Thesis Abstract

Mathieu equation is a special type of differential equation that arises in various fields of physics, engineering, and applied mathematics. The equation is named after Émile Léonard Mathieu, a French mathematician who studied its properties in the 19th century. The Mathieu equation is a second-order linear differential equation with periodic coefficients. Its general form is written as \[ \frac{d^2y}{dx^2} + (a - 2q \cos(2x))y = 0 \] where \( a \) and \( q \) are constants. The Mathieu equation is known for its mathematical complexity due to the presence of trigonometric functions in the coefficient terms. Despite its challenging nature, the Mathieu equation has found numerous applications in diverse fields. One of the key areas where the Mathieu equation finds application is in the study of vibrational modes of elliptical membranes and circular plates. The solutions to the Mathieu equation provide insights into the natural frequencies and modes of vibration of these structures, which are crucial in engineering design and analysis. In the field of quantum mechanics, the Mathieu equation appears in the study of the motion of charged particles in periodic potentials. The solutions to the Mathieu equation in this context help in understanding the behavior of particles in crystal lattices and other periodic structures, contributing to the development of materials science and solid-state physics. Moreover, the Mathieu equation plays a significant role in the analysis of stability and resonance in mechanical systems. By solving the Mathieu equation, researchers can predict the stability regions of periodic solutions in systems subjected to parametric excitation, leading to advancements in control theory and mechanical engineering. In recent years, the Mathieu equation has also found applications in the field of nonlinear dynamics and chaos theory. The study of Mathieu equation with nonlinear terms has revealed rich dynamical behaviors such as bifurcations, chaos, and strange attractors, shedding light on the complex dynamics of physical systems. Overall, the Mathieu equation represents a fundamental mathematical model with wide-ranging applications in various scientific and engineering disciplines. Its analytical and numerical solutions continue to provide valuable insights into the behavior of linear and nonlinear systems, making it a versatile tool for researchers and practitioners alike.

Thesis 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.