Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations
This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions spa...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/565067 |
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| _version_ | 1832565537148764160 |
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| author | Jan Čermák Tomáš Kisela Luděk Nechvátal |
| author_facet | Jan Čermák Tomáš Kisela Luděk Nechvátal |
| author_sort | Jan Čermák |
| collection | DOAJ |
| description | This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and q-difference calculus. Some of our results are new also in these particular discrete settings. |
| format | Article |
| id | doaj-art-d5367d39b51b4207b91f945d1311b274 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-d5367d39b51b4207b91f945d1311b2742025-02-03T01:07:29ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/565067565067Discrete Mittag-Leffler Functions in Linear Fractional Difference EquationsJan Čermák0Tomáš Kisela1Luděk Nechvátal2Institute of Mathematics, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech RepublicInstitute of Mathematics, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech RepublicInstitute of Mathematics, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech RepublicThis paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and q-difference calculus. Some of our results are new also in these particular discrete settings.http://dx.doi.org/10.1155/2011/565067 |
| spellingShingle | Jan Čermák Tomáš Kisela Luděk Nechvátal Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations Abstract and Applied Analysis |
| title | Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations |
| title_full | Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations |
| title_fullStr | Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations |
| title_full_unstemmed | Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations |
| title_short | Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations |
| title_sort | discrete mittag leffler functions in linear fractional difference equations |
| url | http://dx.doi.org/10.1155/2011/565067 |
| work_keys_str_mv | AT jancermak discretemittaglefflerfunctionsinlinearfractionaldifferenceequations AT tomaskisela discretemittaglefflerfunctionsinlinearfractionaldifferenceequations AT ludeknechvatal discretemittaglefflerfunctionsinlinearfractionaldifferenceequations |