Approximation of the Riesz–Caputo Derivative by Cubic Splines
Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/15/2/69 |
| _version_ | 1797483531305222144 |
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| author | Francesca Pitolli Chiara Sorgentone Enza Pellegrino |
| author_facet | Francesca Pitolli Chiara Sorgentone Enza Pellegrino |
| author_sort | Francesca Pitolli |
| collection | DOAJ |
| description | Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate. |
| first_indexed | 2024-03-09T22:48:23Z |
| format | Article |
| id | doaj.art-5da682bb751a4072af446a44529c31e4 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-09T22:48:23Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-5da682bb751a4072af446a44529c31e42023-11-23T18:24:32ZengMDPI AGAlgorithms1999-48932022-02-011526910.3390/a15020069Approximation of the Riesz–Caputo Derivative by Cubic SplinesFrancesca Pitolli0Chiara Sorgentone1Enza Pellegrino2Department SBAI, Università di Roma “La Sapienza”, Via Antonio Scarpa 16, 00161 Rome, ItalyDepartment SBAI, Università di Roma “La Sapienza”, Via Antonio Scarpa 16, 00161 Rome, ItalyDepartment DIIIE, University of L’Aquila, E. Pontieri 2, Roio Poggio, 67040 L’Aquila, ItalyDifferential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate.https://www.mdpi.com/1999-4893/15/2/69fractional differential equationRiesz derivativecubic splinecollocation method |
| spellingShingle | Francesca Pitolli Chiara Sorgentone Enza Pellegrino Approximation of the Riesz–Caputo Derivative by Cubic Splines Algorithms fractional differential equation Riesz derivative cubic spline collocation method |
| title | Approximation of the Riesz–Caputo Derivative by Cubic Splines |
| title_full | Approximation of the Riesz–Caputo Derivative by Cubic Splines |
| title_fullStr | Approximation of the Riesz–Caputo Derivative by Cubic Splines |
| title_full_unstemmed | Approximation of the Riesz–Caputo Derivative by Cubic Splines |
| title_short | Approximation of the Riesz–Caputo Derivative by Cubic Splines |
| title_sort | approximation of the riesz caputo derivative by cubic splines |
| topic | fractional differential equation Riesz derivative cubic spline collocation method |
| url | https://www.mdpi.com/1999-4893/15/2/69 |
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