(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the fundamental solution or Green function for the diffusion/heat equation in [itex](-\infty, \infty )[/itex] to determine the fundamental solution to [itex]\frac{\partial u }{ \partial t } =k^2 \frac{\partial ^2 u }{ \partial x ^2 }[/itex] in the semi-line [itex](0, \infty )[/itex] with initial condition [itex]u(x,0)= f(x)[/itex] for [itex]x>0[/itex] and the boundary conditions:

1)Dirichlet: [itex]u(0,t)=0[/itex]

2)Neumann: [itex]\frac{\partial u }{ \partial x } (0,t)=0[/itex]. With [itex]t \geq 0[/itex].

For it (method of images), consider the extension of the problem to [itex](-\infty, \infty )[/itex] taking the parity (odd or even) extension of the function according to 1) and 2).

Describe a method of solving:

3) [itex]u(0,t)=f(t)[/itex], [itex]t\geq 0[/itex].

4) [itex]\frac{\partial u }{ \partial x } (0,t)=g(t)[/itex], [itex]t \geq 0[/itex].

2. Relevant equations

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

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 [itex]L= \frac{\partial }{\partial t }-k^2 \frac{\partial ^2 }{\partial x ^2 }[/itex] applied to Green's function gives the Dirac delta function [itex]\delta (x-s)[/itex].

Any tip will be greatly appreciated!

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# Homework Help: Long exercise about PDE and Green's function

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