Mathieu Equation And Its Application

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MATHIEU EQUATION AND ITS APPLICATION

Mathieu Equation

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.

The Mathieu differential equation d2u . , (1) -h (A – í2cos2z}» = 0, dz2 also commonly known as the equation of the elliptic cylinder functions, is too well known to require any introduction. Its solutions govern problems of the greatest diversity in astronomy and theoretical physics, and have accordingly been the subjects of a vast number of investigations.f The differential equation as such depends upon two independent parameters, designated in the form written above by A and Ü. In the present discussion these are to be taken real but are to be numerically unrestricted except that at least one is to be large. The variable will be permitted to range over the complex plane. Since the coefficient of the differential equation is an even simply periodic analytic function of z, it is known from Floquet’s theory of such equations that the solutions are in general of the structure u(z) = cxC”(z) + c2e-“l4>(- z)

 

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