CHAPTER ONE
INTRODUCTION
Brief Review on Mathieu Equation
Mathieu equation is a special case of a linear second order homogeneous differential equation (Ruby1995). The equation was first discussed in 1868, 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 defined the periodic Mathieu Angular functions of integer 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 continued 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 early1880’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’. Stephenson 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). TheMathieufunctionwasfurtherinvestigatedbynumberofresearcherswho found a considerable amount of mathematical results that were collected more than 60 years ago by Mc-Lachlan (Gutiérrez-Vegaaetal2002).Whittaker and other scientist derived in1900s derived the higher-order terms of the Mathieu differential equation. A variety of the equation exist in text book written by Abramowitz and Stegun (1964) Mathieu differential equation occurs in two main categories of physical problems. First, applications involving elliptical geometries such as, analysis of vibrating modes. In ellipticm embrane, 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
+ y =0, (1.1)
dy 2
dx2 [a-2qcos(2x)](x)
where a and qarereal constants known as the characteristic value and parameter respectively.
Closely related to the Mathieu differential equation is the Modified Mathieu Differential equation given by:
– y =0, (1.2) dy 2 du2 [a-2qcosh (2u)](u)
where u=ixis substituted in to 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:
(1-t) -t + y =0. (1.3) 2 dy 2
dt2
dy
dt
[a+2q(1-2t2)](t)
This has two singularities at=1,-1andoneir regulars ingularity at infinity, which implies that in general (unlike many other special functions), the solution of Mathieu differential equation cannot be expressed in terms of hyper geometric functions (Mritunjay2011).
Purpose of the Study
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 3 Differential equation and how it arises from the elliptical coordinate system. Also, we talked about the Modified Mathieu differential equation and the Mathieu differential equation 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, Radio frequency quadrupole, Frequency modulation, Stability of a floating body, Alternating Gradient Focusing, the Paul trap for charged particles and the Quantum Pendulum.