Mathieu Wavelet - Mathieu Differential Equations

Mathieu Differential Equations

Mathieu's equation is related to the wave equation for the elliptic cylinder. In 1868, the French mathematician Émile Léonard Mathieu introduced a family of differential equations nowadays termed Mathieu equations.

Given, the Mathieu equation is given by

The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency.

The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions. They have long been applied on a broad scope of wave-guide problems involving elliptical geometry, including:

1. analysis for weak guiding for step index elliptical core optical fibres
2. power transport of elliptical wave guides
3. evaluating radiated waves of elliptical horn antennas
4. elliptical annular microstrip antennas with arbitrary eccentricity )
5. scattering by a coated strip.