MATHIEU EQUATION AND ITS APPLICATION
CHAPTER ONE
INTRODUCTION
1.1Brief Review on Mathieu equation
Mathieu equation is a special case of a linear second order homogeneous
differential equation(Ruby1995). The equation was first discussed in1868, by Emile
Leonard Mathieu in 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. A decade later, He in edefined the periodic Mathieu Angular
Function so finteger order as Fourier cosine and sine series; furthermore, without
evaluating the corresponding coefficient, He obtained a transcendental equation for
characteristic numbers expressed in terms of infinite on tinued fractions; and also
showed that one set of periodic functions of integer order could be in a series of
Bessel function(Chaos-CadorandLey-Koo2002).
In the early 1880’s, Floquet went further to publish a theory and thus a solution
to the Mathieu differential equation; his work was named after him as, ‘Floquet’s
Theorem ’or‘ Floquet’s Solution’. Stephens on used an approximate Mathieu equation,
and proved, that it is possible to stabilize the upper position of a rigid pendulum by
vibrating its pivot point vertically at a specific high frequency. (Stépán and Insperger
2003).There exists an extensive literature on these equations; and in particular, a
well-high exhaustive compendium was given by Mc-Lachlan(1947).
The Mathieu function was further investigated by number of researchers who
found a considerable amount of mathematical results that were collected more than
60years ago by Mc-Lachlan(Gutiérrez-Vegaaetal2002). Whittaker and other
scientist derived in 1900s derived the higher-order terms of the Mathieu differential
equation. Avariety of the equation exist in textbook written by Abramowitzand
Stegun(1964).
Mathieu differential equation occurs in two main categories of physical problems.
First, applications involving elliptical geometries such as, analysis of vibrating modes
1in the elliptic membrane, the propagating modes of elliptic pipes and the oscillations of
water in a lake of elliptic shape. Mathieu equation arises after separating the wave
equation using elliptic coordinates. Secondly, problems involving periodic motion
examples are, the trajectory of an electron in a periodic array of atoms, the
mechanics of the quantum pendulum and the oscillation of floating vessels.
The canonical form for the Mathieu differential equation is given by
2
y
d
x
a-2qcos
2x
,(1.1)+y=0
((
))
[
]
2
dx
whereandarerealconstantsknownasthecharacteristicvalueandparameteraq
respectively.
CloselyrelatedtotheMathieudifferentialequationistheModifiedMathieu
differentialequationgivenby:
2
y
d
u
a-2qcosh
2u
(1.2)-y=0,
((
))
[
]
2
du
where u=ix is substituted into equation(1.1).
The substitution of t=cos (x) In the canonical Mathieu differential equation(1.1)
above transforms the equation into its algebraic form as given below:
2
y
ddy
2
2
a+2q
t
(1-t
(1.3))-t+y=0.
(
)
[
]
(
)
t
1-2
2
dt
dt
This has two singularities at t=1,-1 and one irregular singularity at infinity, which
implies that in general(un-like many other special functions), the solution of Mathieu
differential equation cannot be expressed in terms of hyper geometric functions
(Mritunjay2011).
The purpose of the study is to facilitate the understanding of some of the
properties of Mathieu functions and their applications. We believe that this study will
be helpful in achieving a better comprehension of their basic characteristics. This
study is also intended to enlighten students and researchers who are unfamiliar with
Mathieu functions. In the chapter two of this work, we discussed the Mathieu
2differential equation and how It arises from the elliptical coordinate system. Also, we
talked about the Modified Mathieu differential equation and the Mathieu differential
equation in an algebraic form. The chapter three was based on the solutions to the
Mathieu equation known as Mathieu functions and also the Floquet’s theory. In the
chapter four, we showed how Mathieu functions can be applied to describe the
inverted pendulum, elliptic drum head, Radiofrequency quadrupole, Frequency
modulation, Stability of a floating body, Alternating Gradient Focusing, the Paul trap
for charged particles and the Quantum Pendulum.
.