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1D advection with damping/forcing

  1. Jan 29, 2010 #1

    I'm looking for a solution to

    u_t + A u_x + B (u - f) = 0

    where f is a given function linear in x and constant in t.

    u(x,0) = f(x)

    u(0,t) = f(0)

    A and B constant. Does this equation have a name? It's almost the inviscid Burger, but has the damping term (B u) and force term (B f).

    Also, any ideas on a solution? I've tried method of characteristics but I trip over the third term and/or the boundary conditions

    Regards, John
  2. jcsd
  3. Jan 30, 2010 #2
    Your PDE is first order, so you need only one boundary condition.

    The general solution to your PDE is as follows

    [tex]u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -F(-At+x)],[/tex]

    where F is an arbitrary function.

    If the boundary condition is u(x,0) = f(x) then

    [tex]u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -f(-At+x)][/tex]
  4. Jan 30, 2010 #3
    Wow! That's a lot more elegant than what I eventually struggled through... :-)

    Using a Laplace' transform on the time, then I solved the ODE in x and transformed back. That did use both the boundaries though...

    I think the reason I have two boundary statements is that I forgot to mention it is u(x,0) for x>0 and u(0,t) for t>0 so effectively forms a single boundary condition on the "quarter-plane". The solution takes one form for x-vt >0 and another for x-vt<0.

    Appreciate the help, regards, John
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