I am relatively new to the domain of PDE and enjoying at the moment.
Now, I am trying to study domain of influence for the follwoing 1D homogenous wave equation:
$\partial_{t}^{2}u-\partial_{x}^{2}u = 0$ in the domain $t>x>0$ with boundary condition at diagonal $u(t, t) = \phi(t)$, and $\partial_{x}u(t,0)=\psi(t)$ given that $t \geq 0$.
With a general solution in form of $F(x-t)+G(x+t)$, I found the solution in terms of $\psi$ and $\phi$ as follows: $u(t,x) = \phi(\frac{t-x}{2})-\phi(0)+\phi(\frac{t+x}{2})-\int_{0}^{t-x}\psi(s) ds$.
Now, assume that $\phi$ and $\psi$ are compactly supported over [0,1]. However, somehow when I tried to compute the domain of influence and due to the integral above, I am keep getting the entire upper triangle of first quadrant (i.e. entire domain where solution is defined). My logic is as follows: in order to find the domain where perturbation at $x_{0} \in D$ where $D$ is domain, I need to check when does the range of integral over $\psi$ contains some value on the support [0,1] (since domain of influence will be present when compact support [0,1] intersects with [0, t-x] there).
My computation above keeps giving me the suspicious answer depicted above. Can anyone clarify / verify how to compute domain of influence and what error I am making if I am doing something wrong? Thanks!
Your original formula is incorrect. It is rather: $$ u(t,x) = \frac{\phi(x-t)+\phi(x+t)}{2}+\frac12\int_{x-t}^{x+t}\psi(s)ds $$ You can check this by plugging it into the equation. Therefore to find $u(t,x)$, you need to know $\phi$ at the boundary of the light cone and $\psi$ in its interior.
Hope this helps.