Question

Solve the initial/moving-boundary problem

u_xx-u_tt = 0 , 0 < x < infinity, 0 < t < 2x

u(x,0) =u₀(x) , 0 <= x < infinity

u_t(x,0)=v₀(x) , 0 <= x < infinity

u(x,2x)=h(x) , x >= 0

where u₀(x), v₀(x), h(x) are twice continuously differentiable on the domain

Answer 1

solution of the one-dimensional wave equation $c^2u_{xx}-u_{tt} = 0$ is

$u(x,t)=\phi(x+ct)+\psi(x-ct)$

we have $c=1$

then:

$u(x,t)=\phi(x+t)+\psi(x-t)$

$u(x,0)=\phi(x)+\psi(x)=u_0(x)$

$u_t(x,0)=\phi'(x)-\psi'(x)=v_0(x)$

$u(x,2x)=\phi(3x)+\psi(-x)=h(x)$