International Scientific Journal

Thermal Science - Online First

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Exact solution to nonlinear biological population model with fractional order

In this paper, Optimal Homotopy Asymptotic Method (OHAM) has been extended to seek out the exact solution of fractional generalized biological population models. The time fractional derivatives are described in the Caputo sense. It (OHAM) is a new approach for fractional models. The proposed approach presents a procedure by that we have transferred the model to a series of simpler problems which are solvable by hand work applying the Riemann-Liouville fractional integral operator and obtained exact solution of fractional the generalized biological population by adding the solutions of first three simple problems of the series of simpler problems. The new approach provides exact solution in the way of smoothly convergent series.
PAPER REVISED: 2017-12-22
PAPER ACCEPTED: 2018-01-03
  1. K.B. Oldham, J. Spanier, the Fractional Calculus, Academic Press, New York, 1974.
  2. M. M. Meerschaert, H. P. Scheffler and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Physic. 211 (2006) 249 -261.
  3. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physic, 37 (2004) 161-208.
  4. I. Podlubny, Fractional Differential Equations, Academic Press New York 1999.
  5. W. R. Schneider and W. Wyess, Fractional diffusion and wave equations, J. Math. Physic. 30 (1989) 134-144.
  6. S.Z.Rida, A. M.A. El-Sayed and A. AM Arafa. Effect of bacterial memory dependent growth by usingfractional derivatives reaction-diffusion chemotactic model, Journal of Statistical Physics, 140(4) (2010) 797-811.
  7. A.M.A. El-Sayed, S. Z. Rida, and A. A. M. Arafa. On the solutions of the generalized reactiondiffusion model for bacterial colony, Acta Applicandae Mathematicae, 110(3 ) (2010) 1501-1511.
  8. G.W. Wang, and T. Z. Xu, The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations, Romanian Journal of Physics, 66 (2014) 636-645.
  9. S.Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, A fractional model for bacterial chemo attractant in a liquid medium, Nonlinear Science Letters A, 14 (2010) 415-420.
  10. A. M. A .El-Sayed, et al, Numerical behavior of a fractional order dynamical model of RNA silencing, International Journal of Scientific World, 4(2) (2016) 52-56.
  11. A. M. A,. El-Sayed, S. Z. Rida, and A. A. M. Arafa, Exact solutions of fractional-order biological population model, Commun. Theor. Physics, 52(6) (2009), 992-1002.
  12. A. M. A . El-Sayed, S. Z. Rida, and A. A. M. Arafa,On the solutions of time-fractional bacterial chemo taxis in a diffusion gradient chamber, Int. J. Nonlinear Sci. 7(4) (2009) 485-492.
  13. S.Z. Rida, H. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional nonlinear Schrödinger equation, Physics Letters A, 372(5) (2008) 553-558.
  14. A. Karbalaie, M. M. Montazeri and H. H. Muhammed. New approach to find the exact solution of fractional partial differential equation, WSEAS Transactions on Mathematics, 11(10) (2012) 908-917.
  15. Caputo, M, and F. Mainardi, Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, 1(2) (1971) 161-198.
  16. R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in: Fractals Fractional Calculus in Continuum Mechanics, eds. A. Carpinteri and F. Mainar, Springer Verlag, Wien and New York (1997) pp. 223-276.
  17. R. Gorenflo and R. Rutman, On Ultraslow and Intermediate Processes, in Transform Methods and Special Functions, eds. P. Rusev, I. Dimovski, and V. Kiryakova, Science Culture Technology Publishing (SCTP), Singapore (1995) pp. 61-81.
  18. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  19. V. Marinca, N. Herisanu , The Optimal Homotopy Perturbation Method: Nonlinear Dynamical Systems in Engineering. Springer, Berlin, Heidelberg, 2012.