elliptic equation method for solving fractional partial differential equations, Advances in Difference Equations, 228, 1–11p.
32. Zheng, B., Wen, C., 2013, Exact solutions for fractional partial differential equations by a new fractional sub-equation method, Advances in Difference Equations, 199, 1–12p.