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Math Help - System of PDEs question

  1. #1
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    System of PDEs question

    The system of PDEs below are a model for linearised compressible fluid flow

    u_x + u_t = 0
    (\rho u)_x + \rho_t = 0
    (E u)_x + E_t + p u_x = 0

    where u=u(x,t) ,  \rho=\rho(x,t) , E= E(x,t) represent velocity, density and a measure of energy per unit volume of a fluid and assume p is constant.
    Find a solution that satisfies the initial conditions

     u(x,0) = x
     \rho(x,0) = (x-1)^2
    E(x,0) = E_0 (x) where  E_0 (x) is a given function.

    Any tips on where to start for this one, should I solve each equation independently or equate them and then solve?
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  2. #2
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    Re: System of PDEs question

    Since the first is de-couple from the other two, I would start here. With the IC give the explicit solution. Then consider the enxt two. Also, since p is constant then the second can be written as

    \left[E+p\right]_t +\left[ \left(E+p\right)u\right]}_x = 0

    Making the second and third PDEs the same. Try that.
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  3. #3
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    Re: System of PDEs question

    May be this helps.
    You may try to solve the first equation (PDE the first order) and go to the second one.
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