MATHIEU EQUATION AND ITS APPLICATION

MATHIEU EQUATION AND ITS APPLICATION

1.1Brief Reviewon Mathieuequation


Mathieu equation is a special case of a linear second order homogeneous differential equation(Ruby1995).The equation was first discussedin1868,by Emile Leonard Mathieuin connection with problem of vibrations in elliptical membrane. He developed the leading terms of the series solution known as Mathieu function of the
elliptical membranes. Adecadelater,Heine defined the periodic Mathieu Angular
Functions of integer order as Fourier cosine and sineseries; 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
2
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
3
differentialequationandhowitarisesfromtheellipticalcoordinatesystem.Also,we
talkedabouttheModifiedMathieudifferentialequationandtheMathieudifferential
equationinanalgebraicform.Thechapterthreewasbasedonthesolutionstothe
MathieuequationknownasMathieufunctionsandalsotheFloquet’stheory.Inthe
chapterfour,weshowedhowMathieufunctionscanbeappliedtodescribethe
invertedpendulum,ellipticdrumhead,Radiofrequencyquadrupole,Frequency
modulation,Stabilityofafloatingbody,AlternatingGradientFocusing,thePaultrap
for charged particles and the  Quantum Pendulum.

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