# Solving a simple differential equation

I have an equation which I wish to solve

$\dfrac{\partial\rho(x,t)}{\partial t}=-\dfrac{\partial\left(\rho(x,t)\bar{v}(x,t)\right)}{\partial x}$

where I know the $\rho(x,t)$ and want to find $\bar{v}(x,t)$, but I do not know what boundary conditions to impose on $\bar{v}(x,t)$. Is there a way I can still find a solution for $\bar{v}$?

rho is defined as $\rho(x,t)=\sqrt{\frac{1}{2\pi}}\dfrac{\Gamma\left(1+\dfrac{\kappa_T(t)}{\kappa}\right)}{\Gamma\left(0.5+\dfrac{\kappa_T(t)}{\kappa}\right)}\left(1+\kappa_T(t)\dfrac{x^2}{2}\right)^{-1-\frac{\kappa}{\kappa_T(t)}}$

with $\kappa_T(t)=A+B\sin(\omega t)$.

So far I was thinking of trying to numerically integrate $\int_{-\infty}^x\dfrac{\partial\rho}{\partial t}dx'$ (assuming $\rho\bar{v}=0$ at $x=-\infty$) to get $-\rho(x,t)\bar{v}(x,t)$, and maybe just tabulate a load of values like this. I haven't actually been able to properly implement this so far, but also a better method would be useful.

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Is this a question about Mathematica? –  Szabolcs Mar 25 at 15:04
Please show what you tried already. And it would probably help if you showed what $rho$ is. –  murray Mar 25 at 15:05
Regarding the boundary conditions: this is the continuity equation, so it probably came from physics. You can come up with a boundary condition based on the physical interpretation of the problem. –  Szabolcs Mar 25 at 15:09
OK yeah sorry I didn't give enough information before. It is a continuity equation so $\rho$ goes to zero at x = +- Infinity, but actually it's not completely obvious what the average velocity does. My feeling is that it gets larger for large |x| (but oscillates). –  user12244 Mar 25 at 15:23
This is a continuity equation in a velocity field. You are giving a profile for rho and wonder what the velocity field is. So you are observing changes in the density, and you know that the flow is moving the medium (that rho describes), and the question is how the flow looks like that leads to these changes. This is not a mathematica question. Yet, I wonder, is the solution unique? Can you prove that? –  zorank Mar 25 at 18:24