System of PDEs question

**mel240** · September 12th 2011, 02:45 AM

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?

**Danny** · September 12th 2011, 03:22 AM

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.

**zzzoak** · September 12th 2011, 11:38 AM

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

Thread: System of PDEs question

LinkBack

Thread Tools

Display

System of PDEs question

Re: System of PDEs question

Re: System of PDEs question

Similar Math Help Forum Discussions

Unconditionally stable scheme for solving first order system of PDEs

Harmonic functions - pdes

Quick PDEs Question! BVPs: time-dependent heat gen in a bar

Hilbert Methods for PDEs

PDEs

Search Tags