Chaos on the Vallis Model for El Niño with Fractional Operators
The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first tim...
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| Format: | Article |
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MDPI AG
2016-03-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/18/4/100 |
| _version_ | 1817993187985195008 |
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| author | Badr Saad T. Alkahtani Abdon Atangana |
| author_facet | Badr Saad T. Alkahtani Abdon Atangana |
| author_sort | Badr Saad T. Alkahtani |
| collection | DOAJ |
| description | The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo–Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo–Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo–Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo–Fabrizio presents more information that were not revealed in the model with local derivative. |
| first_indexed | 2024-04-14T01:35:35Z |
| format | Article |
| id | doaj.art-486f258229d045f7958e379a70c559c9 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T01:35:35Z |
| publishDate | 2016-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-486f258229d045f7958e379a70c559c92022-12-22T02:19:56ZengMDPI AGEntropy1099-43002016-03-0118410010.3390/e18040100e18040100Chaos on the Vallis Model for El Niño with Fractional OperatorsBadr Saad T. Alkahtani0Abdon Atangana1Department of Mathematics, Colleges of Sciences, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi ArabiaInstitute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9301, South AfricaThe Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo–Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo–Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo–Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo–Fabrizio presents more information that were not revealed in the model with local derivative.http://www.mdpi.com/1099-4300/18/4/100Vallis modelchaotic behaviorCaputo–Fabrizio fractional derivativeanalysisnumerical simulations |
| spellingShingle | Badr Saad T. Alkahtani Abdon Atangana Chaos on the Vallis Model for El Niño with Fractional Operators Entropy Vallis model chaotic behavior Caputo–Fabrizio fractional derivative analysis numerical simulations |
| title | Chaos on the Vallis Model for El Niño with Fractional Operators |
| title_full | Chaos on the Vallis Model for El Niño with Fractional Operators |
| title_fullStr | Chaos on the Vallis Model for El Niño with Fractional Operators |
| title_full_unstemmed | Chaos on the Vallis Model for El Niño with Fractional Operators |
| title_short | Chaos on the Vallis Model for El Niño with Fractional Operators |
| title_sort | chaos on the vallis model for el nino with fractional operators |
| topic | Vallis model chaotic behavior Caputo–Fabrizio fractional derivative analysis numerical simulations |
| url | http://www.mdpi.com/1099-4300/18/4/100 |
| work_keys_str_mv | AT badrsaadtalkahtani chaosonthevallismodelforelninowithfractionaloperators AT abdonatangana chaosonthevallismodelforelninowithfractionaloperators |