Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays
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 utili...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/764894 |
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| author | N. H. Sweilam M. M. Khader A. M. S. Mahdy |
| author_facet | N. H. Sweilam M. M. Khader A. M. S. Mahdy |
| author_sort | N. H. Sweilam |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e594866a262a4917b7a9804f79f6a834 |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-e594866a262a4917b7a9804f79f6a8342025-08-20T02:21:19ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/764894764894Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different DelaysN. H. Sweilam0M. M. Khader1A. M. S. Mahdy2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, Faculty of Science, Benha University, Benha 15318, EgyptDepartment of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptA 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.http://dx.doi.org/10.1155/2012/764894 |
| spellingShingle | N. H. Sweilam M. M. Khader A. M. S. Mahdy Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays Journal of Applied Mathematics |
| title | Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays |
| title_full | Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays |
| title_fullStr | Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays |
| title_full_unstemmed | Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays |
| title_short | Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays |
| title_sort | numerical studies for fractional order logistic differential equation with two different delays |
| url | http://dx.doi.org/10.1155/2012/764894 |
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