A Non-homogenous convection equation

[member 428835]

Does anyone know if there are solutions to the following PDE

$$u_t + u_x = f(x,t)$$

If not in a general context, what if $f(x,t) = \delta(x-at)/(x-at)$? Please let me know if you have any information.

Method of characteristics?

 

[fresh_42]

I would substitute $\dfrac{x-at}{\varepsilon} =:y$ and with $\dfrac{dy}{dx}\; , \;\dfrac{dy}{dt}$ known, there should be a way to solve it. Just an idea.

 

[member 428835]

I would substitute $\dfrac{x-at}{\varepsilon} =:y$ and with $\dfrac{dy}{dx}\; , \;\dfrac{dy}{dt}$ known, there should be a way to solve it. Just an idea.

A good idea, but this just implies any function of the form $g(x-at)$ is a homogenous solution. How do you deal with the inhomogenous RHS?

 

[bigfooted]

Well the method of characteristics gives you
$$ \frac{dt}{ds}=1 \rightarrow t(r,s)=s+C_1(r)$$
$$\frac{dx}{ds}=1 \rightarrow x(r,s)=s+ C_2(r)$$
$$\frac{du}{ds}=f(x,t) \rightarrow u(r,s)=\int f(s+C_2(r),s+C_1(r)) ds + C_3(r)$$

So if you know the boundary conditions you can substitute them and try to solve all ode's explicitly. If the BC is $$u(x,0)=\phi(x)$$ your BC's can be written as
$$x(r,0) = r \rightarrow C_2(r) = r $$
$$t(r,0) = 0 \rightarrow C_1(r) = 0$$
$$u(r,0) = \phi(r) \rightarrow C_3(r) = \phi(r) - \int f(r,0)ds$$

and you can substitute $s=t$ and $r=x-t$ in the equation for $u$. Hope this helps. I have used the notation from:
https://web.stanford.edu/class/math220a/handouts/firstorder.pdf