A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

ABSTRACT

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

KEYWORDS

Exact Solutions, Complex Transformation, Modified Simple Equation Method, Nonlinear Equations of Fractional Order, Fractional Calculus Theory

Exact Solutions, Complex Transformation, Modified Simple Equation Method, Nonlinear Equations of Fractional Order, Fractional Calculus Theory

Cite this paper

Younis, M. (2014) A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method.*Applied Mathematics*, **5**, 1927-1932. doi: 10.4236/am.2014.513186.

Younis, M. (2014) A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method.

References

[1] Johnson, R.S. (1970) A Non-Linear Equation Incorporating Damping and Dispersion. Journal of Fluid Mechanics, 42, 49-60.

http://dx.doi.org/10.1017/S0022112070001064

[2] Glöckle, W.G. and Nonnenmacher, T.F. (1995) A Fractional Calculus Approach to Self Similar Protein Dynamics. Biophysical Journal, 68, 46-53.

http://dx.doi.org/10.1016/S0006-3495(95)80157-8

[3] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[4] He, J.H. (1999) Some Applications of Nonlinear Fractional Differential Equations and Their Applications. Bulletin of Science and Technology, 15, 86-90.

[5] Wang, Q. (2006) Numerical Solutions for Fractional KDV-Burgers Equation by Adomian Decomposition Method. Applied Mathematics and Computation, 182, 1048-1055.

http://dx.doi.org/10.1016/j.amc.2006.05.004

[6] Rahman, M., Mahmood, A. and Younis, M. (2014) Improved and More Feasible Numerical Methods for Riesz Space Fractional Partial Differential Equations. Applied Mathematics and Computation, 237, 264-273.

http://dx.doi.org/10.1016/j.amc.2014.03.103

[7] Liu, J. and Hou, G. (2011) Numerical Solutions of the Space- and Time-Fractional Coupled Burgers Equations by Generalized Differential Transform Method. Applied Mathematics and Computation, 217, 7001-7008.

http://dx.doi.org/10.1016/j.amc.2011.01.111

[8] Iftikhar, M., Rehman, H.U. and Younis, M. (2013) Solution of Thirteenth Order Boundary Value Problems by Differential Transformation Method. Asian Journal of Mathematics and Applications, 2014, 11 p.

[9] Wang, M., Li, X. and Zhang, J. (2008) The -Expansion Method and Travelling Wave Soltions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423.

http://dx.doi.org/10.1016/j.physleta.2007.07.051

[10] Bin, Z. (2012) -Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics. Communications in Theoretical Physics, 58, 623-630.

http://dx.doi.org/10.1088/0253-6102/58/5/02

[11] Younis, M. and Zafar, A. (2014) Exact Solution to Nonlinear Differential Equations of Fractional Order via - Expansion Method. Applied Mathematics, 5, 1-6.

http://dx.doi.org/10.4236/am.2014.51001

[12] Liu, S.K., Fu, Z.T., Liu, S.D. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74.

http://dx.doi.org/10.1016/S0375-9601(01)00580-1

[13] Parkes, E.J. and Duffy, B.R. (1996) An Automated Tanh-Function Method for Finding Solitary Wave Solutions to Non-Linear Evolution Equations. Computer Physics Communications, 98, 288-300.

http://dx.doi.org/10.1016/0010-4655(96)00104-X

[14] Gepreel, K.A. (2011) The Homotopy Perturbation Method Applied to the Nonlinear Fractional Kolmogorov Petrovskii Piskunov Equations. Applied Mathematics Letters, 24, 1428-1434.

http://dx.doi.org/10.1016/j.aml.2011.03.025

[15] Younis, M. (2013) The First Integral Method for Time-Space Fractional Differential Equations. Journal of Advanced Physics, 2, 220-223.

http://dx.doi.org/10.1166/jap.2013.1074

[16] Younis, M. and Ali, S. (2014) New Applications to Solitary Wave Ansatz. Applied Mathematics, 5, 969-974.

http://dx.doi.org/10.4236/am.2014.56092

[17] Li, Z.-B. and He, J.-H. (2010) Fractional Complex Transform for Fractional Differential Equations. Computers & Mathematics with Applications, 15, 970-973.

[18] Jawad, A.J.M., Petkovic, M.D. and Biswas, A. (2010) Modified Simple Equation Method for Nonlinear Evolution Equations. Applied Mathematics and Computation, 217, 869-877.

http://dx.doi.org/10.1016/j.amc.2010.06.030

[19] Younis, M., Iftikhar, M. and Rehman, H.U. (2014) Exact Solutions to the Nonlinear Schrdinger and Eckhaus Equations by Modied Simple Equation Method. Journal of Advanced Physics, (accepted).

[20] Jumarie, G. (2006) Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results. Computers & Mathematics with Applications, 51, 1367-1624.

http://dx.doi.org/10.1016/j.camwa.2006.02.001

[21] Jumarie, G. (2009) Laplace Transform of Fractional Order via the Mittag Leffler Function and Modified Riemannan Liouville Derivative. Applied Mathematics Letters, 22, 1659-1664.

http://dx.doi.org/10.1016/j.aml.2009.05.011

[1] Johnson, R.S. (1970) A Non-Linear Equation Incorporating Damping and Dispersion. Journal of Fluid Mechanics, 42, 49-60.

http://dx.doi.org/10.1017/S0022112070001064

[2] Glöckle, W.G. and Nonnenmacher, T.F. (1995) A Fractional Calculus Approach to Self Similar Protein Dynamics. Biophysical Journal, 68, 46-53.

http://dx.doi.org/10.1016/S0006-3495(95)80157-8

[3] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[4] He, J.H. (1999) Some Applications of Nonlinear Fractional Differential Equations and Their Applications. Bulletin of Science and Technology, 15, 86-90.

[5] Wang, Q. (2006) Numerical Solutions for Fractional KDV-Burgers Equation by Adomian Decomposition Method. Applied Mathematics and Computation, 182, 1048-1055.

http://dx.doi.org/10.1016/j.amc.2006.05.004

[6] Rahman, M., Mahmood, A. and Younis, M. (2014) Improved and More Feasible Numerical Methods for Riesz Space Fractional Partial Differential Equations. Applied Mathematics and Computation, 237, 264-273.

http://dx.doi.org/10.1016/j.amc.2014.03.103

[7] Liu, J. and Hou, G. (2011) Numerical Solutions of the Space- and Time-Fractional Coupled Burgers Equations by Generalized Differential Transform Method. Applied Mathematics and Computation, 217, 7001-7008.

http://dx.doi.org/10.1016/j.amc.2011.01.111

[8] Iftikhar, M., Rehman, H.U. and Younis, M. (2013) Solution of Thirteenth Order Boundary Value Problems by Differential Transformation Method. Asian Journal of Mathematics and Applications, 2014, 11 p.

[9] Wang, M., Li, X. and Zhang, J. (2008) The -Expansion Method and Travelling Wave Soltions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423.

http://dx.doi.org/10.1016/j.physleta.2007.07.051

[10] Bin, Z. (2012) -Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics. Communications in Theoretical Physics, 58, 623-630.

http://dx.doi.org/10.1088/0253-6102/58/5/02

[11] Younis, M. and Zafar, A. (2014) Exact Solution to Nonlinear Differential Equations of Fractional Order via - Expansion Method. Applied Mathematics, 5, 1-6.

http://dx.doi.org/10.4236/am.2014.51001

[12] Liu, S.K., Fu, Z.T., Liu, S.D. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74.

http://dx.doi.org/10.1016/S0375-9601(01)00580-1

[13] Parkes, E.J. and Duffy, B.R. (1996) An Automated Tanh-Function Method for Finding Solitary Wave Solutions to Non-Linear Evolution Equations. Computer Physics Communications, 98, 288-300.

http://dx.doi.org/10.1016/0010-4655(96)00104-X

[14] Gepreel, K.A. (2011) The Homotopy Perturbation Method Applied to the Nonlinear Fractional Kolmogorov Petrovskii Piskunov Equations. Applied Mathematics Letters, 24, 1428-1434.

http://dx.doi.org/10.1016/j.aml.2011.03.025

[15] Younis, M. (2013) The First Integral Method for Time-Space Fractional Differential Equations. Journal of Advanced Physics, 2, 220-223.

http://dx.doi.org/10.1166/jap.2013.1074

[16] Younis, M. and Ali, S. (2014) New Applications to Solitary Wave Ansatz. Applied Mathematics, 5, 969-974.

http://dx.doi.org/10.4236/am.2014.56092

[17] Li, Z.-B. and He, J.-H. (2010) Fractional Complex Transform for Fractional Differential Equations. Computers & Mathematics with Applications, 15, 970-973.

[18] Jawad, A.J.M., Petkovic, M.D. and Biswas, A. (2010) Modified Simple Equation Method for Nonlinear Evolution Equations. Applied Mathematics and Computation, 217, 869-877.

http://dx.doi.org/10.1016/j.amc.2010.06.030

[19] Younis, M., Iftikhar, M. and Rehman, H.U. (2014) Exact Solutions to the Nonlinear Schrdinger and Eckhaus Equations by Modied Simple Equation Method. Journal of Advanced Physics, (accepted).

[20] Jumarie, G. (2006) Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results. Computers & Mathematics with Applications, 51, 1367-1624.

http://dx.doi.org/10.1016/j.camwa.2006.02.001

[21] Jumarie, G. (2009) Laplace Transform of Fractional Order via the Mittag Leffler Function and Modified Riemannan Liouville Derivative. Applied Mathematics Letters, 22, 1659-1664.

http://dx.doi.org/10.1016/j.aml.2009.05.011