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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| Subjects: | |
| 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 http://eprints.uthm.edu.my/8455/ |
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| Summary: | 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. |
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