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Green's formula

  1. Nov 8, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider the wave operator
    [tex] \Box^2 = \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}[/tex]
    Write Green's forumula for an arbitrary domain [itex] \Omega [/itex] with boundary [itex] \Gamma [/itex]


    2. Relevant equations
    define a vector q, where
    [tex] q \cdot e_t = n \cdot e_t [/tex]
    [tex] q \cdot e_x = -n \cdot e_x [/tex]
    where e_x, e_t are unit vectors in x, t dimensions, n is the normal.
    Show Green's formula takes the form
    [tex] \int_\Omega (v\Box^2u - u\Box^2 v)dx dt = \int_\Gamma \left( v\frac{\partial u}{\partial q} -u\frac{\partial u}{\partial q} \right)dl[/tex]
    where dl is an element of arclength along the bounding curve [itex] \Gamma [/itex]
    (u and v are arbitrary functions in R1)

    3. The attempt at a solution
    I'm starting with Green's formula for an arbitrary linear operator:
    [tex] \int_\Omega (vLu - L^*v) = \int_\Gamma div(J(u,v)) [/tex]
    Where L is an ODE operator, and J is the conjunct of u and v. For the operator above, this takes the form of
    [tex] \int_\Omega (vLu - L*v) = \int_\Gamma div \left[e_t \left( v\frac{\partial u}{\partial t} -u\frac{\partial u}{\partial t}\right) + u grad_x v - v grad_x u \right][/tex]
    Where grad_x is the gradient along x. Now, the div operator should give us the same a the dot product of the normal to the line space:
    [tex] = \int_\Gamma n\cdot\left[e_t \left( v\frac{\partial u}{\partial t} -u\frac{\partial u}{\partial t}\right) + u grad_x v - v grad_x u\right] [/tex]

    I believe that [itex] n \cdot grad_x = \frac{\partial}{\partial n}[/itex]. So, substituting the form of q above, I have
    [tex] = \int_\Gamma \left( q\cdot e_t \left( v\frac{\partial u}{\partial t} -u\frac{\partial u}{\partial t}\right) + \left(v\frac{\partial u}{\partial q} - u\frac{\partial u}{\partial q}\right)\right) [/tex]

    I'm almost there, I'm just not sure how [itex] \left( q\cdot e_t \left( v\frac{\partial u}{\partial t} -u\frac{\partial u}{\partial t}\right) [/itex] disappears.
     
  2. jcsd
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