Applied and Computational Mathematics
Volume 3, Issue 1, February 2014, Pages: 27-31
Received: Dec. 8, 2013;
Published: Feb. 28, 2014
Views 3452 Downloads 208
Mohamed S. Mohamed, Mathematics Department, Faculty of Science, Al-Azhar University, Egypt；Mathematics Department, Faculty of Science, Taif University, Saudi Arabia
In this paper, we use the optimal homotopy analysis method (OHAM) for approximate solutions of the fractional order Logistic equation. The numerical results obtained are compared with the results obtained by using variational iteration method (VIM) and Adomian decomposition method (ADM). The fractional derivatives are described by Caputo's sense. Exact and/or approximate analytical solutions of these equations are obtained. The results reveal that this method is very effective and powerful to obtain the approximate solutions.
Mohamed S. Mohamed,
Application of Optimal HAM for Solving the Fractional Order Logistic Equation, Applied and Computational Mathematics.
Vol. 3, No. 1,
2014, pp. 27-31.
K. B. Oldham, J. Spanier, The fractional calculus, Academic press, New York 1974.
K. S. Miller, B. Ross, An introduction to the fractional and fractional differential equations, John Wiley and Sons, New York, 1993.
Y. Luchko, R. Gorenflo, The initial-value problem for some fractional differential equations with Caputo derivative, Preprint Series A08-98, Fachbereich Mathematik and Informatic, Freie Universitat,Berlin, 1998.
I. Podlubny, Fractional differential equations, Academic press, New York,1999.
S S Ray, RK Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput.(167)(2005)561-571.
O. Abdulaziz, I. Hashim, Chowdhury MSH, Zulkifle Ak, Assessment of decomposition method for linear and nonlinear fractional differential equations. Far East J Appl. Math. 28(1)(2007)95-112.
S. Momani, Z. Odibat , Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A365 (2007)345-350.
Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitions Fractals. 36(2008)167-174.
Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equation of fractional order. Int J Nonlinear Sci. Num. Simul. 1(7)(2006)271-279.
S. Momani, Z. Odibat. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals.31(2007)1248-1255.
SJ Liao, The proposed homotopy analysis technique for the solution of nonlinear problem. Ph.D thesis, Shanghai Jiao Tong University;1992.
SJ Liao, An approximate solution technique which does not depend upon small parameters: a special example. Int. J. Nonlinear Mech. 30(1995)371-380.
SJ Liao, An approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int. J. Nonlinear Mech. 32(1997)815-822.
SJ Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlinear Mech. 34(4)(1999)759-778.
SJ Liao, Beyond perturbation: introduction to the homotopy analysis method,CRC Press, Boca Raton: Chapman& Hall, 2003.
SJ Liao, Notes on the homotopy analysis method: Some defintions and theorems, Commun Nonlinear Sci Numer Simulat. 14(2009)983-997.
K. Hemida, M. S. Mohamed, Numerical simulation of the generalized Huxley equation by homotopy analysis method, Journal of applied functional analysis, 5(4)(2010)344-350.
M. S. Mohamed, Application of homotopy analysis method to fractional order generalized Huxley equation, Journal of applied functional analysis, 7(4)(2012), pp. 367-373.
M. S. Mohamed, H. Ghany, Analytic approximations for fractional-order hyperchaotic system, Journal of advanced research in Dynamical and Control Systems, 3(2011), pp. 1-12
H. A. Ghany and M. S. Mohamed, White noise functional solutions for the wick-type stochastic fractional Kdv-Burgers-Kuramoto equations, Journal of the Chinese Journal of Physics, 50(4)(2012), pp. 619-626.
K. A. Gepreel and M. S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chinese physics B, 22(1)( 2013), pp. 010201-6.
A.M.A. El-Sayed, E.M. El-Mesiry and H.A.A. El- Saka , On the fractional-order logistic equation, Applied Mathematics-Letters 20(2007)817-823.
A. M. A. El-Sayed, H. A. A. El-Saka, and E. M. El-Maghrabi, "On the fractional-order logistic equationwith two different delays," Zeitschrift fur Naturforschung. Section A , vol. 66, no. 3-4, pp. 223--227, 2011.
N. H. Sweilam , M. M. Khader, A. M. S. Mahdy, Numerical studies for solving fractional order logistic equation,International Journal of Pure and Applied Mathematics , 78( 8) 2012, 1199-1210
N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays, Journal of Applied Mathematics,Volume 2012, Article ID 764894, 14 pages
S J Liao. An optimal homotopy analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Science and Numerical Simulation, 15(8)(2010) 2003-2016.
K. Yabushita , M. Yamashita and K. Tsuboi. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A. Math.Gen., 40(2007) 8403.
S. M. Abo-Dahab, M. S. Mohamed and T. A. Nofal. A One Step Optimal Homotopy Analysis Method for propagation of harmonic waves in nonlinear generalized magneto-thermo elasticity with two relaxation times under influence of rotation. Journal of in Abstract and Applied Analysis, (2013) 1-14.
K. A. Gepreel, and M. S. Mohamed. An optimal homotopy analysis method nonlinear fractional differential equation. Journal of Advanced Research in Dynamical and Control Systems, 6(2014)1-10.