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Afrika Matematika

, Volume 30, Issue 7–8, pp 1133–1143 | Cite as

Exact solution for the fractional partial differential equation by homo separation analysis method

  • M. ZuriqatEmail author
Article
  • 34 Downloads

Abstract

In this paper, the homo separation analysis method is used to obtain the exact solution for linear and nonlinear fractional partial differential equation (FPDE). This analytical method is a combination of the homotopy analysis method with the separation of variables method. By using this method, FPDE to be solved is changed into FODE. However, this method depends on several important topics and definitions such as Riemann–Liouville fractional integral, Caputo’s definition and Mittag-leffler function. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given.

Keywords

Homotopy analysis method Separation of variables Partial differential equations Mittag-leffler function Caputo derivative Relaxation–oscillation equation 

Mathematics Subject Classification

35M11 

Notes

References

  1. 1.
    Gorenflo, R., Mainardi, F.: Fractional Calculus. Springer, Vienna (1977)zbMATHGoogle Scholar
  2. 2.
    Miller, K.S., Ross, B.: An Inroduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)Google Scholar
  3. 3.
    Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 81, 435–493 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    El-Sayed, A.: Nonlinear fractional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999)zbMATHGoogle Scholar
  6. 6.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore (2009)zbMATHGoogle Scholar
  7. 7.
    Zaslavsky, G.M.: Chaos fractional kenetics and anomalous transport. Phys. Rep. 371(6), 461–580 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Al-Smadi, M., Abu Arqub, O.: Computational algorithm for solving fredholm time-fractional partial integro differential equations of dirichlet functions type with error estimates. Appl. Math. Comput. 342, 280–294 (2019)MathSciNetGoogle Scholar
  9. 9.
    Abuteen, E., Freihat, A., Al-Smadi, M., Khalil, H., Khan, R.A.: Approximate series solution of nonlinear, fractional Klein–Gordon equations using fractional reduced differential transform method. J. Math. Stat. 12(1), 23–33 (2016)CrossRefGoogle Scholar
  10. 10.
    Hirota, R.: Exact enve lope-soliton solutions of a non linear wave. J. Math. Phys. 14(7), 805–809 (1973)CrossRefGoogle Scholar
  11. 11.
    Maliet, W.: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. Comput. Appl. Math. 164, 529–541 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Odibat, Z., Bertelle, C., Aziz-Alaoui, M.A., Duchamp, G.: A multi-step diffrential transform method and application to non-chaotic or chaotic systems. Comput. Math. Appl. 59(4), 1462–1472 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Erturk, V.S., Momani, S., Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 13, 1642–1654 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Odibat, Z., Momani, S.: A generalized differential transform method for linear partial differential equations of fractional order. Appl. Math. Lett. 21, 194–199 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Freihat, A.A., Zuriqat, M.: Analytical solution of fractional Burgers–Huxley equations via residual power series method. Lobachevskii J. Math. 40(2), 174–182 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365, 345–350 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang, Q.: Homotopy perturbation method for fractional order KdV equation. Appl. Math. Comput. 190(2), 1795–1802 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ghandehari, M.A.M., Ranjbar, M.: A numerical method for solving a fractional partial differential equation through converting it into an NLP problem. Comput. Math. Appl. 65, 975–982 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ghandehari, M.A.M., Ranjbar, M.: Solving the fractional Volterra integro-differential equations by an extremum problem. J. Adv. Res. Sci. Comput. 7, 38–49 (2015)MathSciNetGoogle Scholar
  20. 20.
    Zhang, J.L., Wang, M.L., Li, X.R.: The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrö dinger equation. Chaos Solitons Fractals 33, 1450–1457 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wazwaz, A.M.: The tanh–coth and the sech methods for exact solutions of the Jaulent–Miodek equation. Phys. Lett. A 366, 85–90 (2007)CrossRefGoogle Scholar
  22. 22.
    Zuriqat, M.: The homo separation analysis method for solving the partial differential equation. Ital. J. Pure Appl. Math. 40, 535–543 (2018)zbMATHGoogle Scholar
  23. 23.
    Yang, G., Chen, R., Yao, L.: On exact solutions to partial differential equations by the modified homotopy perturbation method. Acta Math. Appl. Sin. 28, 91–98 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ghandehari, M., Ranjbar, M.: Using homo-separation of variables for pricing European option of the fractional Black–Scholes model in financial markets. Math. Sci. 5(2), 181–187 (2016)Google Scholar
  25. 25.
    Cang, J., Tan, Y., Xu, H., Liao, S.: Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals 40, 1–9 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Song, L., Zhang, H.: Application of homotopy analysis method to fractional KdV-Burgers–Kuramoto equation. Phys. Lett. A 367, 88–94 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Al-Smadi, M., Gumah, G.: On the homotopy analysis method for fractional SEIR epidemic model. Res. J. Appl. Sci. Eng. Technol. 7(18), 3809–3820 (2014)CrossRefGoogle Scholar
  28. 28.
    Zurigat, M., Momani, S., Alawneh, A.: Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method. Comput. Math. Appl. 59, 1227–1235 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Samko, S.G., Kilber, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publshers, Yverdon (1993)Google Scholar
  30. 30.
    Cheng, J., Chu, Y.: Solution to the linear fractional differential equation using Adomian decomposition method. Math. Probl. Eng. 2011, 14 (2014). (Article ID 587068)zbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAl al-Bayt UniversityAl MafraqJordan

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