# Is it possible to solve a hyperbolic moving boundary problem?

J. L. Davies says in his book,

"The basic principle in PDEs is that boundary value problems are associated with elliptic equations while initial value problems, mixed problems, and problems with radiation effects at boundaries are associated with hyperbolic and parabolic equations."

John Crank attests to this in his book saying,

"A free-boundary value problem requires the solution of an elliptic partial differential equation. For a moving boundary problem the equation is of parabolic type."

I might have unterstood these texts wrong, but it seems to imply that I cannot solve for a moving boundary problem in the case of a hyperbolic system. But why? For example, if I ask you to study the refraction of light through a moving slab of glass, then why shouldn't you be able to solve it?

In other words, what is the basic principle in PDE talking about? How do I overcome its limitations in case of hyperbolic boundary value problems?

Let us consider the simplest hyperbolic equation: the linear advection equation $u_t + c u_x = 0$. The initial data is zero and the moving boundary condition is $u(v t,t)=f(t)$. The method of characteristics gives
• $\frac{\text d t}{\text d s} = 1$, letting $t(0)=t_0$ gives $t=s+t_0$.
• $\frac{\text d x}{\text d s} = c$, letting $x(0)=v t_0$ gives $x=cs+vt_0$.
• $\frac{\text d u}{\text d s} = 0$, letting $u(0)=f(t_0)$ gives $u=f(t_0)$.
Therefore, $$u = f\left(\frac{x-ct}{v-c}\right) \qquad\text{if}\qquad v\neq c \, .$$ One can observe that there is a problem if $v=c$, as suggested in the answer by @timur.