Advertisement

Analytical Solution of Two Dimensional Nonlinear Space–Time Fractional Burgers–Huxley Equation Using Fractional Sub-Equation Method

  • Neeraj Kumar Tripathi
Short Communication
  • 3 Downloads

Abstract

The analytical solution of nonlinear two dimensional space–time fractional Burgers–Huxley equation involving Jumarie’s modified Riemann–Liouville derivative is derived using the tool fractional sub equation method. The obtained results clearly exhibit that the mathematical tool is quite effective, efficient and appropriate for solving various kinds of nonlinear fractional order physical models compared to other existing schemes.

Keywords

Fractional order derivative Burgers–Huxley equation Fractional Ricatti equation Fractional sub-equation method 

Notes

Acknowledgements

The present research work is carried out under the project scheme of Science and Engineering Research Board (SERB), Government of India sanctioned to Dr. S. Das, Department of Mathematical Sciences, IIT (BHU) as Principal Investigator.

References

  1. 1.
    Podlubny I (1999) Fractional differential equations. Academic Press, CambridgeMATHGoogle Scholar
  2. 2.
    Oldham KB, Spanier J (1974) The fractional calculus. Theory and Applications of differentiation and integration to arbitrary order. Academic Press, CambridgeMATHGoogle Scholar
  3. 3.
    Miller KS, Ross B (1993) An introduction to fractional calculus and fractional differential equations. Wiley, HobokenMATHGoogle Scholar
  4. 4.
    Hilfer R (2000) Applications of fractional calculus in physics. World Scientific, LondonCrossRefMATHGoogle Scholar
  5. 5.
    West BJ, Bolognab M, Grigolini P (2003) Physics of fractal operators. Springer, New YorkCrossRefGoogle Scholar
  6. 6.
    Diethelm K (2004) The analysis of fractional differential equation. Springer, BerlinGoogle Scholar
  7. 7.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamMATHGoogle Scholar
  8. 8.
    Liu S, Fu Z, Liu S, Zhao Q (2001) Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 289:69–74ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    He JH, Wu XH (2006) Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 30:700–708ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Tascan F, Bekir A, Koparan M (2009) Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun Nonlinear Sci Numer Simul 10:1810–1815ADSCrossRefGoogle Scholar
  11. 11.
    Wanga M, Lia X, Zhanga J (2008) The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A 372:417–423ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer Academic, BostonCrossRefMATHGoogle Scholar
  13. 13.
    Bobolian E, Javadi S (2004) New method for calculating adomian polynomials. Appl Math Comput 153:253MathSciNetMATHGoogle Scholar
  14. 14.
    Soliman AA, Abdou MA (2008) The decomposition method for solving the coupled modified KdV equations. Math Comput Model 47:1035–1041MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rashidi MM, Ganji DD, Dinarvand S (2009) Explicit analytical solutions of the generalized Burger and Burger Fisher equations by homotopy perturbation method. Numer Math Partial Differ Equ 25:409–417MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Das S (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57(3):483–487MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Molliq RY, Noorani MSM, Hashim I (2009) Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal Real World Appl 10:1854–1869MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liao SJ (1992) The proposed homotopy analysis technique for solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, ChinaGoogle Scholar
  19. 19.
    Liao SJ (2009) Notes on the homotopy analysis method: some definitions and theorems. Commun Nonlinear Sci Numer Simul 14:983–997ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. CRC Press, Boca RatonCrossRefGoogle Scholar
  21. 21.
    Tripathi NK, Das S, Ong SH, Jafari H, Qurashi MA (2016) Solution of higher order nonlinear time-fractional reaction diffusion equation. Entropy 18(9):329.  https://doi.org/10.3390/e18090329 ADSCrossRefGoogle Scholar
  22. 22.
    Odibat Z, Momani S (2008) A generalized differential transform method for linear partial differential equations of fractional order. Appl Math Lett 21:194–199MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Celik I (2012) Haar wavelet method for solving generalized Burgers–Huxley equation. Arab J Math Sci 18:25–37MathSciNetMATHGoogle Scholar
  24. 24.
    Chen CF, Hsiao CH (1997) Haar wavelet method for solving lumped and distributed-parameter systems. IEEE Proc Control Theory Appl 144:87–94CrossRefMATHGoogle Scholar
  25. 25.
    Liu N, Lin E (1998) Legendre wavelet method for numerical solutions of partial differential equations. SIAM J Math Anal 29:1040–1065MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bateman H (1915) Some recent researches on the motion of fluids. Mon Weather Rev 43:163–170ADSCrossRefGoogle Scholar
  27. 27.
    Burgers JM (1939) Mathematical example illustrating relations occurring in the theory of turbulent fluid motion. Trans R Neth Acad Sci Amst 17:1–53MathSciNetMATHGoogle Scholar
  28. 28.
    Efimova OY, Kudryashov NA (2004) Exact solutions of the Burgers–Huxley equation. J Appl Math Mech 68(3):413–420MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Estevez PG, Gordoa PR (1990) Painlevé analysis of the generalized Burgers–Huxley equation. J Phys A 23:4831–4837ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Javidi M (2006) A numerical solution of the generalized Burgers–Huxley equation by spectral collocation method. Appl Math Comput 178(2):338–344MathSciNetMATHGoogle Scholar
  31. 31.
    Wazwaz AM (2008) Analytic study on Burgers–Fisher Huxley equations and combined forms of these equations. Appl Math Comput 195(2):754–761MathSciNetMATHGoogle Scholar
  32. 32.
    Bazeia D (1998) Chiral solutions to generalized Burgers and Burgers–Huxley equations. MIT-CTP 2174Google Scholar
  33. 33.
    Bajunaid I, Fahmay ES (2007) Approximation solution for the generalized time-delayed Burgers–Huxley equation. Far East J Appl Math 28:81–94MathSciNetMATHGoogle Scholar
  34. 34.
    Satsuma JA, Blowitz MF, Uchssteiner BK, Ruskal M (1987) Topics in soliton theory and exactly solvable nonlinear equations. World Scientific, SingaporeGoogle Scholar
  35. 35.
    Zhang S, Zhang HQ (2011) Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys Lett A 375:1069–1073ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Jumarie G (2006) Modified Riemann–Liouville derivative and fractional Taylor series of non differentiable functions further results. Comput Math Appl 51:1367–1376MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ekici M, Duran D, Sonmezoglu A (2014) Constructing of exact solutions to the (2 + 1)-dimensional breaking soliton equations by the multiple (G′/G)-expansion method. J Adv Math Stud 7:27–44MathSciNetMATHGoogle Scholar
  38. 38.
    Guo S, Zhou Y (2010) The extended (G′/G)-expansion method and its applications to Whitham–Broer–Kaup-like equations and coupled Hirota–Satsuma KdV equations. Appl Math Comput 215:3214–3221MathSciNetMATHGoogle Scholar
  39. 39.
    Guo S, Mei LQ, Li Y, Sun YF (2012) The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics. Phys Lett A 376:407–411ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Jafari H, Tajadodi H, Baleanu D, Al-Zahrani AA, Alhamed YA, Zahid AH (2013) Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma–Tasso–Olver equation. Cent Eur J Phys 11(10):1482–1490Google Scholar
  41. 41.
    Zhang S, Zong QA, Liu D, Gao Q (2010) A generalized exp-function method for fractional Riccati differential equation. Commun Fract Calc 1:48–51Google Scholar
  42. 42.
    Hammad DA, El-Azab MS (2015) 2 N order compact finite difference scheme with collocation method for solving the generalized Burger’s–Huxley and Burger’s–Fisher equations. Appl Math Comput 258:296–311MathSciNetMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (Banaras Hindu University)VaranasiIndia

Personalised recommendations