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MATHIEU EQUATION AND ITS APPLICATION

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MATHIEU EQUATION AND ITS APPLICATION

CHAPTERONE

INTRODUCTION

1.1BriefReviewonMathieuequation

Mathieuequationisaspecialcaseofalinearsecondorderhomogeneous

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

1
inellipticmembrane,thepropagatingmodesofellipticpipesandtheoscillationsof

waterinalakeofellipticshape.Mathieuequationarisesafterseparatingthewave

equationusingellipticcoordinates.Secondly,problemsinvolvingperiodicmotion

examplesare,thetrajectoryofanelectroninaperiodicarrayofatoms,the

mechanicsofthequantumpendulumandtheoscillationoffloatingvessels.

ThecanonicalformfortheMathieudifferentialequationisgivenby

2
y
d
x
a-2qcos
2x
,(1.1)+y=0
((
))
[
]
2
dx

whereandarerealconstantsknownasthecharacteristicvalueandparameteraq

respectively.

CloselyrelatedtotheMathieudifferentialequationistheModifiedMathieu

differentialequationgivenby:

2
y
d
u
a-2qcosh
2u
(1.2)-y=0,
((
))
[
]
2
du

whereu=ixissubstitutedintoequation(1.1).

Thesubstitutionoft=cos(x)inthecanonicalMathieudifferentialequation(1.1)

abovetransformstheequationintoitsalgebraicformasgivenbelow:

2
y
ddy
2
2
a+2q
t
(1-t
(1.3))-t+y=0.
(
)
[
]
(
)
t
1-2
2
dt
dt

Thishastwosingularitiesatt=1,-1andoneirregularsingularityatinfinity,which

impliesthatingeneral(un-likemanyotherspecialfunctions),thesolutionofMathieu

differentialequationcannotbeexpressedintermsofhypergeometricfunctions

(Mritunjay2011).

Thepurposeofthestudyistofacilitatetheunderstandingofsomeofthe

propertiesofMathieufunctionsandtheirapplications.Webelievethatthisstudywill

behelpfulinachievingabettercomprehensionoftheirbasiccharacteristics.This

studyisalsointendedtoenlightenstudentsandresearcherswhoareunfamiliarwith

Mathieufunctions.Inthechaptertwoofthiswork,wediscussedtheMathieu

2
differentialequationandhowitarisesfromtheellipticalcoordinatesystem.Also,we

talkedabouttheModifiedMathieudifferentialequationandtheMathieudifferential

equationinanalgebraicform.Thechapterthreewasbasedonthesolutionstothe

MathieuequationknownasMathieufunctionsandalsotheFloquet’stheory.Inthe

chapterfour,weshowedhowMathieufunctionscanbeappliedtodescribethe

invertedpendulum,ellipticdrumhead,Radiofrequencyquadrupole,Frequency

modulation,Stabilityofafloatingbody,AlternatingGradientFocusing,thePaultrap

forchargedparticlesandtheQuantumPendulum..

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