Asymptotic Expansions of Fractional Derivatives andTheir Applications
We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-04-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/3/2/171 |
| _version_ | 1819319583976194048 |
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| author | Tohru Morita Ken-ichi Sato |
| author_facet | Tohru Morita Ken-ichi Sato |
| author_sort | Tohru Morita |
| collection | DOAJ |
| description | We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD. |
| first_indexed | 2024-12-24T11:06:00Z |
| format | Article |
| id | doaj.art-d3cec642f7e0414f9d905fce3f31c6e4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-24T11:06:00Z |
| publishDate | 2015-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d3cec642f7e0414f9d905fce3f31c6e42022-12-21T16:58:37ZengMDPI AGMathematics2227-73902015-04-013217118910.3390/math3020171math3020171Asymptotic Expansions of Fractional Derivatives andTheir ApplicationsTohru Morita0Ken-ichi Sato1Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, JapanCollege of Engineering, Nihon University, Koriyama 963-8642, JapanWe compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD.http://www.mdpi.com/2227-7390/3/2/171fractional derivativeasymptotic expansionKummer’s differential equationconfluent hypergeometric function |
| spellingShingle | Tohru Morita Ken-ichi Sato Asymptotic Expansions of Fractional Derivatives andTheir Applications Mathematics fractional derivative asymptotic expansion Kummer’s differential equation confluent hypergeometric function |
| title | Asymptotic Expansions of Fractional Derivatives andTheir Applications |
| title_full | Asymptotic Expansions of Fractional Derivatives andTheir Applications |
| title_fullStr | Asymptotic Expansions of Fractional Derivatives andTheir Applications |
| title_full_unstemmed | Asymptotic Expansions of Fractional Derivatives andTheir Applications |
| title_short | Asymptotic Expansions of Fractional Derivatives andTheir Applications |
| title_sort | asymptotic expansions of fractional derivatives andtheir applications |
| topic | fractional derivative asymptotic expansion Kummer’s differential equation confluent hypergeometric function |
| url | http://www.mdpi.com/2227-7390/3/2/171 |
| work_keys_str_mv | AT tohrumorita asymptoticexpansionsoffractionalderivativesandtheirapplications AT kenichisato asymptoticexpansionsoffractionalderivativesandtheirapplications |