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Wave equation with inhomogeneous boundary conditions

  1. Mar 27, 2009 #1
    I'm reposting this because there was a problem with the title/LaTeX last time.

    1. The problem statement, all variables and given/known data

    Solve the wave equation (1) on the region 0<x<2 subject to the boundary conditions (2) and the initial condition (3) by separation of variables.


    2. Relevant equations

    (1) [itex]\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}[/itex]

    (2) [itex]\frac{\partial u}{\partial x}(0,t)=1[/itex] ; [itex]\frac{\partial u }{\partial x}(2,t)=1[/itex]

    (3) [itex]\frac{\partial u}{\partial t}(x,0)=0[/itex]


    3. The attempt at a solution

    I've defined [itex]\theta(x,t)=u(x,t)-u_{st}(x) = u(x,t)-x-h(t)[/itex] where u_st is the steady state solution (4). I've used this to create a new PDE with homogeneous boundary conditions.

    The PDE is:

    [itex]\frac{\partial^2 \theta}{\partial t^2} + h''(t)=c^2 \frac{\partial^2 \theta}{\partial x^2}[/itex].

    By subbing in [itex]\theta=f(t)g(x)[/itex] I get:

    [itex]f''(t)g(x)+h''(t)=c^2 f(t) g''(x)[/itex]

    I'm not sure how to transform this into two ODEs. Can someone help?

    (4): The solution to [itex]\frac{\partial^2 u}{\partial x^2} = 0[/itex] subject to (2).
     
  2. jcsd
  3. Mar 27, 2009 #2
    I think you just need to introduce new function v: u(x,t) = v(x,t) + x. The equation for v(x,t) will be the same but boundary conditions will become homogeneous =) and btw, you are missing one more initial condition like u(x,t=0) = ...
     
  4. Apr 23, 2009 #3
    Im also stuck on this one, for me its the initial conditions and boundary conditions being partial derivitives that throws me.
     
    Last edited: Apr 23, 2009
  5. Apr 25, 2009 #4
    The eqn. given for boundary conditions is a partial derivitive so do i intergrate it to find u(x,t)
    in which case is it that; u(o,t) = x and u(2,t) = x. so that.
    u(x,t) = v(x,t) + x so that in both cases.... umm then i get stuck.
     
  6. Apr 25, 2009 #5
    No-one at all?
     
  7. Apr 26, 2009 #6
    So far ive got that:
    [tex] U_{st} (x) = x [/tex]
    (a) Find the steady state solution, ust(x), by solving [tex]\frac{\delta ^{2} U_{st} }{\delta x^{2}} = 0 [/tex] and applying the boundary
    conditions. (You will only be able to determine one of the two arbitrary constants).

    confused here, surely [tex] U_{st} (x) = x + c[/tex] (as boundary conditions given are derivitives)

    then initial condition:

    [tex]
    \frac{\partial u}{\partial t}(x,0)=0
    [/tex]

    given means that c = 0? :S
     
  8. Apr 26, 2009 #7
    Latex is fail :(

    So far ive got that:
    U_st (x) = x
    (a) Find the steady state solution, u_st(x), by solving (partial second derivitive u by x = 0) and applying the boundary
    conditions. (You will only be able to determine one of the two arbitrary constants).

    confused here, surely U_st (x) = x + c (as boundary conditions given are derivitives)

    then initial condition:

    (as above... in first post)

    given means that c = 0? :S
     
  9. Apr 27, 2009 #8
    Can no one point me in the right direction?
     
  10. May 10, 2009 #9
    Instead of integrating the BC, you may differentiate the solution you get and then apply the BC.
     
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