Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications
Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0<α≤1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extensio...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/4149320 |
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| author | Thabet Abdeljawad Qasem M. Al-Mdallal Mohamed A. Hajji |
| author_facet | Thabet Abdeljawad Qasem M. Al-Mdallal Mohamed A. Hajji |
| author_sort | Thabet Abdeljawad |
| collection | DOAJ |
| description | Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0<α≤1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order 2<α≤3 is proved and the ordinary difference Lyapunov inequality then follows as α tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given. |
| format | Article |
| id | doaj-art-a2575259df5c4f50ab59286693038dd7 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a2575259df5c4f50ab59286693038dd72025-02-03T01:32:13ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/41493204149320Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and ApplicationsThabet Abdeljawad0Qasem M. Al-Mdallal1Mohamed A. Hajji2Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaDepartment of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al Ain, Abu Dhabi, UAEDepartment of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al Ain, Abu Dhabi, UAERecently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0<α≤1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order 2<α≤3 is proved and the ordinary difference Lyapunov inequality then follows as α tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.http://dx.doi.org/10.1155/2017/4149320 |
| spellingShingle | Thabet Abdeljawad Qasem M. Al-Mdallal Mohamed A. Hajji Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications Discrete Dynamics in Nature and Society |
| title | Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications |
| title_full | Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications |
| title_fullStr | Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications |
| title_full_unstemmed | Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications |
| title_short | Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications |
| title_sort | arbitrary order fractional difference operators with discrete exponential kernels and applications |
| url | http://dx.doi.org/10.1155/2017/4149320 |
| work_keys_str_mv | AT thabetabdeljawad arbitraryorderfractionaldifferenceoperatorswithdiscreteexponentialkernelsandapplications AT qasemmalmdallal arbitraryorderfractionaldifferenceoperatorswithdiscreteexponentialkernelsandapplications AT mohamedahajji arbitraryorderfractionaldifferenceoperatorswithdiscreteexponentialkernelsandapplications |