Project Abstract

Project Overview

<p>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<br>elliptical membranes. Adecadelater,Heine defined the periodic Mathieu Angular<br>Functions of integer order as Fourier cosine and sineseries; furthermore, without<br>evaluatingthecorrespondingcoefficient,Heobtainedatranscendentalequationfor<br>characteristicnumbersexpressedintermsofinfinitecontinuedfractions;andalso<br>showedthatonesetofperiodicfunctionsofintegerordercouldbeinaseriesof<br>Besselfunction(Chaos-CadorandLey-Koo2002).<br>Intheearly1880’s,Floquetwentfurthertopublishatheoryandthusasolution<br>totheMathieudifferentialequation;hisworkwasnamedafterhimas,‘Floquet’s<br>Theorem’or‘Floquet’sSolution’.StephensonusedanapproximateMathieuequation,<br>andproved,thatitispossibletostabilizetheupperpositionofarigidpendulumby<br>vibratingitspivotpointverticallyataspecifichighfrequency.(StépánandInsperger<br>2003).Thereexistsanextensiveliteratureontheseequations;andinparticular,a<br>well-highexhaustivecompendiumwasgivenbyMc-Lachlan(1947).<br>TheMathieufunctionwasfurtherinvestigatedbynumberofresearcherswho<br>foundaconsiderableamountofmathematicalresultsthatwerecollectedmorethan<br>60yearsagobyMc-Lachlan(Gutiérrez-Vegaaetal2002).Whittakerandother<br>scientistderivedin1900sderivedthehigher-ordertermsoftheMathieudifferential<br>equation.AvarietyoftheequationexistintextbookwrittenbyAbramowitzand<br>Stegun(1964).<br>Mathieudifferentialequationoccursintwomaincategoriesofphysicalproblems.<br>First,applicationsinvolvingellipticalgeometriessuchas,analysisofvibratingmodes<br>2<br>inellipticmembrane,thepropagatingmodesofellipticpipesandtheoscillationsof<br>waterinalakeofellipticshape.Mathieuequationarisesafterseparatingthewave<br>equation using ellipticcoordinates.Secondly,problemsinvolving periodicmotion<br>examplesare,thetrajectoryofan electron in aperiodicarrayofatoms,the<br>mechanicsofthequantumpendulumandtheoscillationoffloatingvessels.<br>ThecanonicalformfortheMathieudifferentialequationisgivenby<br>+ y =0, (1.1)<br>dy 2<br>dx2 [a-2qcos(2x)] (x)<br>whereaandqarerealconstantsknownasthecharacteristicvalueandparameter<br>respectively.<br>Closely related to the Mathieu differentialequation is the Modified Mathieu<br>differentialequationgivenby:<br>– y =0, (1.2)<br>dy 2<br>du2 [a-2qcosh(2u)] (u)<br>whereu=ixissubstitutedintoequation(1.1).<br>Thesubstitutionoft=cos(x)inthecanonicalMathieudifferentialequation(1.1)<br>abovetransformstheequationintoitsalgebraicformasgivenbelow:<br>(1-t) -t + y =0. (1.3) 2 dy 2<br>dt2<br>dy<br>dt<br>[a+2q(1-2t2)] (t)<br>Thishastwosingularitiesatt=1,-1andoneirregularsingularityatinfinity,which<br>impliesthatingeneral(un-likemanyotherspecialfunctions),thesolutionofMathieu<br>differentialequationcannotbeexpressedintermsofhypergeometricfunctions<br>(Mritunjay2011).<br>Thepurposeofthestudyistofacilitatetheunderstandingofsomeofthe<br>propertiesofMathieufunctionsandtheirapplications.Webelievethatthisstudywill<br>behelpfulinachievingabettercomprehensionoftheirbasiccharacteristics.This<br>studyisalsointendedtoenlightenstudentsandresearcherswhoareunfamiliarwith<br>Mathieufunctions.Inthechaptertwoofthiswork,wediscussedtheMathieu<br>3<br>differentialequationandhowitarisesfromtheellipticalcoordinatesystem.Also,we<br>talkedabouttheModifiedMathieudifferentialequationandtheMathieudifferential<br>equationinanalgebraicform.Thechapterthreewasbasedonthesolutionstothe<br>MathieuequationknownasMathieufunctionsandalsotheFloquet’stheory.Inthe<br>chapterfour,weshowedhowMathieufunctionscanbeappliedtodescribethe<br>invertedpendulum,ellipticdrumhead,Radiofrequencyquadrupole,Frequency<br>modulation,Stabilityofafloatingbody,AlternatingGradientFocusing,thePaultrap<br>for charged particles and the Quantum Pendulum.<br><br><br></p>