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Long exercise about PDE and Green's function

  1. Jan 11, 2012 #1

    fluidistic

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    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!
     
  2. jcsd
  3. Jan 18, 2012 #2

    fluidistic

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    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?
     
  4. Feb 24, 2012 #3

    fluidistic

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    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!. :smile:
     
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