Mathieu equation and its application

1.1BriefReviewonMathieuequation

Mathieu equation isa specialcase of a linear second order homogeneousdifferentialequation(Ruby1995).Theequationwasfirstdiscussedin1868,byEmileLeonardMathieuinconnectionwithproblemofvibrationsinellipticalmembrane.HedevelopedtheleadingtermsoftheseriessolutionknownasMathieufunctionoftheellipticalmembranes.Adecadelater,HeinedefinedtheperiodicMathieuAngularFunctionsofintegerorderasFouriercosineandsineseries;furthermore,withoutevaluatingthecorrespondingcoefficient,Heobtainedatranscendentalequationforcharacteristicnumbersexpressedintermsofinfinitecontinuedfractions;andalsoshowedthatonesetofperiodicfunctionsofintegerordercouldbeinaseriesofBesselfunction(Chaos-CadorandLey-Koo2002).Intheearly1880’s,FloquetwentfurthertopublishatheoryandthusasolutiontotheMathieudifferentialequation;hisworkwasnamedafterhimas,‘Floquet’sTheorem’or‘Floquet’sSolution’.StephensonusedanapproximateMathieuequation,andproved,thatitispossibletostabilizetheupperpositionofarigidpendulumbyvibratingitspivotpointverticallyataspecifichighfrequency.(StépánandInsperger2003).Thereexistsanextensiveliteratureontheseequations;andinparticular,awell-highexhaustivecompendiumwasgivenbyMc-Lachlan(1947).TheMathieufunctionwasfurtherinvestigatedbynumberofresearcherswhofoundaconsiderableamountofmathematicalresultsthatwerecollectedmorethan60yearsagobyMc-Lachlan(Gutiérrez-Vegaaetal2002).Whittakerandotherscientistderivedin1900sderivedthehigher-ordertermsoftheMathieudifferentialequation.AvarietyoftheequationexistintextbookwrittenbyAbramowitzandStegun(1964).Mathieudifferentialequationoccursintwomaincategoriesofphysicalproblems.First,applicationsinvolvingellipticalgeometriessuchas,analysisofvibratingmodes2inellipticmembrane,thepropagatingmodesofellipticpipesandtheoscillationsofwaterinalakeofellipticshape.Mathieuequationarisesafterseparatingthewaveequation using ellipticcoordinates.Secondly,problemsinvolving periodicmotionexamplesare,thetrajectoryofan electron in aperiodicarrayofatoms,themechanicsofthequantumpendulumandtheoscillationoffloatingvessels.ThecanonicalformfortheMathieudifferentialequationisgivenby+ y =0, (1.1)dy 2dx2 [a-2qcos(2x)](x)whereaandqarerealconstantsknownasthecharacteristicvalueandparameterrespectively.Closely related to the Mathieu differentialequation is the Modified Mathieudifferentialequationgivenby- y =0, (1.2)dy 2du2 [a-2qcosh(2u)](u)whereu=ixissubstitutedintoequation(1.1).Thesubstitutionoft=cos(x)inthecanonicalMathieudifferentialequation(1.1)abovetransformstheequationintoitsalgebraicformasgivenbelow(1-t) -t + y =0. (1.3) 2 dy 2dt2dydt[a+2q(1-2t2)](t)Thishastwosingularitiesatt=1,-1andoneirregularsingularityatinfinity,whichimpliesthatingeneral(un-likemanyotherspecialfunctions),thesolutionofMathieudifferentialequationcannotbeexpressedintermsofhypergeometricfunctions(Mritunjay2011).ThepurposeofthestudyistofacilitatetheunderstandingofsomeofthepropertiesofMathieufunctionsandtheirapplications.Webelievethatthisstudywillbehelpfulinachievingabettercomprehensionoftheirbasiccharacteristics.ThisstudyisalsointendedtoenlightenstudentsandresearcherswhoareunfamiliarwithMathieufunctions.Inthechaptertwoofthiswork,wediscussedtheMathieu3differentialequationandhowitarisesfromtheellipticalcoordinatesystem.Also,wetalkedabouttheModifiedMathieudifferentialequationandtheMathieudifferentialequationinanalgebraicform.ThechapterthreewasbasedonthesolutionstotheMathieuequationknownasMathieufunctionsandalsotheFloquet’stheory.Inthechapterfour,weshowedhowMathieufunctionscanbeappliedtodescribetheinvertedpendulum,ellipticdrumhead,Radiofrequencyquadrupole,Frequencymodulation,Stabilityofafloatingbody,AlternatingGradientFocusing,thePaultrapforchargedparticlesandtheQuantumPendulum.