# The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation

#### Authors:

Md. Tarikul Islam, M. Ali Akbar,Md. Abul Kalam Azad,

#### DOI NO:

https://doi.org/10.26782/jmcms.2018.06.00004

#### Keywords:

The fractional generalized (D G/G)-expansion method, the expfunction method,the extended tanh method,nonlinear fractional PDEs,conformable fractional derivative, composite transformation,closed form solutions,

#### Abstract

Abstract In this paper, the analytical solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation involving conformable fractional derivative in science and engineering are examined by using the proposed fractional generalized (D G/G)-expansion method, the Exp-function method and the extended tanh method. The suggested equation is converted into ordinary differential equation of fractional order with the aid of a suitable composite transformation and then the methods are applied to construct the solutions. The methods successfully provide many new and more general closed form traveling wave solutions. The obtained solutions may be more effective to analyze the nonlinear physical phenomena relevance to science and engineering than the existing results in literature. The performance of the proposed method is highly noticeable and this method will be used in further works to establish more entirely new solutions for other kinds of nonlinear fractional PDEs.

#### Refference:

I.Alam, M. N. and Akbar, M. A. “The new approach of the generalized )/(GG-expansion method for nonlinear evolution equations”. Ain Shams Eng. J., Vol. 5 PP 595-603 (2014)

II.Alzaidy, J. F. “The fractional sub-equation method and exact analytical solutions for some fractional PDEs”. Amer. J. Math. Anal., PP 14-19 (2013)

III.Baleanu, D. Diethelm, K. Scalas, E. and Trujillo, J. J. “Fractional Calculus: Models and Numerical Methods”. Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA, Vol. 3 (2012)

IV.Deng, W. “Finite element method for the space and time fractional Fokker-Planck equation”. SIAM J. Numer. Anal., Vol. 47, PP 204-226 (2008)

V.Diethelm, K. “The analysis of fractional differential equations”. Springer-Verlag, Berlin, (2010)

VI.El-Sayed, A. M. A. and Gaber, M. “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains”. Phys. Lett. A, Vol. 359, PP 175-182 (2006)

VII.El-Sayed, A. M. A., Behiry, S. H. and Raslan, W. E. “Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation”. Comput. Math. Appl., Vol. 59, PP 1759–1765 (2010)

VIII.Feng, J., Li, W. and Wan, Q. “Using )/(GG-expansion method to seek traveling wave solution of Kadomtsev-Petviashvili-Piskkunov equation”. Appl. Math. Comput., Vol. 217, PP 5860-5865 (2011)

IX.Gao, G. H., Sun, Z. Z. and Zhang, Y. N. “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions”. J. Comput. Phys.,Vol. 231, PP 2865-2879 (2012)

X.Gepreel, K. A. “The homotopy perturbation method applied to nonlinear fractional Kadomtsev-Petviashvili-Piskkunov equations”. Appl. Math. Lett., Vol. 24, PP 1458-1434 (2011)

XI.Gepreel, K. A. and Omran, S. “Exact solutions for nonlinear partial fractional differential equations”. Chin. Phys. B, Vol. 21, PP 110204-110207 (2012)

XII.Guo, S., Mei, L., Li, Y. and Sun, Y. “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics”. Phys. Lett. A, Vol. 376, PP 407-411 (2012)

XIII.Hilfer, R. “Applications of Fractional Calculus in Physics”.World Scientific Publishing, River Edge, NJ, USA, (2000)

XIV.Hu, Y., Luo, Y. and Lu, Z. “Analytical solution of the linear fractional differential equation by Adomiandecomposition method”. J. Comput. Appl. Math., Vol. 215, PP 220-229 (2008)

XV.Huang, Q., Huang, G. and Zhan, H. “A finite element solution for the fractional advection-dispersion equation”. Adv. Water Resour., Vol. 31, 1578-1589 (2008)

XVI.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. and Appl.Vol. 345, PP 476-484 (2008)

XVII.Islam, M. T., Akbar, M. A. and Azad, A. K. “A Rational )/(GG-expansion method and its application to the modified KdV-Burgers equation and the (2+1)-dimensional Boussinesq equation”. Nonlinear Studies, Vol. 6, PP 1-11 (2015)

XVIII.Khalil, R., Horani, M. A., Yousef, A. and Sababheh, M. “A new definition of fractional derivative”. J. Comput. Appl. Math., Vol. 264, PP 65-67 (2014)

XIX.Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. “Theory and Applications of Fractional Differential Equations”.North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, Vol. 204 (2006)

XX.Kiryakova, V. “Generalized Fractional Calculus and Applications”.Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Harlow, UK, Vol. 301 (1994)

XXI.Li, C., Chen, A. and Ye, J. “Numerical approaches to fractional calculus and fractional ordinary differential equation”. J. Comput. Phys.,Vol. 230, PP 3352-3368 (2011)

XXII.Mainardi, F. “Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models”.Imperial College Press, London, UK, (2010)

XXIII.Miller, K. S. and Ross, B. “An Introduction to the Fractional Calculus and Fractional Differential Equations”.John Wiley & Sons, New York, NY, USA, (1993)

XXIV.Momani, S., Odibat, Z. and Erturk, V. S. “Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation”. Phys. Lett. A, Vol. 370, PP 379-387 (2007)

XXV.Odibat, Z. and Momani, S. “A generalized differential transform method for linear partial differential equations of fractional order”. Appl. Math. Lett.,Vol. 21, PP 194-199 (2008)

XXVI.Odibat, Z. and Momani, S. “Fractional Green functions for linear time fractional equations of fractional order”. Appl. Math. Lett., Vol. 21, PP 194-199 (2008)

XXVII.Odibat, Z. and Momani, S. “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics”. Comput. Math. Appl., 58, PP 2199–2208 (2009)

XXVIII.Oldham, K. B. and Spanier, J. “The Fractional Calculus”.Academic Press, New York, NY, USA, (1974)

XXIX.Podlubny, I. “Fractional Differential Equations”.Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, Vol. 198 (1999)

XXX.Sabatier, J., Agrawal, O. P. and Machado, J. A. T. “Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering”.Springer, New York, NY, USA, (2007)

XXXI.Samko, S. G., Kilbas, A. A. and Marichev, O. I. “Fractional Integrals and Derivatives”.Gordon and Breach Science, Yverdon, Switzerland, (1993)

XXXII.West, B. J., Bologna, M. and Grigolini, P. “Physicsof Fractal Operators”.Springer, New York, NY, USA, (2003)

XXXIII.Wu, G. C. and Lee, E. W. M. “Fractional variational iteration method and its application”. Phys. Let. A, Vol. 374, PP 2506-2509 (2010 )

XXXIV.Yang, X. J. “Local Fractional Functional Analysis and Its Applications”.Asian Academic Publisher, Hong Kong, (2011)

XXXV.Yang, X. J. “Advanced Local Fractional Calculus and Its Applications”.World Science Publisher, New York NY, USA, (2012)

XXXVI.Zhang S. and Zhang, H. Q. “Fractional sub-equation method and itsapplications to nonlinear fractional PDEs”. Phys. Lett. A, Vol. 375, PP1069-1073 (2011)

XXXVII.Zhang, Y. and Feng, Q. “Fractional Riccati equation rational expansionmethod for fractional differential equations”.Appl. Math. Inf. Sci., Vol.PP 1575-1584 (2013)

XXXVIII Zheng, B. “)/(GG-expansion method for solving fractional partialdifferentialequations in the theory of mathematical physics”. Commu.Theore. Phys., Vol. 58, PP 623-630 (2012)

Author(s): Md. Tarikul Islam, M. Ali Akbar and Md. Abul Kalam Azad