Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems
We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundame...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2017/3720471 |
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| author | Mohammed Al-Refai Thabet Abdeljawad |
| author_facet | Mohammed Al-Refai Thabet Abdeljawad |
| author_sort | Mohammed Al-Refai |
| collection | DOAJ |
| description | We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems. |
| format | Article |
| id | doaj-art-20a06dde7cba4dce8b1af268a511b13c |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-20a06dde7cba4dce8b1af268a511b13c2025-02-03T01:01:36ZengWileyComplexity1076-27871099-05262017-01-01201710.1155/2017/37204713720471Fundamental Results of Conformable Sturm-Liouville Eigenvalue ProblemsMohammed Al-Refai0Thabet Abdeljawad1Department of Mathematical Sciences, UAE University, P.O. Box 15551, Al Ain, Abu Dhabi, UAEDepartment of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaWe suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.http://dx.doi.org/10.1155/2017/3720471 |
| spellingShingle | Mohammed Al-Refai Thabet Abdeljawad Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems Complexity |
| title | Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems |
| title_full | Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems |
| title_fullStr | Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems |
| title_full_unstemmed | Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems |
| title_short | Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems |
| title_sort | fundamental results of conformable sturm liouville eigenvalue problems |
| url | http://dx.doi.org/10.1155/2017/3720471 |
| work_keys_str_mv | AT mohammedalrefai fundamentalresultsofconformablesturmliouvilleeigenvalueproblems AT thabetabdeljawad fundamentalresultsofconformablesturmliouvilleeigenvalueproblems |