Two computational approaches for solving a fractional obstacle system in Hilbert space
Abstract The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<α≤1 $0<\alpha\leq1$, concerned wit...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-02-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-019-1996-5 |
| _version_ | 1818286377735815168 |
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| author | Shatha Hasan Mohammed Al-Smadi Asad Freihet Shaher Momani |
| author_facet | Shatha Hasan Mohammed Al-Smadi Asad Freihet Shaher Momani |
| author_sort | Shatha Hasan |
| collection | DOAJ |
| description | Abstract The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<α≤1 $0<\alpha\leq1$, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions. |
| first_indexed | 2024-12-13T01:23:38Z |
| format | Article |
| id | doaj.art-92262986c25c412f930743c24cc9d6d2 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-13T01:23:38Z |
| publishDate | 2019-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-92262986c25c412f930743c24cc9d6d22022-12-22T00:04:11ZengSpringerOpenAdvances in Difference Equations1687-18472019-02-012019111710.1186/s13662-019-1996-5Two computational approaches for solving a fractional obstacle system in Hilbert spaceShatha Hasan0Mohammed Al-Smadi1Asad Freihet2Shaher Momani3Department of Applied Science, Ajloun College, Al-Balqa Applied UniversityDepartment of Applied Science, Ajloun College, Al-Balqa Applied UniversityDepartment of Applied Science, Ajloun College, Al-Balqa Applied UniversityDepartment of Mathematics, Faculty of Science, The University of JordanAbstract The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<α≤1 $0<\alpha\leq1$, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.http://link.springer.com/article/10.1186/s13662-019-1996-5Reproducing-kernel methodResidual power series methodInner product spacesObstacle problemsCaputo-fractional derivative |
| spellingShingle | Shatha Hasan Mohammed Al-Smadi Asad Freihet Shaher Momani Two computational approaches for solving a fractional obstacle system in Hilbert space Advances in Difference Equations Reproducing-kernel method Residual power series method Inner product spaces Obstacle problems Caputo-fractional derivative |
| title | Two computational approaches for solving a fractional obstacle system in Hilbert space |
| title_full | Two computational approaches for solving a fractional obstacle system in Hilbert space |
| title_fullStr | Two computational approaches for solving a fractional obstacle system in Hilbert space |
| title_full_unstemmed | Two computational approaches for solving a fractional obstacle system in Hilbert space |
| title_short | Two computational approaches for solving a fractional obstacle system in Hilbert space |
| title_sort | two computational approaches for solving a fractional obstacle system in hilbert space |
| topic | Reproducing-kernel method Residual power series method Inner product spaces Obstacle problems Caputo-fractional derivative |
| url | http://link.springer.com/article/10.1186/s13662-019-1996-5 |
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