partial differential problem with boundary conditions so I have got this problem and I have no idea how to solve it. please help, or give more guidelines. u=u(x,t)
a robin problem for the wave equation in a quartet plane
1.for any alpha that does not equal 0, find a solution to this robin problem for the wave equation in quarter plane.
2.what are the conditions for f(x) and g(x), so the solution would be real?
3.is there only one specific solution?
guiding given:
-for dirichlet boundary conditions we should require u would be odd in x
-for neumann boundary conditions we should require du/dx is odd in x(that means u is even in x)
-hint: what should we require for the given robin boundary condition?
any solution would be good, even if you don't use the guiding
 A: We will seek the solution in the form
$$
u(x,t) = q_1(x - c t) + q_2(x + c t), \quad (x,t) \in [0,\infty)\times[0,\infty). \quad (1)
$$
We need define $q_1(x)$ for any real $x$ and $q_2$ for any nonnegative real $x$.
After substituting (1) in the Cauchy conditions we obtain
$$
q_1(x) + q_2(x) = f(x),\quad -c q_1'(x) + c q_2'(x) = g(x),\quad x \in [0,\infty).
$$
After solving it we get
$$
q_1(x) = \frac{f(x)}{2} - \frac{1}{2 c} \int\limits_0^x g(\xi) d\xi - C,\quad x \in [0,\infty). \\
q_2(x) = \frac{f(x)}{2} + \frac{1}{2 c} \int\limits_0^x g(\xi) d\xi + C,\quad x \in [0,\infty).
$$
where $C$ is an arbitrary real constant.
In order to define $q_1(x)$ for negative $x$ we will use the boundary condition. After substituting (1) in the boundary condition we obtain
$$
\alpha  \left(q_1(-c t)+q_2(c t)\right)+q_1'(-c t)+q_2'(c t) = 0, t \in [0,\infty).
$$
Now we have ODE for $q_1$, when its argument is negative
$$
\alpha  \left(q_1(z)+q_2(-z)\right)+q_1'(z)+q_2'(-z) = 0, z \in (-\infty,0].
$$
Solve it
$$
q_1(z) = \exp(-z \alpha) \left(C_2 - \int\limits_{0}^{z} \exp(\alpha \xi) \left(\alpha q_2(-\xi)+q_2'(-\xi)\right) d\xi \right), z \in (-\infty,0],
$$
where $C_2$ is an arbitrary real constant.
Now, collect all and write the solution
$$
u(x, t) = \frac{f(x-c t) + f(x-c t)}{2} + \frac{1}{2 c} \int\limits_{x- c t}^{x + c t} g(\xi) d\xi,\quad x \geqslant 0, t \geqslant 0, x - c t \geqslant 0,\\
u(x, t) = (C + C_2) \exp(\alpha 
   (c t-x))+\frac{f(x + c
   t)}{2} +\frac{1}{2 c} \int\limits_0^{x + c t} g(\xi) d\xi - \frac{\exp({\alpha  (c t-x)})}{2 c} \times \\
\times \int\limits_0^{x-c t}
   \exp(\alpha \xi) \left(c
   f'(- \xi)+\alpha c f(- \xi)+\alpha 
   \int\limits_0^{-\xi} g(\tau) d\tau
   +g(-\xi)\right) \, d \xi,\\
\quad x \geqslant 0, t \geqslant 0, x - c t \leqslant 0.
$$
We see that the solution $u$ depends on an arbitrary constant $C + C_2$, but it will be continuous if and only if $-C-C_2+\dfrac{f(0)}{2} = 0$.
The question 'is there only one specific solution?' is incorrect, because the class (or space), where we should find the solution, is not specified.
3.1) There are no two or more solutions from the class, which with its partial derivatives have the same discontinuities on the characteristic $x-at=0$.
3.2) This problem cannot have more that one classical solution.
