Mathieu differential equation

A special linear ordinary differential equation of the second order is called a Mathieu differential equation . The DGL is named after the mathematician Émile Léonard Mathieu and is a special case of Hill's differential equation with the parameter function

{\ displaystyle q (x) = q_ {o} + \ Delta q \ cdot \ cos (x)}

Normal form

The equation is presented in different forms in the literature. An equation called normal form has the form

{\ displaystyle \ y '' (x) + [\ lambda + \ gamma \ cos (x)] \ cdot y (x) = 0.}

Is a function of time ${\ displaystyle x}$

{\ displaystyle x = \ Omega \ cdot t}

so the abbreviations and stand for ${\ displaystyle \ lambda}$ ${\ displaystyle \ gamma}$

{\ displaystyle \ lambda = q_ {0} / \ Omega ^ {2} \ ;; \; \ gamma = \ Delta q / \ Omega ^ {2}}

Alternative representation

The DGL is also given as follows, among other things

{\ displaystyle \ y '' (x) + [a-2q \ cos (2x)] \ cdot y (x) = 0}

or

{\ displaystyle \ {\ ddot {x}} (t) + \ omega _ {0} ^ {2} [1 + h \ cos (\ Omega t)] \ cdot x (t) = 0.}

Solution properties

The Mathieu differential equation can be represented as a linear system of differential equations of the first order with two equations:

{\ displaystyle {\ begin {pmatrix} 0 & 1 \\\ lambda + \ gamma \ cos (x) & 0 \\\ end {pmatrix}} {\ begin {pmatrix} u (x) \\ v (x) \\\ end {pmatrix}} = {\ begin {pmatrix} u (x) \\ v (x) \\\ end {pmatrix}} '}

The coefficient matrix is here -periodic. According to Floquet's theorem , the fundamental matrix can be described as ${\ displaystyle 2 \ pi}$

{\ displaystyle \ Phi (x) = P (x) \ exp (xR)}

Here, and also -periodic. The calculation of the Jordanian normal form of the matrix results in two cases: ${\ displaystyle R \ in \ mathbb {C} ^ {2 \ times 2}}$ ${\ displaystyle P: \ mathbb {R} \ rightarrow GL (m; \ mathbb {C})}$ ${\ displaystyle 2 \ pi}$ ${\ displaystyle R}$

${\ displaystyle R}$ has two different (complex) eigenvalues : In this case the solutions are of the form and , where each is -periodic. ${\ displaystyle \ gamma _ {1} \ neq \ gamma _ {2}}$ ${\ displaystyle e ^ {\ gamma _ {1} x} \ phi _ {1} (x)}$ ${\ displaystyle e ^ {\ gamma _ {2} x} \ phi _ {2} (x)}$ ${\ displaystyle \ phi _ {1}, \ phi _ {2}}$ ${\ displaystyle 2 \ pi}$
${\ displaystyle R}$ has a single eigenvalue : Here the solutions are of the form and with a -periodic function . ${\ displaystyle \ gamma}$ ${\ displaystyle e ^ {\ gamma x} \ phi (x)}$ ${\ displaystyle xe ^ {\ gamma x} \ phi (x)}$ ${\ displaystyle 2 \ pi}$ ${\ displaystyle \ phi}$

Individual evidence

↑ Kurt Magnus: Vibrations: An introduction to the physical principles and the theoretical treatment of vibration problems. 8., revised. Edition, Vieweg + Teubner, 2008, Chapter 4, ISBN 3-8351-0193-5 .
↑ NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation (English)
↑ Wolfgang Demtröder: Experimentalphysik 1: Mechanics and heat. Springer, 2008, Chapter 11.7, ISBN 3-540-79294-5 .

Web links

List of equations and identities for Mathieu Functions functions.wolfram.com (English)
E. Mathieu: Mémoire sur Le Mouvement Vibratoire d'une Membrane de forme Elliptique . In: Journal de Mathématiques Pures et Appliquées . 1868, pp. 137-203.

[1] Kurt Magnus: Vibrations: An introduction to the physical principles and the theoretical treatment of vibration problems. 8., revised. Edition, Vieweg + Teubner, 2008, Chapter 4, ISBN 3-8351-0193-5 .

[2] NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation (English)

[3] Wolfgang Demtröder: Experimentalphysik 1: Mechanics and heat. Springer, 2008, Chapter 11.7, ISBN 3-540-79294-5 .