Solving Partial Differential Equations: Strategies and Techniques

[aaaa202]

I have, as is normal for anyone working in physics, come across a differential equation describing the system I am looking it. Now I know little about solving partial differential equations, and indeed I am not even sure if an analytical solutions exists for my equation, but here it is anyways:

∂f/∂t = g(t)∂²f/∂x² - xh(t)f

With boundary conditions:

f(x,0)=f(x,n*τ)=0 ,n=1,2,3...
f(x*(t)-ε(t),t)=1, f(x*(t)+ε(t),t)=0
where x*(t) and ε(t) are some known functions describing the "moving" boundary conditions.

Now I am not expecting anyone to be able to immidiatly look at this equation and know whether there exists a closed form solution. Rather I want to ask you, what you would do to find out?
I litterally have no clue what to do. Is there a big book on all kinds of partial differential equations, or what do you guys do when an equation like the above pops up in your work?

I know from my book that no solution can exist if we only have the condition that f(x,0)=0, but I have provided further boundary conditions saying that this must be true again after a certain period τ. Thus it is assumed that f is periodic in time with my assumption. My hope is that this boundary condition makes the above differential equation analytically solvable. Can anyone immidiatly tell if it doesn't help me? Or are there any tricks from Fourier analysis etc. that one can exploit now that f is periodic in time?

 

[SteamKing]

It would be helpful to us if you could provide a little more context about this 'system' you are looking at. PDEs describe many different physical phenomena, and more information about this equation and where it comes from will be a big help in seeing what kinds of solution are available.

 

[az_lender]

There are indeed big books. A convenient online resource is a set of notes from Grigoryan's USCB course Math 124a. Google it, you'll find it. A difficulty I have with his file is that his equals signs display as long dashes on my screen (like, elongated minus signs), but you can make the translation mentally once you know that's happening.