1887
Volume 2015, Issue 1
  • E-ISSN: 2223-506X

Abstract

In this paper we have established the simplest equation method to find approximate solutions for total Burgurs equations with time fractional derivative. This method is effective in finding approximate traveling wave solutions of nonlinear, fractional evolution equations (NLEEs) in mathematical physics.

The effectiveness of this manageable method has been shown by applying it to several examples in time fractional Burgers equations. The present approach has the potential to apply to other nonlinear fractional differential equations. All numerical calculations in the present study have been performed on a PC, applying programs written in and .

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2015-07-07
2019-08-17
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References

  1. [1]. Wazwaz   AM. . Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations. . Applied Mathematics and Computation . 2008; ;195: 2 : 754– 761 .
    [Google Scholar]
  2. [2]. Wadati   M. . Introduction to solitons. . Pramana: Journal of Physics.   2001; ;57: 5/6 : 841– 847 .
    [Google Scholar]
  3. [3]. Malfliet   W. . Solitary wave solutions of nonlinear wave equations. . American Journal of Physics.   1992; ;60: 7 : 650– 654 .
    [Google Scholar]
  4. [4]. Biswas   A. . Topological 1-soliton solution of the nonlinear Schro dingers equation with Kerr law nonlinearity in (12) dimensions. . Communications in Nonlinear Science and Numerical Simulation.   2009; ;14: 7 : 2845– 2847 .
    [Google Scholar]
  5. [5]. Biswas   A. . 1-Soliton solution of the K(m, n) equation with generalized evolution. . Physics Letters A.   2008; ;372: 25 : 4601– 4602 .
    [Google Scholar]
  6. [6]. Nassar   HA., , Abdel-Razek   MA., , Seddeek   AK. . Expanding the tanh-function method for solving nonlinear equations. . Applied Mathematics.   2011; ;2: 9 : 1096– 1104 .
    [Google Scholar]
  7. [7]. Fan   EG. . Extended tanh-function method and its applications to nonlinear equations. . Physics Letters A.   2000; ;277: 4-5 : 212– 218 .
    [Google Scholar]
  8. [8]. Kudryashov   NA. . Simplest equation method to look for exact solutions of nonlinear differential equations. . Chaos, Solitons and Fractals.   2005; ;24: 5 : 1217– 1231 .
    [Google Scholar]
  9. [9]. Vitanov   NK., , Dimitrova   ZI. . Modified method of simplest equation and its application to nonlinear PDEs. . Applied Mathematics and Computation.   2010; ;216: 9 : 2587– 2595 .
    [Google Scholar]
  10. [10]. Zayed   EME., , Zedan   HA., , Gepreel   KA. . On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations. . Chaos, Solitons and Fractals.   2004; ;22: 2 : 285– 303 .
    [Google Scholar]
  11. [11]. Zhou   YB., , Wang   ML., , Wang   YM. . Periodic wave solutions to coupled KdV equations with variable coefficients. . Physics Letters A.   2003; ;308: 1 : 31– 36 .
    [Google Scholar]
  12. [12]. Hirota   R. . The direct method in soliton theory . Volume 155 of Cambrige tracts in Mathematics . UK: : Cambridge University Press;   2004; .
    [Google Scholar]
  13. [13]. Akbar   MA., , Ali   NHM. . Exp-function method for Duffing equation and new solutions of (21) dimensional dispersive long wave equations. . Progress in Applied Mathematics.   2011; ;1: 2 : 30– 42 .
    [Google Scholar]
  14. [14]. Adomian   G. . Solving frontier problems of physics: the decomposition method . Boston: : Kluwer Academic;   1994; .
    [Google Scholar]
  15. [15]. Wu   GC., , Baleanu   D. . Variational iteration method for the Burgers flow with fractional derivatives New Lagrange multipliers. . Applied Mathematical Modelling.   2013; ;37: 9 : 6183– 6190 .
    [Google Scholar]
  16. [16]. Abdou   MA. . The extended tanh-method and its applications for solving nonlinear physical models. . Applied Mathematics and Computation.   2007; ;190: 1 : 988– 996 .
    [Google Scholar]
  17. [17]. Sirendaoreji   E. . New exact travelling wave solutions for the Kawahara and modified Kawahara equations. . Chaos, Solitons and Fractals.   2004; ;19: 1 : 147– 150 .
    [Google Scholar]
  18. [18]. Shi   LM., , Zhang   LF., , Meng   H., , Zhao   HW., , Zhou   SP. . A method to construct Weierstrass elliptic function solution for nonlinear equations. . International Journal of Modern Physics B.   2011; ;25: 4 : 1931– 1939 .
    [Google Scholar]
  19. [19]. He   Y., , Li   S., , Long   Y. . Exact solutions of the Klein-Gordon equation by modified Exp-function method. . International Mathematical Forum.   2012; ;7: 4 : 175– 182 .
    [Google Scholar]
  20. [20]. Zayed   EME., , Ibrahim   SAH. . Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method. . Chinese Physics Letters.   2012; ;29: 6 : 060201 .
    [Google Scholar]
  21. [21]. Kudryashov   NA. . Simplest equation method to look for exact solutions of nonlinear differential equations. . Chaos, Solitons and Fractals.   2005; ;24: 5 : 1217– 1231 .
    [Google Scholar]
  22. [22]. Vitanov   NK., , Dimitrova   Zi. . Modified method of simplest equation and its application to nonlinear PDEs. . Applied Mathematics and Computation . 2010; ;216: 9 : 2587– 2595 .
    [Google Scholar]
  23. [23]. Bateman   H. . Some recent researches on the motion of fluids. . Monthly Weather Reviews.   1915; ;43: 4 : 163– 170 .
    [Google Scholar]
  24. [24]. Burger   JM. . A mathematical model illustrating the theory of turbulence. . Advances in Applied Mechanics.   1948; ; 171– 199 .
    [Google Scholar]
  25. [25]. Cole   JD. . On a quasi-linear parabolic equation occurring in aerodynamics. . Quarterly Applied Mathematics.   1951; ;9: 3 : 225– 236 .
    [Google Scholar]
  26. [26]. Jumarie   G. . Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. . Computers and Mathematics with Applications.   2006; ;51: 9-10 : 1367– 1376 .
    [Google Scholar]
  27. [27]. Kong   C., , Wang   D., , Song   L., , Zhang   H. . New exact solutions to MKDV-Burgers equation and (21)-dimensional dispersive long wave equation via extended Riccati equation method. . Chaos, Solitons and Fractals.   2009; ;39: 2 : 697– 706 .
    [Google Scholar]
  28. [28]. Malfliet   W., , Hereman   W. . The tanh method: Exact solutions of nonlinear evalution and wave equations. . Physica Scripta.   1996; ;54: 6 : 563– 568 .
    [Google Scholar]
  29. [29]. Feng   Z., , Chen   G. . Solitary wave solutions of the compound Burgers Korteweg de Vries equation. . Physica A: Statistical mechanics and its applications.   2005; ;352: 2-4 : 419– 435 .
    [Google Scholar]
  30. [30]. Gao   H., , Zhao   RX. . New exact solutions to the generalized Burgers-Huxley equation. . Applied Mathematics and Computation.   2010; ;217: 4 : 1598– 1603 .
    [Google Scholar]
  31. [31]. El-Wakil   SA., , Abdou   MA. . New exact traweling wave solutions using modified extended tanh-function method. . Chaos, Solitons and Fractals.   2007; ;31: 4 : 840– 852 .
    [Google Scholar]
  32. [32]. Esen   A., , Yagmurlu   M., , Tasbozm   O. . Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations. . Applied Mathematics and Information Sciences.   2013; ;7: 5 : 1951– 1956 .
    [Google Scholar]
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