I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched to solve this uses separation of variables to generate two ODE's, and this is referred to as the "eigenvalue problem". My question is, is there any other way to do this that does not use separation of variables? I want to find approximate eigenvalues using a numerical method (like power method) but not sure if it is possible.
Any information would be appreciated
If you are looking for an analytic solution (i.e., some formula for eigenvalues), then separation of variables is the way to go. It will work only in domains of special geometry, which fortunately include disks and squares. For an arbitrary domain, there is no practical analytic way to get the eigenvalues, and one uses numerical methods instead.
Since a numerical method is what you want, I recommend the site Computational Science, and especially the questions Library to compute eigenvalues of the Laplace operator in a polyhedral domain and Solving PDE or eigenvalue problems without FEM.