# Long exercise about PDE and Green's function

1. Jan 11, 2012

### fluidistic

1. The problem statement, all variables and given/known data
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) $\frac{\partial u }{ \partial x } (0,t)=g(t)$, $t \geq 0$.
2. Relevant equations
I don't really know where to look for the "Green function". In wikipedia it looks like $u(x)= \int G(x,s)f(s)ds$.

3. 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!

2. Jan 18, 2012

### 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?

3. Feb 24, 2012

### 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!.