Project Abstract

Abstract
The Mathieu equation is a differential equation of the form \[ \frac{{d^2y}}{{dx^2}} + (a - 2q \cos(2x))y = 0 \] where \( a \) and \( q \) are real parameters. This equation is a special case of a more general class of equations known as Hill's equation. The Mathieu equation has a wide range of applications in various fields such as physics, engineering, and mathematics. One of the key features of the Mathieu equation is its periodic solutions, which arise due to the periodic nature of the cosine function in the equation. These periodic solutions play a crucial role in understanding the behavior of physical systems governed by the Mathieu equation. For example, in the field of physics, the Mathieu equation arises in the study of the motion of charged particles in a periodic electric field or the oscillations of a stretched elastic membrane. In addition to its periodic solutions, the Mathieu equation also exhibits instability regions, where the solutions grow exponentially with time. These instability regions are of significant interest in the study of stability and bifurcation phenomena in dynamical systems. Understanding the boundaries of these instability regions is essential for predicting the behavior of systems described by the Mathieu equation. Furthermore, the Mathieu equation has applications in the field of mathematics, particularly in the theory of special functions. The solutions of the Mathieu equation are given in terms of Mathieu functions, which are a special class of functions that arise in many areas of mathematics, including number theory and differential equations. Moreover, the Mathieu equation finds applications in engineering, such as in the design of control systems and electronic circuits. By studying the solutions of the Mathieu equation, engineers can analyze the stability and performance of systems subject to periodic disturbances. Overall, the Mathieu equation is a powerful mathematical tool with diverse applications in physics, engineering, and mathematics. Its periodic solutions, instability regions, and connection to special functions make it a valuable subject of study for researchers across different disciplines. Further research on the Mathieu equation and its applications will continue to advance our understanding of complex dynamical systems and lead to new technological developments.

Project Overview

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
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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
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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.