Функции Матьё

Функции Матьё — математические специальные функции, являющиеся периодическими решениями уравнения Матьё. Используются при решении различных задач математической физики, в частности, при описании волнового движения с эллиптическими граничными условиями, при изучении явления параметрического резонанса, при изучении нелинейных колебаний в различных разделах теоретической и экспериментальной физики и т. д.

Уравнение Матьё

Уравнением Матьё называется дифференциальное уравнение вида (каноническая форма):

{\frac {d^{2}y}{dx^{2}}}+[a-2q\cos(2x)]y=0,

где $a$ и $q$ - параметры, от которых зависит поведение решения (устойчивое или неустойчивое), данную зависимость иллюстрирует диаграмма Айнса-Стретта.

Решения уравнения Матьё

Согласно теореме Флоке, всегда существуют решения уравнения Матьё в виде: $y(x)=\exp(\mu x)\phi (x)$ ,

где $\phi (x)$ имеет период $2\pi$ . При $\mu =0$ эти решения являются периодическими с периодом $2\pi$ и называются функциями Матьё. Они обозначаются как: $\mathrm {ce} _{0}(x),\mathrm {ce} _{1}(x),\mathrm {se} _{1}(x),\mathrm {ce} _{2}(x),\mathrm {se} _{2}(x),...$ . Функции Матьё можно представить в виде сумм косинусов или синусов: $\mathrm {ce} _{2n}(x)=\sum _{k}A_{k}\cos 2kx,\mathrm {ce} _{2n+1}(x)=\sum _{k}B_{k}\cos(2k+1)x,\mathrm {se} _{2n}(x)=\sum _{k}C_{k}\sin 2kx,\mathrm {se} _{2n+1}(x)=\sum _{k}D_{k}\sin(2k+1)x,$ где величины $A_{k},B_{k},C_{k},D_{k}$ являются функциями от величин $a,q$ в уравнении Матьё. Значения $A_{k},B_{k},C_{k},D_{k}$ можно получить, подставляя решение уравнения Матьё в виде разложения по ряду Фурье в уравнение и приравнивая подобные члены.

См. также

Уравнение Хилла

Литература

Мэтьюз Дж., Уокер Р. Математические методы в физике / Пер. с англ. — М.: Атомиздат, 1972. — 392 с.
Уравнение Матье, EqWorld

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