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I am trying to solve the following PDE. I suspect it is a transport equation but I am not sure.

$$ \partial_t u(x,t) + \partial_x\left[f(x,t)u(x,t)\right] = -\left(\dfrac{\phi_1}{f(x,t)} + \dfrac{\phi_2}{g(x,t)}\right) $$

where $\partial_t u(x,t)$ denotes the partial derivative with respect to time and $\partial_x$ represents the derivative with respect to $x$. I am trying to solve for the function $u(x,t)$. $\phi_1$ and $\phi_2$ are just parameters.

How can I solve this equation analytically? Does it have a name?

P.S.: I am thinking of using the following initial condition.

$$u(x,0) = \dfrac{c}{f(x,0)}exp\left(-\int \left(\dfrac{\phi_1}{f(x,0)}+\dfrac{\phi_2}{g(x,0)}\right)dx\right)$$

If this is not a possibility, can somebody illustrate how this would work with a simpler initial condition or provide a sketch of the general solution?

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  • $\begingroup$ What are $f(x,t),g(x,t)$? What are the conditions on them? $\endgroup$ – Yuriy S Oct 17 '16 at 14:36
  • $\begingroup$ Do you need the general solution, since there are no initial/boundary conditions listed? $\endgroup$ – Yuriy S Oct 17 '16 at 14:38
  • $\begingroup$ @YuriyS those are functions of time $t$ and "space" $x$. I have not thought on any conditions on them really. $\endgroup$ – Sophie Oct 17 '16 at 14:47
  • $\begingroup$ @YuriyS I do have an initial condition. It would be that $u(x,0) = \dfrac{u(0,0)f(0,0)}{f(x,0)}exp\left(-\int \left(\dfrac{\phi_1}{f(x)}+\dfrac{\phi_2}{g(x)}\right)dx\right)$. And $u(0,0)*f(0,0)=c$. Do you have any suggestions? $\endgroup$ – Sophie Oct 17 '16 at 14:52

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