The modified Kudryashov method for solving some fractional-order nonlinear equations

Springer Science and Business Media LLC - Tập 2014 - Trang 1-13 - 2014
Serife Muge Ege1, Emine Misirli1
1Department of Mathematics, Ege University, Bornova, Turkey

Tóm tắt

In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie’s modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

Tài liệu tham khảo

Zhang S: Application of Exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 2007, 365: 448–453. 10.1016/j.physleta.2007.02.004 Misirli E, Gurefe Y: Exp-function method to solve the generalized Burgers-Fisher equation. Nonlinear Sci. Lett. A 2010, 1: 323–328. Misirli E, Gurefe Y: Exp-function method for solving nonlinear evolution equations. Math. Comput. Appl. 2011, 16: 258–266. Liu S, Fu Z, Liu S, Zhao Q: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289: 69–74. 10.1016/S0375-9601(01)00580-1 Yan Z:Abundant families of Jacobi elliptic function solutions of the (2+1) -dimensional integrable Davey-Stewartson-type equation via a new method. Chaos Solitons Fractals 2003, 18: 299–309. 10.1016/S0960-0779(02)00653-7 Tascan F, Bekir A, Koparan M: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Numer. Simul. 2009, 10: 1810–1815. Abbasbandy S, Shirzadi A: The first integral method for modified Benjamin-Bona-Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 1759–1765. 10.1016/j.cnsns.2009.08.003 Zayed EME, Gepreel KA:The ( G ′ /G) -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 2009., 50: Article ID 013502 10.1063/1.3033750 Wanga M, Lia X, Zhanga J:The ( G ′ /G) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372: 417–423. 10.1016/j.physleta.2007.07.051 Soliman AA, Abdo HA: New exact solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method. Int. J. Nonlinear Sci. 2009, 7(3):274–282. Salas AH, Gomez CA: Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Math. Probl. Eng. 2010., 2010: Article ID 194329 10.1155/2010/194329 Podlubny I Math. Sci. Eng. In Fractional Differential Equations. Academic Press, New York; 1999. Bhrawy AH, Baleanu D: A spectral Legendre-Gauss-Labatto collacation method for a space-time fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 2013, 72: 219–233. 10.1016/S0034-4877(14)60015-X Jafari H, Nazari M, Baleanu D, Khalique CM: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 2013, 66: 838–843. 10.1016/j.camwa.2012.11.014 Mehdinejadiani B, Naseri AA, Jafari H, Ghanbarzadeh A, Baleanu D: A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives. Comput. Math. Appl. 2013, 66: 785–794. 10.1016/j.camwa.2013.01.002 Momani S, Odibat Z, Erturk VS: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation. Phys. Lett. A 2007, 370: 379–387. 10.1016/j.physleta.2007.05.083 El-Sayed AMA, Behiry SH, Raslan WE: Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation. Int. J. Nonlinear Sci. 2010, 59: 1759–1765. Hu Y, Luo Y, Lu Z: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl. Math. 2008, 215: 220–229. 10.1016/j.cam.2007.04.005 Saadatmandi A, Dehghan M: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 2010, 59: 1326–1336. 10.1016/j.camwa.2009.07.006 Inc M: The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 2008, 345: 476–484. 10.1016/j.jmaa.2008.04.007 Wua G, Lee EWM: Fractional variational iteration method and its application. Phys. Lett. A 2010, 374: 2506–2509. 10.1016/j.physleta.2010.04.034 Elbeleze AA, Kilicman A, Taib BM: Fractional variational iteration method and its application to fractional partial differential equation. Math. Probl. Eng. 2013., 2013: Article ID 543848 10.1155/2013/543848 Zhang S, Zhang HQ: Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 2011, 375: 1069–1073. 10.1016/j.physleta.2011.01.029 Meng F, Feng Q: A new fractional sub-equation method and its applications for space-time fractional partial differential equations. J. Appl. Math. 2013., 2013: Article ID 481729 10.1155/2013/481729 Alzaidy JF: Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics. Br. J. Math. Comput. Sci. 2013, 3: 153–163. Alzaidy JF: The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. Am. J. Math. Anal. 2013, 11: 14–19. Guo S, Mei L, Li Y, Sun Y: The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Phys. Lett. A 2012, 376: 407–411. 10.1016/j.physleta.2011.10.056 Zheng B: Exp-function method for solving fractional partial differential equations. Sci. World J. 2013., 2013: Article ID 465723 10.1155/2013/465723 Lu B: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 2012, 395: 684–693. 10.1016/j.jmaa.2012.05.066 Younis M: The first integral method for time-space fractional differential equations. J. Adv. Phys. 2013, 2: 220–223. 10.1166/jap.2013.1074 Meng F: A new approach for solving fractional partial differential equations. J. Appl. Math. 2013., 2013: Article ID 256823 10.1155/2013/256823 Zayed EME, Amer YA, Shohib RMA:Exact traveling wave solutions for nonlinear fractional partial differential equations using the improved ( G ′ /G) -expansion method. Int. J. Eng. Appl. Sci. 2014, 7: 18–31. Jumarie G: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51: 1367–1376. 10.1016/j.camwa.2006.02.001 Jumarie G: Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution. J. Appl. Math. Comput. 2007, 24: 31–48. 10.1007/BF02832299 Kudryashov NA: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 2248–2253. 10.1016/j.cnsns.2011.10.016 Ege SM, Misirli E: The modified Kudryashov method for solving some evolution equations. AIP Conf. Proc. 2012, 1470: 244–246. Ege SM, Misirli E: Solutions of the space-time fractional foam-drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method. Int. J. Res. Advent Technol. 2014, 2(3):384–388. Kabir MM: Modified Kudryashov method for generalized forms of the nonlinear heat conduction equation. Int. J. Phys. Sci. 2011, 6: 6061–6064. Kabir MM, Khajeh A, Aghdam EA, Koma AY: Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations. Math. Methods Appl. Sci. 2011, 34: 244–246. Stakhov A, Rozin B: On a new class of hyperbolic functions. Chaos Solitons Fractals 2005, 23: 379–389. 10.1016/j.chaos.2004.04.022