An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space
Abstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we u...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-05-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13662-021-03428-3 |
| _version_ | 1818647945848815616 |
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| author | Mohammed Al-Smadi Nadir Djeddi Shaher Momani Shrideh Al-Omari Serkan Araci |
| author_facet | Mohammed Al-Smadi Nadir Djeddi Shaher Momani Shrideh Al-Omari Serkan Araci |
| author_sort | Mohammed Al-Smadi |
| collection | DOAJ |
| description | Abstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space H 2 [ a , b ] $\mathcal{H}^{2}[a,b]$ . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences. |
| first_indexed | 2024-12-17T01:10:36Z |
| format | Article |
| id | doaj.art-469b0687337c4a2ab9fd3b9a60eff935 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-17T01:10:36Z |
| publishDate | 2021-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-469b0687337c4a2ab9fd3b9a60eff9352022-12-21T22:09:09ZengSpringerOpenAdvances in Difference Equations1687-18472021-05-012021111810.1186/s13662-021-03428-3An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert spaceMohammed Al-Smadi0Nadir Djeddi1Shaher Momani2Shrideh Al-Omari3Serkan Araci4Department of Applied Science, Ajloun College, Al-Balqa Applied UniversityDepartment of Mathematics, Faculty of Science, The University of JordanNonlinear Dynamics Research Center (NDRC), Ajman UniversityDepartment of Mathematics, Faculty of Science, Al Balqa Applied UniversityDepartment of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu UniversityAbstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space H 2 [ a , b ] $\mathcal{H}^{2}[a,b]$ . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.https://doi.org/10.1186/s13662-021-03428-3Caputo–Fabrizio fractional derivativeAbel-type differential equationReproducing kernel algorithmNumerical solutionError analysis |
| spellingShingle | Mohammed Al-Smadi Nadir Djeddi Shaher Momani Shrideh Al-Omari Serkan Araci An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space Advances in Difference Equations Caputo–Fabrizio fractional derivative Abel-type differential equation Reproducing kernel algorithm Numerical solution Error analysis |
| title | An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space |
| title_full | An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space |
| title_fullStr | An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space |
| title_full_unstemmed | An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space |
| title_short | An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space |
| title_sort | attractive numerical algorithm for solving nonlinear caputo fabrizio fractional abel differential equation in a hilbert space |
| topic | Caputo–Fabrizio fractional derivative Abel-type differential equation Reproducing kernel algorithm Numerical solution Error analysis |
| url | https://doi.org/10.1186/s13662-021-03428-3 |
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