American Journal of Mathematical Analysis
ISSN (Print): 2333-8490 ISSN (Online): 2333-8431 Website: Editor-in-chief: Grigori Rozenblum
Open Access
Journal Browser
American Journal of Mathematical Analysis. 2013, 1(1), 14-19
DOI: 10.12691/ajma-1-1-3
Open AccessArticle

The Fractional Sub-Equation Method and Exact Analytical Solutions for Some Nonlinear Fractional PDEs

J. F. Alzaidy1,

1Mathematics Department, Faculty of Science, Taif University, Kingdom of Saudi Arabia

Pub. Date: February 26, 2013

Cite this paper:
J. F. Alzaidy. The Fractional Sub-Equation Method and Exact Analytical Solutions for Some Nonlinear Fractional PDEs. American Journal of Mathematical Analysis. 2013; 1(1):14-19. doi: 10.12691/ajma-1-1-3


In the present paper, a fractional sub-equation method is proposed to solve fractional differential equations. Being concise and straightforward, this method is applied the space–time fractional Potential Kadomtsev–Petviashvili (PKP) equation and the space–time fractional Symmetric Regularized Long Wave (SRLW) equation. As a result, many exact analytical solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear fractional PDEs arising in mathematical physics.

fractional sub-equation method fractional differential equation modified Riemann–Liouville derivative Mittag-Leffler function analytical solutions

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


[1]  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.
[2]  R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ,USA, 2000.
[3]  B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
[4]  K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY,USA, 1993.
[5]  S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[6]  I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, San Diego, Calif, USA, 1999.
[7]  K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
[8]  V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994.
[9]  I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, New York, NY, USA, 1999.
[10]  J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
[11]  F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010.
[12]  D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA,2012.
[13]  X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong.
[14]  X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
[15]  A. H. A. Ali, The modified extended tanh-function method for solving coupled MKdV and coupled Hirota-Satsuma coupled KdV equations, Phys. Lett.A, 363(2007) 420.
[16]  C. Li, A. Chen and J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput .Phys, 230(2011) 3352.
[17]  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, 231(2012) 2865.
[18]  W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J . Numer. Anal, 47(2008/09) 204.
[19]  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.
[20]  Z. Odibat and S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett, 21(2008)194.
[21]  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.
[22]  A. M. A. El-Sayed and M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A, 359(2006)175.
[23]  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.
[24]  Z. Odibat and S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput . Math. Appl, 58(2009)2199.
[25]  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, 345(2008)476.
[26]  G. C. Wu and E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A, 374(2010)2506.
[27]  J.-H.He, Homotopy perturbation technique, Comput Methods. Appl. Mech. Eng, 178(1999)257.
[28]  E. Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A, 282(2001)18.
[29]  A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional- order differential equations, Comput. Math. Appl, 59 (2010) 1326.
[30]  Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Anal, 71 (2009) 2724.
[31]  S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375(2011)1069.
[32]  M. L. Wang, Solitary wave solutions for variant Boussinesq equations,Phys. Lett. A, 199(1995)169.
[33]  G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl, 51(2006)1367.
[34]  G. Jumarie, Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Comput, 24 (2007) 31.
[35]  S. Guo, L. Mei, Y. Li and Y. Sun, 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.
[36]  B. Lu, Bä cklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A, 376 (2012) 2045.
[37]  Y. B. Zhou, M. L. Wang and Y. M. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys.Lett.A, 308(2003)31.
[38]  S. Zhang, Q. A. Zong, D. Liu and Q. Gao, A generalized Exp-function method for fractional Riccati differential equations,Commun. Fract. Calc, 1 (2010) 48.
[39]  Z. S. Lü and H. Q. Zhang, On a new modified extended tanh-function method, Commun. Theor. Phys. (Beijing, China), 39(2003) 405.
[40]  A. Borhanifar and M.M. Kabir, New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations, J. Comput. Appl. Math, 229 (2009) 158.
[41]  Fei .Xu, Application of Exp-function method to Symmetric Regularized Long Wave (SRLW) equation. Phys. Lett. A, 372 (2008) 252.