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 Literature Review2.2 Theoretical Framework
2.3 Historical Perspectives
2.4 Empirical Studies
2.5 Conceptual Framework
2.6 Current Trends
2.7 Critical Analysis
2.8 Research Gaps
2.9 Methodological Approaches
2.10 Comparative Review
Chapter THREE
3.1 Research Methodology Overview3.2 Research Design
3.3 Data Collection Methods
3.4 Sampling Techniques
3.5 Data Analysis Procedures
3.6 Research Instruments
3.7 Ethical Considerations
3.8 Validity and Reliability
Chapter FOUR
4.1 Introduction to Findings Analysis4.2 Demographic Analysis
4.3 Quantitative Results Interpretation
4.4 Qualitative Results Interpretation
4.5 Statistical Analysis
4.6 Data Visualization
4.7 Comparative Analysis
4.8 Discussion of Findings
Chapter FIVE
5.1 Conclusion and Summary5.2 Summary of Findings
5.3 Implications of the Study
5.4 Recommendations for Future Research
5.5 Conclusion Statement
Project 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.
Project Overview
Blazingprojects Mobile App
📚 Over 50,000 Project Materials
📱 100% Offline: No internet needed
📝 Over 98 Departments
🔍 Software coding and Machine construction
🎓 Postgraduate/Undergraduate Research works
📥 Instant Whatsapp/Email Delivery
Related Research
Predictive Modeling of Stock Market Trends Using Machine Learning Algorithms...
The project topic "Predictive Modeling of Stock Market Trends Using Machine Learning Algorithms" involves the application of advanced statistical tech...
Analysis of Factors Affecting Student Performance in Online Learning Environments: A...
The project on "Analysis of Factors Affecting Student Performance in Online Learning Environments: A Statistical Approach" aims to investigate the var...
Predictive Modeling of Customer Churn in Telecommunication Industry Using Machine Le...
The research project on "Predictive Modeling of Customer Churn in Telecommunication Industry Using Machine Learning Techniques" aims to address the cr...
Analysis of Factors Influencing Student Performance in Online Learning Environments:...
The project titled "Analysis of Factors Influencing Student Performance in Online Learning Environments: A Statistical Approach" aims to investigate a...
Analysis of factors influencing customer satisfaction in online retail using statist...
The research project titled "Analysis of factors influencing customer satisfaction in online retail using statistical techniques" aims to investigate ...
Predictive Modeling of Customer Churn using Machine Learning Algorithms...
The project topic, "Predictive Modeling of Customer Churn using Machine Learning Algorithms," focuses on utilizing advanced machine learning technique...
Analysis of Factors Influencing Student Performance in Higher Education Using Machin...
The project on "Analysis of Factors Influencing Student Performance in Higher Education Using Machine Learning Algorithms" aims to explore the various...
Analysis of Factors Affecting Student Performance in Higher Education Using Machine ...
The project "Analysis of Factors Affecting Student Performance in Higher Education Using Machine Learning Techniques" aims to investigate the various ...
Predictive Modeling of Stock Prices Using Time Series Analysis...
The project topic "Predictive Modeling of Stock Prices Using Time Series Analysis" involves utilizing advanced statistical methods to forecast and pre...
