THERMAL SCIENCE

International Scientific Journal

M. Ali Akbar

,

Norhashidah Hj. Mohd. Ali

,

Abdul-Majid Wazwaz

CLOSED FORM TRAVELING WAVE SOLUTIONS OF NON-LINEAR FRACTIONAL EVOLUTION EQUATIONS THROUGH THE MODIFIED SIMPLE EQUATION METHOD

ABSTRACT
In this article, the modified simple equation (MSE) method is introduced to examine the closed form wave solutions of the fractional non-linear Cahn-Allen equation and of the fractional generalized reaction Duffing equation. The fractional derivatives are delineated in the sense of Jumarie’s modified Riemann-Liouville derivative. A fractional complex transformation is used to transform the fractional-order PDE into integer order ODE. The reduced equations are then examined by using the MSE method and some new and further general solutions of these equations are successfully established. The approach of this method is simple, standard and the obtained solutions are highly encouraging. It is also powerful, reliable and effective.
PAPER SUBMITTED: 2017-06-13
PAPER REVISED: 2017-11-21
PAPER ACCEPTED: 2017-12-05
PUBLISHED ONLINE: 2018-03-04
DOI REFERENCE: https://doi.org/10.2298/TSCI170613097A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S341 - S352]
REFERENCES
  1. Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
  2. Oldman KB, Spanier J. The Fractional Calculus: Theory and applications of differentiation and integration to arbitrary order. Academic Press, New York, 1974.
  3. Bekir A, Aksoyb E, Cevikel AC, Exact solutions of nonlinear time fractional partial differential equations by sub-equation method. Math. Meth. Appl. Sci., 38(2014), 1-6.
  4. Taghizadeh N, Mirzazadeh M, Rahimian M, Akbari M. Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Engr. J., 4(2013), 897-902.
  5. Eslami M, Vajargah BF, Mirzazadeh M, Biswas A. Application of first integral method to fractional partial differential equations, Indian J. Phys., 88(2014), 177-184.
  6. Mirzazadeh M, Eslami M, Biswas A. Solitons and periodic solutions to a couple of fractional nonlinear evolution equations, Pramana J. Phys., 82(2014), 465-476.
  7. Mohyud-Din ST, Khan Y, Faraz N, Yildirm A., Exp-function method for solitary and periodic solutions of Fitzhugh Nagumo equations, Int. J. Numer. Meth. Heat Fluid Flow, 22 (2012), 335-341.
  8. Mohyud-Din ST, Noor MA, Waheed T. Exp-function method for generalized traveling solutions of Calogero-Degasperis Fokas equation, J. Phys. Sci., 65a (2010), 78-84.
  9. Liu W, Chen K. The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana J. Phys., 81(2013), 377-384.
  10. Shang N, Zheng B. Exact solutions for three fractional partial differential equations by the (G /G) method, Int. J. Appl. Math., 43(2013) 3.
  11. Bulut H, Pandir Y. Modified trial equation method to the nonlinear fractional Sharma-Tasso-Olever equation. Int. J. Modeling Optimization, 3(2013), 353-357.
  12. Pandir Y, Gurefe Y, Misirli E. New exact solutions of the time-fractional nonlinear dispersive KdV equation. Int. J. Modeling Optimization, 3(2013), 349-352.
  13. Yang XJ, Zhang YD. A new Adomian decomposition procedure scheme for solving local fractional Volterra integral equation. Advances Inform. Tech. Manag., 1( 2012), 158-161.
  14. Seadawy AR. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in plasma, Comp. Math. Appl., 67(2014), 172-180.
  15. Seadawy AR. Three dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in magnetized plasma, Comp. Math. Appl., 71(2016), 201-212.
  16. Seadawy AR, Kalaawy OHE, Aldenari RB. Water wave solutions of Zufiria's higherorder Boussinesq type equations and its stability, Appl. Math. Comput., 280(2016), 57-71.
  17. Seadawy AR. New exact solutions for the KdV equation with higher order nonlinearity by using the variational method, Comput. Math. Appl., 62 (2011), 3741-3755.
  18. Sheikholeslami M. Cuo-water nanofluid free convection in a porous cavity considering Darcy law, Eur. Phys. J. Plus, 132(2017), 132: 29.
  19. Arshad M, Seadawy A, Lu D, Wang J. Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations, Results Phys. 6 (2016), 1136-1145.
  20. Inc M, Aliyu AI, Yusuf A. Solitons and conservation laws to the resonance nonlinear Schrodinger's equation with both spatio-temporal and inter-modal dispersions, Optik-Int. J. Light Electron Optics, 142(2017), 509-522.
  21. Aslan EC, Inc M, Baleanu D. Optical solitons and stability analysis of the NLSE with anti-cubic nonlinearity, Superlattices and Microstructures, 109(2017), 784-793.
  22. Mohyud-Din ST, Bibi S. Exact solutions for nonlinear fractional differential equations using exponential rational function method, Opt. Quant. Electron., 49(2017), 49: 64.
  23. Mohyud-Din ST, Irshad A. Solitary wave solutions of some nonlinear PDEs arising in electronics, Opt. Quant. Electron., 49(2017) 49: 130.
  24. Mohyud-Din ST, Yildirm A, Sariaydin S. Numerical soliton solution of the Kaup-Kupershmidt equation, Int. J. Numer. Meth. Heat Fluid Flow, 21(2011), 272-281.
  25. Mohyud-Din ST, Negahdary E, Usman M. A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations, Int. J. Numer. Meth. Heat Fluid Flow, 2(2012), 641-658.
  26. Mohyud-Din ST, Noor MA, Noor KI. Traveling wave solutions of seventh-order generalized KdV equations using He's polynomials, Int. J. Nonlinear Sci. Numer. Simulat., 10(2009), 223-229.
  27. Mohyud-Din ST, Noor MA, Noor KI, Hosseini MM. Variational iteration method for reformulated partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 11 (2010), 87-92.
  28. Khan K, Akbar MA. Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method. Ain Shams Engr. J., 5(2014), 274-256.
  29. Jawad AJM, Petkovic MD, Biswas A. Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217(2010), 869-877.
  30. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouvillie derivative for non-differentiable functions, Appl. Math. Lett., 22(2009), 378-385.
  31. Li ZB, He JH. Application of the fractional complex transform to fractional differential equations. Nonlin. Sci. Lett. A-Math., Phys. Mech., 2(2011), 121-126.
  32. Esen A, Yagmurlu NM, Tasbozan O. Approximate analytical solution to time-fractional damped burger and Cahn-Allen equations. Appl. Math. Inform. Sci., 7(5)( 2013), 1951-1956.
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Closed form traveling wave solutions of non-linear fractional evolution equations through the modified simple equation method

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