AM  Vol.5 No.1 , January 2014
Exact Solution to Nonlinear Differential Equations of Fractional Order via (G’/G)-Expansion Method
Abstract: In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.
Cite this paper: M. Younis and A. Zafar, "Exact Solution to Nonlinear Differential Equations of Fractional Order via (G’/G)-Expansion Method," Applied Mathematics, Vol. 5 No. 1, 2014, pp. 1-6. doi: 10.4236/am.2014.51001.

[1]   R. S. Johnson, “A Non-Linear Equation Incorporating Damping and Dispersion,” Journal of Fluid Mechanics, Vol. 42, No. 1, 1970, pp. 49-60.

[2]   W. G. Glockle and T. F. Nonnenmacher, “A Fractional Calculus Approach to Self-Similar Protein Dynamics,” Biophysical Journal, Vol. 68, No. 1, 1995, pp. 46-53.

[3]   I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[4]   J. H. He, “Some Applications of Nonlinear Fractional Differential Equations and Their Applications,” Bulletin of Science, Technology & Society, Vol. 15, No. 2, 1999, pp. 86-90.

[5]   M. Wang, X. Li and J. Zhang, “The (G'/G)-Expansion Method and Travelling Wave Soltions of Nonlinear Evolution Equations in Mathematical Physics,” Physical Letter A, Vol. 372, 2008, pp. 417-423.

[6]   Z. Feng, “On Explicit Excact Solutions to the Compound Burgers-KdV Equation,” Physical Letter A, Vol. 293, No. 1-2, 2002, pp. 57-66.

[7]   S. K. Liu, Z. T. Fu, S. D. Liu and Q. Zhao, “Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations,” Physical Letter A, Vol. 289, No. 1-2, 2001, pp. 69-74.

[8]   M. Younis and A. Zafar, “The Modified Simple Equation Method for Solving Nonlinear Phi-Four Equation,” International Journal of Innovation and Applied Studies, Vol. 2 No. 4, 2013, pp. 661-664.

[9]   K. A. Gepreel, “The Homotopy Perturbation Method Applied to the Nonlinear Fractional Kolmogorov Petrovskii Piskunov Equations,” Applied Mathematics Letters, Vol. 24, No. 8, 2011, pp. 1428-1434.

[10]   G.-C. Wu, “A Fractional Characteristic Method for Solving Fractional Partial Differential Equations,” Applied Mathematics Letters, Vol. 24, No. 7, 2011, pp. 1046-1050.

[11]   M. Younis, “The First Integral Method for Time-Space Fractional Differential Equations,” Journal of Advanced Physics, Vol. 2, No. 3, 2013, pp. 220-223.

[12]   Q. Wang, “Numerical Solutions for Fractional KDV-Burgers Equation by Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 1048-1055.

[13]   G. Jumarie, “Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results,” Computers & Mathematics with Applications, Vol. 51, No. 9-10, 2006, pp. 1367-1624.

[14]   G. Jumarie, “Laplaces Transform of Fractional Order via the Mittag Leffler Function and Modified Riemann-Liouville Derivative,” Applied Mathematics Letters, Vol. 22, No. 11, 2009, pp. 1659-1664.

[15]   M. Younis, A. Zafar, K. Ul-Haq and M. Rahman, “Travelling Wave Solutions of Fractional Order Coupled Burger’s Equation by (G'/G)-Expansion Method,” American Journal of Computational and Applied Mathematics, Vol. 3, No. 2, 2013, pp. 81-85.

[16]   K. A. Gepreel and S. Omran, “Exact Solutions for Nonlinear Partial Fractioanl Differential Equations,” Chinese Physics B, Vol. 21, No. 11, 2012, Article DI: 110204.

[17]   Z.-B. Li and J.-H. He, “Fractional Complex Transform for Fractional Differential Equations,” Computers & Mathematics with Applications, Vol. 15, No. 5, 2010, pp. 970-973.