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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). $

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