Mathematical Analysis and Applications of Mathieu's Equation Revisited

Abstract

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation. \'{E}mile L\'{e}onard Mathieu, modelled vibrating elliptical drumheads by Mathieu's equation. This equation has wide applications in many fields such as~optics, quantum mechanics and~general relativity. They occur in many problems involving periodic motion, in analysis of boundary value problems having elliptic symmetry. Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called~Floquet theory. Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed in this paper.