# Assigning Well-Posed Boundary Conditions for the Wave Equation

This is not a homework question. I came across this while studying for an exam and I don't know how to answer it. The question is this:

Consider the slightly modified wave equation $$u_{tt} - u_{xx} + u =0, \,\,\,\,\,\, \text{ for } (x,t) \in (0,1) \times (0,\infty)$$ with smooth initial conditions $u(x,0) = \phi(x)$, $u_t(x,0) = \psi(x)$, $x \in (0,1).$ For what choices of real constants $a,b,c,d$ do the boundary conditions \begin{align*} au_x(0,t) + bu_t(0,t) &= 0, \\ cu_x(1,t) + du_t(1,t) &= 0, \,\,\,\,\,\,\, t > 0 \end{align*} lead to a well-posed equation?

## 1 Answer

The equation you consider is called the Klein-Gordon equation. To solve the given boundary value problem, you need the associated Green's function (see here and here for more information). Since you want to answer questions about which boundary conditions are possible, it's probably the most useful to derive the associated Green's function yourself, e.g. along these lines. Well-posedness of the problem is then equivalent with the existence of a Green's function.

• Thank you so much! I was struggling to find any solid references. – User8128 Mar 22 '16 at 17:45