In many Physics courses you solve PDEs like heat or wave on square, circular, or spherical domains with separation of variables. Are there ways to solve PDEs and Boundary value problems on irregular domains given by some function analytically? For example the wave equation inside a waveguide where the cross sectional shape is given by the intersection between 2 defined curves a parabola and a cubic for example. Another example could be the Schrödinger equation for a particle in a 3D cavity given by the intersection of 2 surfaces. Keep in mind the irregular boundaries are defined by functions.
I know separation of variables wont work but are there other methods for irregular domains given by functions? Would eigenfunction expansions or variational methods work? What I am asking is are there methods to solve Linear PDEs on irregular domains where the boundaries are defined by functions? I know there are numerical methods for PDEs on irregular domains what I want to know is are there Analytical methods approximate or exact for PDEs on irregular domains?
From my knowledge, one of the most used tool to deal with irregular domains in such cases is to implement conformal mapping (and therefore reduce the problem to the problem defined in e.g. a circle). You can google 'conformal mappings equations irregular domains' and find some examples of how it is done in particular cases.