Computing Mathieu function solutions for linear ODEs

Edgardo Cheb-Terrab

MITACS-CECM, Simon Fraser University and Maplesoft

    Mathieu functions were first introduced by Mathieu (1868) to 
    represent the solutions of the equation

                  y" + (a - 2 q cos(2 x)) y = 0

    which arises from the separation of the 2-D or 3-D wave equation
    modeling the motion of an elliptic membrane.

    Mathieu functions are non-elementary nor Liouvillian, nor do they 
    admit a hypergeometric representation, making them difficult to treat. 
    They are perhaps the simplest class of special functions of the Heun 
    type, typically associated with linear ODEs having four regular singular
    points. On the other hand, the fact that these functions appear
    frequently in physical problems involving elliptical shapes or
    periodic potentials has attracted their attention for a long time.

    Mathieu functions were implemented in the Maple system a couple
    of months ago, for its new release. This opened the way for developing
    algorithms to compute linear ODE exact solutions which require the
    presence of Mathieu functions in order to be expressed. This talk
    presents such an algorithm, implemented in Maple, around the idea of
    solving an "equivalence" to Mathieu's ODE under transformations

                                k
                             A x  + B
                        x -> --------,   y(x) -> P(x) y(x)
                                k
                             C x  + D

    where {A,B,C,D,k} are constants with respect to x, and P(x) is any
    function (even arbitrary) different from zero. The algorithm includes
    computing the values of the function parameters {a,q} such that the
    equivalence is possible. During the talk, a brief demo of the Maple
    implementation being described will be performed.