Using the parallelogram identity, I need to solve the following initial boundary value problem for a vibrating semi-infinite string with a nonhomogeneous boundary condition:
$ u_{tt} − u_{xx} = 0 , \ 0 < x < \infty, t > 0 $ $u(0,t) = h(t)$ $u(x,0) = f(x), \ u_{t}(x,0) = g(x)$ where $f, g, h ∈ C_2\{[0, ∞)\}$
I really have try to solve it, be I still dont know how to use the parallelogram identity. Thanks for your help.
Edit: The parallelogram identity is
$u(x_0 − a, t_0 − b) + u(x_0 + a, t_0 + b) = u(x_0 − b, t_0 − a) + u(x_0 + b, t_0 + a). $