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Math Help - Advection Equation

  1. #1
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    Advection Equation

    Solve:

    ut + ((x-1)u)x=0, x E(0,1) & t > 0

    I want to try the solution u(x,t) = f(x + ct), but think this might be incorrect, any ideas?
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  2. #2
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    By the way ut, means differnatiate u with t, and ((x-1)u) is differentiated by x
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  3. #3
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    Quote Originally Posted by ryanhorne View Post
    Solve:

    ut + ((x-1)u)x=0, x E(0,1) & t > 0

    I want to try the solution u(x,t) = f(x + ct), but think this might be incorrect, any ideas?
    If you directly substitute your solution form into the PDE you'll get

     <br />
cf' +(x-1)f' + f = 0.<br />

    If you let x = r - ct this becomes

    cf'(r) + (r - ct - 1)f'(r) + f(r) = 0

    Now differentiate wrt t gives f'(r) = 0. So the only form is f(r) = constant.

    The actual solution of the PDE is u = e^{-t}f\left((x-1)e^{-t}\right).
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  4. #4
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    You a Genius! Cheers
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