  One of the well-studied equations in the theory of ODEs is the Mathieu
differential equation. A common approach for obtaining solutions is to seek
solutions via Fourier series by converting the equation into an infinite system
of linear equations for the Fourier coefficients. We study the asymptotic
behavior of these Fourier coefficients and discuss the ways in which to
numerically approximate solutions. We present both theoretical and numerical
results pertaining to the stability of the Mathieu differential equation and
the properties of solutions. Further, based on the idea of using Fourier
series, we provide a method in which the Mathieu differential equation can be
generalized to be defined on the infinite Sierpinski gasket. We discuss the
stability of solutions to this fractal differential equation and describe
further results concerning properties and behavior of these solutions.
