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Journal of Applied Mathematics
Volume 2012, Article ID 764894, 14 pages
http://dx.doi.org/10.1155/2012/764894
Research Article

Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays

1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2Department of Mathematics, Faculty of Science, Benha University, Benha 15318, Egypt
3Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 10 March 2012; Revised 17 June 2012; Accepted 4 July 2012

Academic Editor: J. Biazar

Copyright © 2012 N. H. Sweilam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.