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.

.


TYPE IN YOUR TOPIC AND CLICK SEARCH.






RESEARCHWAP.COM

Researchwap.com is an online repository for free project topics and research materials, articles and custom writing of research works. We’re an online resource centre that provides a vast database for students to access numerous research project topics and materials. Researchwap.com guides and assist Postgraduate, Undergraduate and Final Year Students with well researched and quality project topics, topic ideas, research guides and project materials. We’re reliable and trustworthy, and we really understand what is called “time factor”, that is why we’ve simplified the process so that students can get their research projects ready on time. Our platform provides more educational services, such as hiring a writer, research analysis, and software for computer science research and we also seriously adhere to a timely delivery.

TESTIMONIES FROM OUR CLIENTS


Please feel free to carefully review some written and captured responses from our satisfied clients.

  • "Exceptionally outstanding. Highly recommend for all who wish to have effective and excellent project defence. Easily Accessable, Affordable, Effective and effective."

    Debby Henry George, Massachusetts Institute of Technology (MIT), Cambridge, USA.
  • "I saw this website on facebook page and I did not even bother since I was in a hurry to complete my project. But I am totally amazed that when I visited the website and saw the topic I was looking for and I decided to give a try and now I have received it within an hour after ordering the material. Am grateful guys!"

    Hilary Yusuf, United States International University Africa, Nairobi, Kenya.
  • "Researchwap.com is a website I recommend to all student and researchers within and outside the country. The web owners are doing great job and I appreciate them for that. Once again, thank you very much "researchwap.com" and God bless you and your business! ."

    Debby Henry George, Massachusetts Institute of Technology (MIT), Cambridge, USA.
  • "I love what you guys are doing, your material guided me well through my research. Thank you for helping me achieve academic success."

    Sampson, University of Nigeria, Nsukka.
  • "researchwap.com is God-sent! I got good grades in my seminar and project with the help of your service, thank you soooooo much."

    Cynthia, Akwa Ibom State University .
  • "Great User Experience, Nice flows and Superb functionalities.The app is indeed a great tech innovation for greasing the wheels of final year, research and other pedagogical related project works. A trial would definitely convince you."

    Lamilare Valentine, Kwame Nkrumah University, Kumasi, Ghana.
  • "Sorry, it was in my spam folder all along, I should have looked it up properly first. Please keep up the good work, your team is quite commited. Am grateful...I will certainly refer my friends too."

    Elizabeth, Obafemi Awolowo University
  • "Am happy the defense went well, thanks to your articles. I may not be able to express how grateful I am for all your assistance, but on my honour, I owe you guys a good number of referrals. Thank you once again."

    Ali Olanrewaju, Lagos State University.
  • "My Dear Researchwap, initially I never believed one can actually do honest business transactions with Nigerians online until i stumbled into your website. You have broken a new legacy of record as far as am concerned. Keep up the good work!"

    Willie Ekereobong, University of Port Harcourt.
  • "WOW, SO IT'S TRUE??!! I can't believe I got this quality work for just 3k...I thought it was scam ooo. I wouldn't mind if it goes for over 5k, its worth it. Thank you!"

    Theressa, Igbinedion University.
  • "I did not see my project topic on your website so I decided to call your customer care number, the attention I got was epic! I got help from the beginning to the end of my project in just 3 days, they even taught me how to defend my project and I got a 'B' at the end. Thank you so much researchwap.com, infact, I owe my graduating well today to you guys...."

    Joseph, Abia state Polytechnic.
  • "My friend told me about ResearchWap website, I doubted her until I saw her receive her full project in less than 15 miniutes, I tried mine too and got it same, right now, am telling everyone in my school about researchwap.com, no one has to suffer any more writing their project. Thank you for making life easy for me and my fellow students... Keep up the good work"

    Christiana, Landmark University .
  • "I wish I knew you guys when I wrote my first degree project, it took so much time and effort then. Now, with just a click of a button, I got my complete project in less than 15 minutes. You guys are too amazing!."

    Musa, Federal University of Technology Minna
  • "I was scared at first when I saw your website but I decided to risk my last 3k and surprisingly I got my complete project in my email box instantly. This is so nice!!!."

    Ali Obafemi, Ibrahim Badamasi Babangida University, Niger State.
  • To contribute to our success story, send us a feedback or please kindly call 2348037664978.
    Then your comment and contact will be published here also with your consent.

    Thank you for choosing researchwap.com.