# The Mathieu Differential Equation and Generalizations to Infinite Fractafolds

## TL;DR

This paper analyzes the Mathieu differential equation's solutions, explores their stability, and extends the framework to fractal structures like the Sierpinski gasket, combining theoretical and numerical approaches.

## Contribution

It introduces a method to generalize the Mathieu equation to fractal domains and studies the stability and properties of solutions in this new setting.

## Key findings

- Fourier series approach effectively approximates solutions
- Stability results for classical Mathieu equation confirmed
- Extension to fractal domain demonstrates similar solution behavior

## Abstract

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.