MATHIEU EQUATION AND ITS APPLICATION


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.

.

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