On the Definitions of Nabla Fractional Operators
We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commuta...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/406757 |
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| _version_ | 1832564902887161856 |
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| author | Thabet Abdeljawad Ferhan M. Atici |
| author_facet | Thabet Abdeljawad Ferhan M. Atici |
| author_sort | Thabet Abdeljawad |
| collection | DOAJ |
| description | We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic
properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced. |
| format | Article |
| id | doaj-art-9d435698fa644fdcb36530e969adb778 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-9d435698fa644fdcb36530e969adb7782025-02-03T01:09:51ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/406757406757On the Definitions of Nabla Fractional OperatorsThabet Abdeljawad0Ferhan M. Atici1Department of Mathematics, Çankaya University, 06530 Ankara, TurkeyDepartment of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USAWe show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.http://dx.doi.org/10.1155/2012/406757 |
| spellingShingle | Thabet Abdeljawad Ferhan M. Atici On the Definitions of Nabla Fractional Operators Abstract and Applied Analysis |
| title | On the Definitions of Nabla Fractional Operators |
| title_full | On the Definitions of Nabla Fractional Operators |
| title_fullStr | On the Definitions of Nabla Fractional Operators |
| title_full_unstemmed | On the Definitions of Nabla Fractional Operators |
| title_short | On the Definitions of Nabla Fractional Operators |
| title_sort | on the definitions of nabla fractional operators |
| url | http://dx.doi.org/10.1155/2012/406757 |
| work_keys_str_mv | AT thabetabdeljawad onthedefinitionsofnablafractionaloperators AT ferhanmatici onthedefinitionsofnablafractionaloperators |