MATHIEU EQUATION AND ITS APPLICATION

1.1BriefReviewonMathieuequation
Mathieu equation isa specialcase of a linear second order homogeneous
differentialequation(Ruby1995).Theequationwasfirstdiscussedin1868,byEmile
LeonardMathieuinconnectionwithproblemofvibrationsinellipticalmembrane.He
developedtheleadingtermsoftheseriessolutionknownasMathieufunctionofthe
ellipticalmembranes.Adecadelater,HeinedefinedtheperiodicMathieuAngular
FunctionsofintegerorderasFouriercosineandsineseries;furthermore,without
evaluatingthecorrespondingcoefficient,Heobtainedatranscendentalequationfor
characteristicnumbersexpressedintermsofinfinitecontinuedfractions;andalso
showedthatonesetofperiodicfunctionsofintegerordercouldbeinaseriesof
Besselfunction(Chaos-CadorandLey-Koo2002).
Intheearly1880’s,Floquetwentfurthertopublishatheoryandthusasolution
totheMathieudifferentialequation;hisworkwasnamedafterhimas,‘Floquet’s
Theorem’or‘Floquet’sSolution’.StephensonusedanapproximateMathieuequation,
andproved,thatitispossibletostabilizetheupperpositionofarigidpendulumby
vibratingitspivotpointverticallyataspecifichighfrequency.(StépánandInsperger
2003).Thereexistsanextensiveliteratureontheseequations;andinparticular,a
well-highexhaustivecompendiumwasgivenbyMc-Lachlan(1947).
TheMathieufunctionwasfurtherinvestigatedbynumberofresearcherswho
foundaconsiderableamountofmathematicalresultsthatwerecollectedmorethan
60yearsagobyMc-Lachlan(Gutiérrez-Vegaaetal2002).Whittakerandother
scientistderivedin1900sderivedthehigher-ordertermsoftheMathieudifferential
equation.AvarietyoftheequationexistintextbookwrittenbyAbramowitzand
Stegun(1964).
Mathieudifferentialequationoccursintwomaincategoriesofphysicalproblems.
First,applicationsinvolvingellipticalgeometriessuchas,analysisofvibratingmodes
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inellipticmembrane,thepropagatingmodesofellipticpipesandtheoscillationsof
waterinalakeofellipticshape.Mathieuequationarisesafterseparatingthewave
equation using ellipticcoordinates.Secondly,problemsinvolving periodicmotion
examplesare,thetrajectoryofan electron in aperiodicarrayofatoms,the
mechanicsofthequantumpendulumandtheoscillationoffloatingvessels.
ThecanonicalformfortheMathieudifferentialequationisgivenby
+ y =0, (1.1)
dy 2
dx2 [a-2qcos(2x)](x)
whereaandqarerealconstantsknownasthecharacteristicvalueandparameter
respectively.
Closely related to the Mathieu differentialequation is the Modified Mathieu
differentialequationgivenby:
- y =0, (1.2)
dy 2
du2 [a-2qcosh(2u)](u)
whereu=ixissubstitutedintoequation(1.1).
Thesubstitutionoft=cos(x)inthecanonicalMathieudifferentialequation(1.1)
abovetransformstheequationintoitsalgebraicformasgivenbelow:
(1-t) -t + y =0. (1.3) 2 dy 2
dt2
dy
dt
[a+2q(1-2t2)](t)
Thishastwosingularitiesatt=1,-1andoneirregularsingularityatinfinity,which
impliesthatingeneral(un-likemanyotherspecialfunctions),thesolutionofMathieu
differentialequationcannotbeexpressedintermsofhypergeometricfunctions
(Mritunjay2011).
Thepurposeofthestudyistofacilitatetheunderstandingofsomeofthe
propertiesofMathieufunctionsandtheirapplications.Webelievethatthisstudywill
behelpfulinachievingabettercomprehensionoftheirbasiccharacteristics.This
studyisalsointendedtoenlightenstudentsandresearcherswhoareunfamiliarwith
Mathieufunctions.Inthechaptertwoofthiswork,wediscussedtheMathieu
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differentialequationandhowitarisesfromtheellipticalcoordinatesystem.Also,we
talkedabouttheModifiedMathieudifferentialequationandtheMathieudifferential
equationinanalgebraicform.Thechapterthreewasbasedonthesolutionstothe
MathieuequationknownasMathieufunctionsandalsotheFloquet’stheory.Inthe
chapterfour,weshowedhowMathieufunctionscanbeappliedtodescribethe
invertedpendulum,ellipticdrumhead,Radiofrequencyquadrupole,Frequency
modulation,Stabilityofafloatingbody,AlternatingGradientFocusing,thePaultrap
forchargedparticlesandtheQuantumPendulum.