Mathieu equation and its application
Table Of Contents
Chapter ONE
1.1 Introduction1.2 Background of study
1.3 Problem Statement
1.4 Objective of study
1.5 Limitation of study
1.6 Scope of study
1.7 Significance of study
1.8 Structure of the research
1.9 Definition of terms
Chapter TWO
2.1 Overview of Mathieu equation2.2 Historical development of Mathieu equation
2.3 Applications of Mathieu equation in physics
2.4 Applications of Mathieu equation in engineering
2.5 Mathematical properties of Mathieu functions
2.6 Stability analysis using Mathieu equation
2.7 Solutions to Mathieu equation
2.8 Comparison of Mathieu equation with other differential equations
2.9 Challenges in solving Mathieu equation
2.10 Future research directions in Mathieu equation
Chapter THREE
3.1 Research design and approach3.2 Data collection methods
3.3 Sampling techniques
3.4 Data analysis procedures
3.5 Research instruments
3.6 Ethical considerations
3.7 Reliability and validity
3.8 Limitations of research methodology
Chapter FOUR
4.1 Overview of research findings4.2 Analysis of data collected
4.3 Interpretation of results
4.4 Comparison with existing literature
4.5 Discussion on key findings
4.6 Implications of research findings
4.7 Recommendations for future research
4.8 Limitations of the study
Chapter FIVE
5.1 Summary of research findings5.2 Conclusions drawn from the study
5.3 Contributions to the field
5.4 Implications for practice
5.5 Recommendations for further research
Thesis Abstract
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.
Thesis Overview
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