An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

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 th...

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Veröffentlicht in:	Advances in difference equations Jg. 2021; H. 1; S. 1 - 18
Hauptverfasser:	Al-Smadi, Mohammed, Djeddi, Nadir, Momani, Shaher, Al-Omari, Shrideh, Araci, Serkan
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 26.05.2021 Springer Nature B.V SpringerOpen
Schlagworte:	Abel-type differential equation Algorithms Analysis Caputo–Fabrizio fractional derivative Convergence Difference and Functional Equations Differential equations Error analysis Exact solutions Functional Analysis Hilbert space Kernels Mathematical models Mathematics Mathematics and Statistics Numerical analysis Numerical solution Ordinary Differential Equations Partial Differential Equations Reproducing kernel algorithm Abel-type differential equation Caputo–Fabrizio fractional derivative Error analysis Reproducing kernel algorithm Numerical solution
ISSN:	1687-1847, 1687-1839, 1687-1847
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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 ] . 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.
AbstractList	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. 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 ] . 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. 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 $\mathcal{H}^{2}[a,b]$ 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. 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 H2[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.
ArticleNumber	271
Author	Araci, Serkan Momani, Shaher Al-Omari, Shrideh Al-Smadi, Mohammed Djeddi, Nadir
Author_xml	– sequence: 1 givenname: Mohammed surname: Al-Smadi fullname: Al-Smadi, Mohammed organization: Department of Applied Science, Ajloun College, Al-Balqa Applied University, Nonlinear Dynamics Research Center (NDRC), Ajman University – sequence: 2 givenname: Nadir surname: Djeddi fullname: Djeddi, Nadir organization: Department of Mathematics, Faculty of Science, The University of Jordan – sequence: 3 givenname: Shaher surname: Momani fullname: Momani, Shaher organization: Nonlinear Dynamics Research Center (NDRC), Ajman University, Department of Mathematics, Faculty of Science, The University of Jordan – sequence: 4 givenname: Shrideh orcidid: 0000-0001-8955-5552 surname: Al-Omari fullname: Al-Omari, Shrideh email: shridehalomari@bau.edu.jo, s.k.q.alomari@fet.edu.jo organization: Department of Mathematics, Faculty of Science, Al Balqa Applied University – sequence: 5 givenname: Serkan surname: Araci fullname: Araci, Serkan organization: Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University
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Snippet	Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based... Abstract Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential...
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SubjectTerms	Abel-type differential equation Algorithms Analysis Caputo–Fabrizio fractional derivative Convergence Difference and Functional Equations Differential equations Error analysis Exact solutions Functional Analysis Hilbert space Kernels Mathematical models Mathematics Mathematics and Statistics Numerical analysis Numerical solution Ordinary Differential Equations Partial Differential Equations Reproducing kernel algorithm
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Title	An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space
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