Long exercise about PDE and Green's function

[fluidistic]

Homework Statement

Use the fundamental solution or Green function for the diffusion/heat equation in (-\infty, \infty ) to determine the fundamental solution to \frac{\partial u }{ \partial t } =k^2 \frac{\partial ^2 u }{ \partial x ^2 } in the semi-line (0, \infty ) with initial condition u(x,0)= f(x) for x>0 and the boundary conditions:
1)Dirichlet: u(0,t)=0
2)Neumann: \frac{\partial u }{ \partial x } (0,t)=0. With t \geq 0.
For it (method of images), consider the extension of the problem to (-\infty, \infty ) taking the parity (odd or even) extension of the function according to 1) and 2).
Describe a method of solving:
3) u(0,t)=f(t), t\geq 0.
4) \frac{\partial u }{ \partial x } (0,t)=g(t), t \geq 0.

Homework Equations

I don't really know where to look for the "Green function". In wikipedia it looks like u(x)= \int G(x,s)f(s)ds.

The Attempt at a Solution

Since I'm self studying PDE's and didn't take the course, I'm having a hard time to start the problem. I think I must find the Green function for which the linear operator L= \frac{\partial }{\partial t }-k^2 \frac{\partial ^2 }{\partial x ^2 } applied to Green's function gives the Dirac delta function \delta (x-s).
Any tip will be greatly appreciated!

 

[fluidistic]

I'm still stuck at starting this exercise.
Shouldn't the equation be inhomogenous first, in order to find Green's function? Should I take a Fourier or Laplace transform of the given PDE, even though they didn't specify it?

 

[fluidistic]

Wow guys, I found the solution (almost 100% sure) of that problem in wikipedia. For those interested, http://en.wikipedia.org/wiki/Heat_equation#Some_Green.27s_function_solutions_in_1D.
Incredible. I didn't understand I had to assume and find from a textbook what was the Green function of the 1D heat equation. I thought I had to derive it which would have been a pain since I don't know how to find it!.