Mathemathical model equation PDE I am studying models in DPE an the professor give us this problem:

$\begin{cases}u_t+au_x=f(x,t)\\ u(x,0)=g(x)\end{cases}$

I've studied the transport equation and the Burger's equation. About heat equation, a little. Hoewever, this one is not included in those kinds of problems.
I will be very mercy if someone could help me with indications to this solution.
Thanks.

We can find the characteristics with this system of ODE's:
$\dfrac{dt}{1}=\dfrac{dx}{a}=\dfrac{du}{f(x,t)}$
From the first proportion, $x=c_1+at$. Now:
$\dfrac{dt}{1}=\dfrac{du}{f(at+c_1,t)}$ or
$f(at+c_1,t)dt=du$ Integrating,
$$u(x,t)=\int_0^tf(ar+c_1,r)dr+c_2=\int_0^tf(ar+x-at,r)dr+c_2$$
For the general solution we have to consider $c_1$ and $c_2$ are somehow related: $c_2=h(c_1)$, with $h$ some single variable differentiable function to determine with th initial conditions: $c_2=h(x-at)$:
$$u(x,t)=\int_0^tf(ar+x-at,r)dr+h(x-at)$$
Finally we can impose the i.c. $u(x,0)=g(x)$
$u(x,0)=g(x)=h(x)$ leading to
$$u(x,t)=\int_0^tf(ar+x-at,r)dr+g(x-at)$$
Added
I just finished and see the link in the comments to your post. There is the answer, but it is proposed and checked, not deduced, so I post mine as it shows a way to get the solution using the method of characteristics.
$$u(x,t)=g(x-at)+\int_0^tf(x-a(t-r),r)dr$$
