Skip to main content

Advertisement

Log in

A new analysis for the Keller-Segel model of fractional order

Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this study, we discuss the application of an analytical technique namely modified homotopy analysis transform method (MHATM) for solving coupled one- dimensional time-fractional Keller-Segel (K-S) equations. The MHATM is a new analytical technique based on homotopy polynomial. We provide a convergence analysis of MHATM and the solution obtained by the proposed method is verified through different graphical representations. The results demonstrate that the proposed methodology is very useful and simple in the determination of the solution of the K-S equations of fractional order.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Atangana, A.: Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Keller-Segel equations. Appl. Math. Modell. 39, 2909–2916 (2015)

    Article  MathSciNet  Google Scholar 

  2. Atangana, A., Alabaraoye, E.: Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations. Adv. diff. equat. 94, 1–14 (2013)

    Google Scholar 

  3. Argyros, I.K.: Convergence and Applications of Newton-type Iterations. Springer-Verlag, New York (2008)

    MATH  Google Scholar 

  4. Anastassiou, G.A.: Fractional differentiation inequalities. Springer, New York (2009)

  5. Anastassiou, G.A.: Fractional representation formulae and right fractional inequalities. Math. Comput. Modell. 54, 3098–3115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Anastassiou, G.A.: Advanced fractional Taylor’s formula. J. Comput. Anal. Appl. 21(7), 1185–1204 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Caputo, V., Mainardi, F.: A new dissipation model based on memory mechanism. Pure. Appl. Geophys. 91, 134–147 (1971)

    Article  MATH  Google Scholar 

  8. Hilfer, R.: Application of Fractional Calculus in Physics World Scientific (2000)

  9. Jafari, H., Golbabai, A., Seifi, S., Sayevand, K.: Homotopy analysis method for solving multi-term linear and nonlinear diffusion—wave equations of fractional order. Comput. Math. Appl. 59(3), 1337–1344 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Bio. 26(3), 399–415 (1970)

    Article  MATH  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, The Netherlands (2006)

    MATH  Google Scholar 

  12. Khan, N.A., Jamil, M., Ara, A.: Approximate solutions to time fractional Schrodinger equation via homotopy analysis method. ISRN Math. Phys. 2012 (2012). Article ID 197068, 11 pages

  13. Kumar, S.: A new analytical modelling for telegraph equation via Laplace transform. Appl. Math. Modell. 38(13), 3154–3163 (2014)

    Article  MathSciNet  Google Scholar 

  14. Kumar, S., Rashidi, M.M.: New analytical method for gas dynamic equation arising in shock fronts. Comput. Phy. Commun. 185(7), 1947–1954 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kumar, S., Yildirim, A., Khan, Y., Leilei, W.: A fractional model of the diffusion equation and its analytical solution using Laplace transform. Sci. Iran 19(4), 1117–1123 (2012)

    Article  Google Scholar 

  16. Liao, S.J.: A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics. Int. J. Nonlinear Mech. 32(5), 815–822 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liao, S.J.: Beyond Perturbation: Introduction to the homotopy analysis method. Chapman and Hall CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  18. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147(2), 499–513 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119(4), 297–354 (2007)

    Article  MathSciNet  Google Scholar 

  20. Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat. 14(4), 983–997 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Magrenan, A.A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Miller, K.S., Ross, B.: An introduction to the fractional integrals and derivativestheory and application (1993)

  23. Momani, S., Odibat, Z.: Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl. Math. Comput. 177(2), 488–494 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, NY, USA (1974)

    MATH  Google Scholar 

  25. Odibat, Z., Momani, S., Xu, H.: A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Appl. Math. Modell. 34(3), 593–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Odibat, Z.: A study on the convergence of homotopy analysis method. Appl. Math. Comput. 217, 782–789 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Odibat, Z., Bataineh, A.S.: An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials. Math. Meth. Appl. Sci. 38, 991–1000 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Podlubny, I.: Fractional differential equations. Academic, New York (1999)

    MATH  Google Scholar 

  29. Pandey, R.K., Singh, Om P., Baranwal, V.K.: An analytic algorithm for the spacetime fractional advectiondispersion equation. Comput. Phys. Commun. 182, 1134–1144 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Magin, R.L.: Fractional calculus in bio-engineering. Inc. Connecticut, Begell House Publisher (2006)

    Google Scholar 

  31. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: Theory and applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  32. Vishal, K., Kumar, S., Das, S.: Application of homotopy analysis method for fractional swift Hohenberg equation-Revisited. Appl. Math. Modell. 36(8), 3630–3637 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, X.J., Baleanu, D., Srivastava, H.M.: Local fractional integral transform and their applications. Elsevier Academic Press (2015)

  34. Yang, X.J., Baleanu, D.: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 17, 625–628 (2013)

    Article  Google Scholar 

  35. Zaslavsky, G.M.: Hamiltonian chaos and fractional dynamics. Oxford University Press (2005)

  36. Zhang, X., Tang, B., He, Y.: Homotopy analysis method for higher-order fractional integro-differential equations. Comput. Math. Appl. 62(8), 3194–3203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sunil Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Kumar, A. & Argyros, I.K. A new analysis for the Keller-Segel model of fractional order. Numer Algor 75, 213–228 (2017). https://doi.org/10.1007/s11075-016-0202-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0202-z

Keywords

Navigation