Solving PDEs: Separation of Vars., Method of Characteristics

[Abraham]

I've taken a first semester course on PDEs. Basically all we learned was separation of variables and method of characteristics. I understand that there are transforms out there, such as laplace and fourier. However, it looks like there aren't many analytical ways of solving PDEs. Mind you, I'm only talking about linear PDEs. I know nothing of nonlinear ones.

Can anyone tell me what other methods there are to solve PDEs?

Anyone with more experience, can you tell me what other types of PDEs are out there? I know the heat, laplace, and wave eq.

Thx

 

[AlephZero]

In "real life" applications, PDEs are almost always solved numerically not analytically.

Even when there are general analytic solutions, it is usually impossible to fit the boundary conditions to "real" regions in space which are not simple rectangles, circles, etc. For example, imagine trying to solve something as "simple" as the wave equation analytically inside a region of 3D space shaped like a real automobile (including the seats, passengers, etc), to decide the most effective places to put sound insulation, loudspeakers for audio equipment, etc.

Numerical solution of PDEs is almost as big a subject area as studying the PDEs themselves. There is a lot more to it than the simple finite difference methods that you find in basic "computational methods" courses.