Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations
Fractional calculusisabranchofmathematicalanalysisinvestigatingthederivatives and integralsofarbitraryorder.Fractionalcalculushasawideapplicationsincemany realistic phenomenaaredefinedinfractionalorderderivativeandintegral.Moreover, fractional differentialequationsprovideanexcellentframeworkfordi...
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| Format: | Thesis |
| Language: | English English English |
| Published: |
2022
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| Online Access: | http://eprints.uthm.edu.my/8455/1/24p%20NG%20YONG%20XIAN.pdf http://eprints.uthm.edu.my/8455/2/NG%20YONG%20XIAN%20COPYRIGHT%20DECLARATION.pdf http://eprints.uthm.edu.my/8455/3/NG%20YONG%20XIAN%20WATERMARK.pdf |
| _version_ | 1825710576496541696 |
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| author | Ng, Yong Xian |
| author_facet | Ng, Yong Xian |
| author_sort | Ng, Yong Xian |
| collection | UTHM |
| description | Fractional calculusisabranchofmathematicalanalysisinvestigatingthederivatives
and integralsofarbitraryorder.Fractionalcalculushasawideapplicationsincemany
realistic phenomenaaredefinedinfractionalorderderivativeandintegral.Moreover,
fractional differentialequationsprovideanexcellentframeworkfordiscussingthe
possibility ofunlimitedmemoryandhereditaryproperties,consideringmoredegrees
of freedom.Inthisthesis,thestabilitycriteriaofthefractionalShimizu-Morioka
system andfractionaloceancirculationmodelinthesenseofCaputoderivative
are developedanalyticallyusingoptimalRouth-Hurwitzconditions.Hence,Routh-
Hurwitz conditionsforcubicandquadraticpolynomialsarepresented.Theadvantage
of Routh-Hurwitzconditionsisthattheyallowonetoobtainstabilityconditions
without solvingthefractionaldifferentialequations.Inthiscase,wefindthecritical
range foradjustablecontrolparameterandfractionalorder �, whichconcludesthat
the equilibriaofsystemsarelocallyasymptoticallystable.Aftermath,thenumerical
results arepresentedtosupportourtheoreticalconclusionsusingtheAdams-type
predictor-correctormethod.Ontheotherhand,wederivetheanalyticalsolutionfor
the inhomogeneoussystemofdifferentialequationswithincommensuratefractional
order 1 < �;�< 2, wherethefractionalorders � and � are uniqueandindependent
of eachother.ThesystemsarefirstwritteninVolterraintegralequationsofthesecond
kind. Further,Picard’ssuccessiveapproximationmethodisperformed,whichisan
explicitanalyticalmethodthatconvergesveryclosetoexactsolutions,andthesolution
is derivedinmultipleseriesandsomespecialfunctionexpressions,suchasGamma
function, Mittag-Lefflerfunctionsandhypergeometricfunctions.Somespecialcases
are discussedwhereallthesolutionsareverifiedusingsubstitution. |
| first_indexed | 2024-03-05T21:59:42Z |
| format | Thesis |
| id | uthm.eprints-8455 |
| institution | Universiti Tun Hussein Onn Malaysia |
| language | English English English |
| last_indexed | 2024-03-05T21:59:42Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | uthm.eprints-84552023-02-27T02:33:56Z http://eprints.uthm.edu.my/8455/ Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations Ng, Yong Xian QA273-280 Probabilities. Mathematical statistics Fractional calculusisabranchofmathematicalanalysisinvestigatingthederivatives and integralsofarbitraryorder.Fractionalcalculushasawideapplicationsincemany realistic phenomenaaredefinedinfractionalorderderivativeandintegral.Moreover, fractional differentialequationsprovideanexcellentframeworkfordiscussingthe possibility ofunlimitedmemoryandhereditaryproperties,consideringmoredegrees of freedom.Inthisthesis,thestabilitycriteriaofthefractionalShimizu-Morioka system andfractionaloceancirculationmodelinthesenseofCaputoderivative are developedanalyticallyusingoptimalRouth-Hurwitzconditions.Hence,Routh- Hurwitz conditionsforcubicandquadraticpolynomialsarepresented.Theadvantage of Routh-Hurwitzconditionsisthattheyallowonetoobtainstabilityconditions without solvingthefractionaldifferentialequations.Inthiscase,wefindthecritical range foradjustablecontrolparameterandfractionalorder �, whichconcludesthat the equilibriaofsystemsarelocallyasymptoticallystable.Aftermath,thenumerical results arepresentedtosupportourtheoreticalconclusionsusingtheAdams-type predictor-correctormethod.Ontheotherhand,wederivetheanalyticalsolutionfor the inhomogeneoussystemofdifferentialequationswithincommensuratefractional order 1 < �;�< 2, wherethefractionalorders � and � are uniqueandindependent of eachother.ThesystemsarefirstwritteninVolterraintegralequationsofthesecond kind. Further,Picard’ssuccessiveapproximationmethodisperformed,whichisan explicitanalyticalmethodthatconvergesveryclosetoexactsolutions,andthesolution is derivedinmultipleseriesandsomespecialfunctionexpressions,suchasGamma function, Mittag-Lefflerfunctionsandhypergeometricfunctions.Somespecialcases are discussedwhereallthesolutionsareverifiedusingsubstitution. 2022-07 Thesis NonPeerReviewed text en http://eprints.uthm.edu.my/8455/1/24p%20NG%20YONG%20XIAN.pdf text en http://eprints.uthm.edu.my/8455/2/NG%20YONG%20XIAN%20COPYRIGHT%20DECLARATION.pdf text en http://eprints.uthm.edu.my/8455/3/NG%20YONG%20XIAN%20WATERMARK.pdf Ng, Yong Xian (2022) Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations. Doctoral thesis, Universiti Tun Hussein Onn Malaysia. |
| spellingShingle | QA273-280 Probabilities. Mathematical statistics Ng, Yong Xian Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title | Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title_full | Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title_fullStr | Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title_full_unstemmed | Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title_short | Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations |
| title_sort | optimal routh hurwitz conditions and picard s successive approximation method for system of fractional differential equations |
| topic | QA273-280 Probabilities. Mathematical statistics |
| url | http://eprints.uthm.edu.my/8455/1/24p%20NG%20YONG%20XIAN.pdf http://eprints.uthm.edu.my/8455/2/NG%20YONG%20XIAN%20COPYRIGHT%20DECLARATION.pdf http://eprints.uthm.edu.my/8455/3/NG%20YONG%20XIAN%20WATERMARK.pdf |
| work_keys_str_mv | AT ngyongxian optimalrouthhurwitzconditionsandpicardssuccessiveapproximationmethodforsystemoffractionaldifferentialequations |