Please, can anyone give me some examples of the exact solution of the following Pde with the corresponding functions f and g. I need these to know if my code of an approximation scheme is correct or not.
$$\left\{\begin{array}{ll} \partial_{t} u+ \partial_{x} u=f(x,t), & t \in(0, 1), x \in(-1, 1) \\ u(0, x)=g(x), & x \in(-1, 1) \\ \frac{\partial u}{\partial n}=0, & x=1,x=-1,t \in(0, 1) \end{array}\right.$$
For $f=0$, the exact solution will be $u(x,t)=g(x−t)$, my problem is that I can't see why this solution should necessarily verify the Neumann boundary conditions.