We have the following simple advection equation: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - \frac{\partial p(x,t)}{\partial x} , \quad0<x<1, \quad t>0 \end{equation}
We also have the following initial and boundary conditions: \begin{equation} p(x,0)=0\\ p(x,t)= f(t) \quad at \quad x=0 \\ \frac{\partial p(x,t)}{\partial x}=0 \quad at \quad x=1 \end{equation}
The PDE should satisfy TWO boundary conditions(BCs). It seems that we can get rid of the inhomogeneous BC by defining $q(x,t)=p(x,t)+f(t)$. Which method could I use to get an explicit solution for this PDE?
