Given:

$$ u_{tt} - c^2 u_{xx} = 0 $$(1) for $ 0<x<\infty $

$$ u |_{t=0} = g(x) , u_t | _{t=0} = h(x) $$(2) for $0<x$, and additionally

$$ (\alpha u + \beta u_t) |_{x=0} = q(t) $$(3) for $ t> 0 $

We are asked to evaluate and find the general solution for both regions $ x > ct $ and $0<x<ct$

Given that the most general solution under the first condition is $$u(x,t) = \phi (x+ct) + \psi(x-ct) $$(I), we will focus on the latter case outlined, i.e. $x<ct$

We know from the definitions that:

$$\phi(x) = \frac{1}{2} g(x) + \frac{1}{2c} \int_{0}^{x} h(x')dx' $$(4) and

$$\psi(x) = \frac{1}{2} g(x) - \frac{1}{2c} \int_{0}^{x} h(x')dx' $$(5)

We can begin examining our boundary conditions. As usual, the particular issue is that $\psi(x-ct)$ is a problem in this region, as the values might be negative, while $\phi(x+ct)$ will not be

Jumping straight into (3), we apply it to (I), to get:

$$ q(t) = \alpha ( \phi(ct) + \psi(-ct) ) + \beta c (\phi(ct) ' - \psi(-ct) ' ) $$(6) and applying the relation $ x = -ct $ to (6):

$$ q(\frac{-x}{c}) = \alpha(\phi(-x) + \psi(x) ) + \beta c (\phi(-x)' - \psi(x)') $$(6')

From here, my steps get a bit more uncertain, where I take the (total) derivative of (4) to be :

$$\phi(x) ' = \frac{1}{2} g'(x) + \frac{1}{2c} (h(x) - h(0) ) + \int_{0}^{x} h'(x')dx' $$(7) using Leibnitz's Rule of Integration

plugging into (6') gives me:

$$ q(\frac{-x}{c}) = \alpha( \frac{1}{2}g(-x) - \frac{1}{2c}\int_{0}^{-x} h(x')dx' + \psi(x)) + \beta c (\frac{1}{2} g'(-x) + \frac{1}{2c} (h(-x) - h(0) ) + \int_{0}^{-x} h'(x')dx - \psi(x)') $$(

NOTE:

I originally misread the example, as I meant to talk about example 4, but I have modified the Robin condition such that it involves $\alpha u $ and $\beta u_t $ as opposed to the original wording of the problem.

This leaves me two questions:

1. Is this the correct procedure to get the conclusion outlined in the example?

2. What steps do I need to take to get the conclusion outlined (an expression for $\psi $ only in terms of functions of $q,\psi', \phi' $)?

3. Is this procedure the correct one, when considering boundary conditions?