# Thread: System of PDEs question

1. ## System of PDEs question

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

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

where $\displaystyle u=u(x,t) , \rho=\rho(x,t) , E= E(x,t)$ 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

$\displaystyle u(x,0) = x$
$\displaystyle \rho(x,0) = (x-1)^2$
$\displaystyle E(x,0) = E_0 (x)$ where $\displaystyle E_0 (x)$ 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?

2. ## 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

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

Making the second and third PDEs the same. Try that.

3. ## 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.