The question is really in the title. I have been seeing many examples of PDE's (heat equation on an infinite domain for example) being solved using Fourier transforms (FT). However, I have been unable to find a theory that says which type of PDE's can be solved by FT. So, my questions would be:
- What does mathematical theory say regarding the type of PDE's that can be solved by FT?
- Is the following PDE solvable by FT : $c \cdot \varphi_{EE} + f(E) \cdot \varphi_{E} + g(E) \cdot \varphi + \varphi_{x} = 0 $, where $\varphi = \varphi(E,x)$, $E \in (-\infty,+\infty)$, $ x \in (0,\infty)$, $\varphi_{EE}=\frac{\partial^2\varphi}{\partial E^2}$ and $\varphi_{x}=\frac{\partial\varphi}{\partial x}$ and $c = constant$
- If the above equation is not solvable by Fourier Transforms, what other methods (aside from numerical solutions) are applicable?
Thank you in advance