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Asymptotic integration of ( 1 + α ) -order fractional differential equations
We establish the long-time asymptotic formula of solutions to the ( 1 + α ) -order fractional differential equation 0 i O t 1 + α x + a ( t ) x = 0 , t > 0 , under some simple restrictions on the functional coefficient a ( t ) , where 0 i O t 1 + α is one of the fractional differential operators...
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| Published in: | Computers & mathematics with applications (1987) 2011-08, Vol.62 (3), p.1492-1500 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
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| cited_by | cdi_FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313 |
| container_end_page | 1500 |
| container_issue | 3 |
| container_start_page | 1492 |
| container_title | Computers & mathematics with applications (1987) |
| container_volume | 62 |
| creator | Băleanu, Dumitru Mustafa, Octavian G. Agarwal, Ravi P. |
| description | We establish the long-time asymptotic formula of solutions to the
(
1
+
α
)
-order fractional differential equation
0
i
O
t
1
+
α
x
+
a
(
t
)
x
=
0
,
t
>
0
, under some simple restrictions on the functional coefficient
a
(
t
)
, where
0
i
O
t
1
+
α
is one of the fractional differential operators
0
D
t
α
(
x
′
)
,
(
0
D
t
α
x
)
′
=
0
D
t
1
+
α
x
and
0
D
t
α
(
t
x
′
−
x
)
. Here,
0
D
t
α
designates the Riemann–Liouville derivative of order
α
∈
(
0
,
1
)
. The asymptotic formula reads as
[
b
+
O
(
1
)
]
⋅
x
s
m
a
l
l
+
c
⋅
x
l
a
r
g
e
as
t
→
+
∞
for given
b
,
c
∈
R
, where
x
s
m
a
l
l
and
x
l
a
r
g
e
represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation
0
i
O
t
1
+
α
x
=
0
,
t
>
0
. |
| doi_str_mv | 10.1016/j.camwa.2011.03.021 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_919908760</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0898122111001738</els_id><sourcerecordid>919908760</sourcerecordid><originalsourceid>FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313</originalsourceid><addsrcrecordid>eNp9kEtOwzAQhi0EEqVwAjbeAUIJ40diZ8ECVbykIjawtlxnglylSWu7oB6Li3Am0pY1q9Fo_m808xFyziBnwMqbee7s4svmHBjLQeTA2QEZMa1EpspSH5IR6EpnjHN2TE5inAOAFBxG5OUubhbL1CfvqO8SfgSbfN_RvqGXlNFr-vNNr2jWhxoDbYJ126ltae2bBgN2yQ8NrtY7Kp6So8a2Ec_-6pi8P9y_TZ6y6evj8-RumjkhipSJSpcKpS1wVs4cZ0oXRQNCotICpdS2sNKCVA2zQoJjyiLnw8WzgqPVgokxudjvXYZ-tcaYzMJHh21rO-zX0VSsqkCrEoak2Cdd6GMM2Jhl8AsbNoaB2bozc7NzZ7buDAgzuBuo2z2FwxOfHoOJzmPnsPYBXTJ17__lfwGdg3eY</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>919908760</pqid></control><display><type>article</type><title>Asymptotic integration of ( 1 + α ) -order fractional differential equations</title><source>Elsevier</source><creator>Băleanu, Dumitru ; Mustafa, Octavian G. ; Agarwal, Ravi P.</creator><creatorcontrib>Băleanu, Dumitru ; Mustafa, Octavian G. ; Agarwal, Ravi P.</creatorcontrib><description>We establish the long-time asymptotic formula of solutions to the
(
1
+
α
)
-order fractional differential equation
0
i
O
t
1
+
α
x
+
a
(
t
)
x
=
0
,
t
>
0
, under some simple restrictions on the functional coefficient
a
(
t
)
, where
0
i
O
t
1
+
α
is one of the fractional differential operators
0
D
t
α
(
x
′
)
,
(
0
D
t
α
x
)
′
=
0
D
t
1
+
α
x
and
0
D
t
α
(
t
x
′
−
x
)
. Here,
0
D
t
α
designates the Riemann–Liouville derivative of order
α
∈
(
0
,
1
)
. The asymptotic formula reads as
[
b
+
O
(
1
)
]
⋅
x
s
m
a
l
l
+
c
⋅
x
l
a
r
g
e
as
t
→
+
∞
for given
b
,
c
∈
R
, where
x
s
m
a
l
l
and
x
l
a
r
g
e
represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation
0
i
O
t
1
+
α
x
=
0
,
t
>
0
.</description><identifier>ISSN: 0898-1221</identifier><identifier>EISSN: 1873-7668</identifier><identifier>DOI: 10.1016/j.camwa.2011.03.021</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Asymptotic integration ; Asymptotic properties ; Constrictions ; Derivatives ; Differential equations ; Linear fractional differential equation ; Mathematical analysis ; Mathematical models ; Operators ; Solution space</subject><ispartof>Computers & mathematics with applications (1987), 2011-08, Vol.62 (3), p.1492-1500</ispartof><rights>2011 Elsevier Ltd</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313</citedby><cites>FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Băleanu, Dumitru</creatorcontrib><creatorcontrib>Mustafa, Octavian G.</creatorcontrib><creatorcontrib>Agarwal, Ravi P.</creatorcontrib><title>Asymptotic integration of ( 1 + α ) -order fractional differential equations</title><title>Computers & mathematics with applications (1987)</title><description>We establish the long-time asymptotic formula of solutions to the
(
1
+
α
)
-order fractional differential equation
0
i
O
t
1
+
α
x
+
a
(
t
)
x
=
0
,
t
>
0
, under some simple restrictions on the functional coefficient
a
(
t
)
, where
0
i
O
t
1
+
α
is one of the fractional differential operators
0
D
t
α
(
x
′
)
,
(
0
D
t
α
x
)
′
=
0
D
t
1
+
α
x
and
0
D
t
α
(
t
x
′
−
x
)
. Here,
0
D
t
α
designates the Riemann–Liouville derivative of order
α
∈
(
0
,
1
)
. The asymptotic formula reads as
[
b
+
O
(
1
)
]
⋅
x
s
m
a
l
l
+
c
⋅
x
l
a
r
g
e
as
t
→
+
∞
for given
b
,
c
∈
R
, where
x
s
m
a
l
l
and
x
l
a
r
g
e
represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation
0
i
O
t
1
+
α
x
=
0
,
t
>
0
.</description><subject>Asymptotic integration</subject><subject>Asymptotic properties</subject><subject>Constrictions</subject><subject>Derivatives</subject><subject>Differential equations</subject><subject>Linear fractional differential equation</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Operators</subject><subject>Solution space</subject><issn>0898-1221</issn><issn>1873-7668</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNp9kEtOwzAQhi0EEqVwAjbeAUIJ40diZ8ECVbykIjawtlxnglylSWu7oB6Li3Am0pY1q9Fo_m808xFyziBnwMqbee7s4svmHBjLQeTA2QEZMa1EpspSH5IR6EpnjHN2TE5inAOAFBxG5OUubhbL1CfvqO8SfgSbfN_RvqGXlNFr-vNNr2jWhxoDbYJ126ltae2bBgN2yQ8NrtY7Kp6So8a2Ec_-6pi8P9y_TZ6y6evj8-RumjkhipSJSpcKpS1wVs4cZ0oXRQNCotICpdS2sNKCVA2zQoJjyiLnw8WzgqPVgokxudjvXYZ-tcaYzMJHh21rO-zX0VSsqkCrEoak2Cdd6GMM2Jhl8AsbNoaB2bozc7NzZ7buDAgzuBuo2z2FwxOfHoOJzmPnsPYBXTJ17__lfwGdg3eY</recordid><startdate>20110801</startdate><enddate>20110801</enddate><creator>Băleanu, Dumitru</creator><creator>Mustafa, Octavian G.</creator><creator>Agarwal, Ravi P.</creator><general>Elsevier Ltd</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20110801</creationdate><title>Asymptotic integration of ( 1 + α ) -order fractional differential equations</title><author>Băleanu, Dumitru ; Mustafa, Octavian G. ; Agarwal, Ravi P.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c335t-39867e4a5eb6bc217855f034e783e448a5a4a047f1a340c17ae22004b52ea8313</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Asymptotic integration</topic><topic>Asymptotic properties</topic><topic>Constrictions</topic><topic>Derivatives</topic><topic>Differential equations</topic><topic>Linear fractional differential equation</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Operators</topic><topic>Solution space</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Băleanu, Dumitru</creatorcontrib><creatorcontrib>Mustafa, Octavian G.</creatorcontrib><creatorcontrib>Agarwal, Ravi P.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Computers & mathematics with applications (1987)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Băleanu, Dumitru</au><au>Mustafa, Octavian G.</au><au>Agarwal, Ravi P.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Asymptotic integration of ( 1 + α ) -order fractional differential equations</atitle><jtitle>Computers & mathematics with applications (1987)</jtitle><date>2011-08-01</date><risdate>2011</risdate><volume>62</volume><issue>3</issue><spage>1492</spage><epage>1500</epage><pages>1492-1500</pages><issn>0898-1221</issn><eissn>1873-7668</eissn><abstract>We establish the long-time asymptotic formula of solutions to the
(
1
+
α
)
-order fractional differential equation
0
i
O
t
1
+
α
x
+
a
(
t
)
x
=
0
,
t
>
0
, under some simple restrictions on the functional coefficient
a
(
t
)
, where
0
i
O
t
1
+
α
is one of the fractional differential operators
0
D
t
α
(
x
′
)
,
(
0
D
t
α
x
)
′
=
0
D
t
1
+
α
x
and
0
D
t
α
(
t
x
′
−
x
)
. Here,
0
D
t
α
designates the Riemann–Liouville derivative of order
α
∈
(
0
,
1
)
. The asymptotic formula reads as
[
b
+
O
(
1
)
]
⋅
x
s
m
a
l
l
+
c
⋅
x
l
a
r
g
e
as
t
→
+
∞
for given
b
,
c
∈
R
, where
x
s
m
a
l
l
and
x
l
a
r
g
e
represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation
0
i
O
t
1
+
α
x
=
0
,
t
>
0
.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.camwa.2011.03.021</doi><tpages>9</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0898-1221 |
| ispartof | Computers & mathematics with applications (1987), 2011-08, Vol.62 (3), p.1492-1500 |
| issn | 0898-1221 1873-7668 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_919908760 |
| source | Elsevier |
| subjects | Asymptotic integration Asymptotic properties Constrictions Derivatives Differential equations Linear fractional differential equation Mathematical analysis Mathematical models Operators Solution space |
| title | Asymptotic integration of ( 1 + α ) -order fractional differential equations |
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