The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order
We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/4/302 |
| _version_ | 1797605401309478912 |
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| author | Lakhlifa Sadek Ahmad Sami Bataineh Hamad Talibi Alaoui Ishak Hashim |
| author_facet | Lakhlifa Sadek Ahmad Sami Bataineh Hamad Talibi Alaoui Ishak Hashim |
| author_sort | Lakhlifa Sadek |
| collection | DOAJ |
| description | We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods. |
| first_indexed | 2024-03-11T05:00:35Z |
| format | Article |
| id | doaj.art-e145fb68e00649acbbc5a70c9b14a136 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T05:00:35Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-e145fb68e00649acbbc5a70c9b14a1362023-11-17T19:19:16ZengMDPI AGFractal and Fractional2504-31102023-03-017430210.3390/fractalfract7040302The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional OrderLakhlifa Sadek0Ahmad Sami Bataineh1Hamad Talibi Alaoui2Ishak Hashim3Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, El Jadida 24000, MoroccoDepartment of Mathematics, Faculty of Science, Al-Balqa Applied University, Al-Salt 19117, JordanDepartment of Mathematics, Faculty of Sciences, Chouaib Doukkali University, El Jadida 24000, MoroccoDepartment of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), Bangi 43650, Selangor, MalaysiaWe present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods.https://www.mdpi.com/2504-3110/7/4/302finite Mittag-Leffler functionfractional differential Riccati equations (FDRE)Caputo fractional derivativeserror analysiserror estimation |
| spellingShingle | Lakhlifa Sadek Ahmad Sami Bataineh Hamad Talibi Alaoui Ishak Hashim The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order Fractal and Fractional finite Mittag-Leffler function fractional differential Riccati equations (FDRE) Caputo fractional derivatives error analysis error estimation |
| title | The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order |
| title_full | The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order |
| title_fullStr | The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order |
| title_full_unstemmed | The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order |
| title_short | The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order |
| title_sort | novel mittag leffler galerkin method application to a riccati differential equation of fractional order |
| topic | finite Mittag-Leffler function fractional differential Riccati equations (FDRE) Caputo fractional derivatives error analysis error estimation |
| url | https://www.mdpi.com/2504-3110/7/4/302 |
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