
Computing Mathieu function solutions for linear ODEsEdgardo ChebTerrabMITACSCECM, 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 2D or 3D wave equation modeling the motion of an elliptic membrane. Mathieu functions are nonelementary 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. 