# Separation of variables for wave equation's boundary-value problem

I am hoping to receive some thoughts on an analytical solution of the following equation, $$u_{tt}-u_{xx}=\gamma u,$$ for $$x\in (0,L)$$ and $$t\in (0,\infty)$$, with initial conditions $$u(0,x)=0, \qquad u_t(0,x)=0,$$ and boundary conditions $$u(t,0)=f(t), \qquad u_x(t,L)+cu(t,L)=g(t),$$ where $$\gamma$$ and $$c$$ are known constants.

I have tried separation of variables, $$u=T(t)X(x)$$, but obtain only a trivial solution for $$T$$.

• What did you expect? If there is no initial wave, what would cause one? Jul 21, 2019 at 18:41

Using separation of variables $$u(x,t) = X(x) T(t)$$, we have $$\frac{T''}{T} = \frac{X''}{X} + \gamma = -\lambda \, ,$$ i.e. $${T''} + \lambda T = 0 \qquad\text{and}\qquad {X''} + (\lambda-\gamma) X = 0$$ for which the Fourier series approach could be applied. However, this might not be an easy task in the case where the boundary conditions are time-dependent.
The problem at hand is very similar to those of a one-dimensional string with vibrating ends. A possible strategy would consist in searching the solution as a sum of eigenfunctions -- so-called normal mode decomposition or expansion. To find the normal modes, one solves the same problem with homogeneous boundary conditions ($$f=0$$, $$g=0$$), see e.g. this link and chap. 4 of (1). Note that this post is closely related.
Alternatively, one may use the method of characteristics for the system \begin{aligned} u_t \qquad &= v\\ \varepsilon_t -v_x &= 0\\ v_t - \varepsilon_x &= \gamma u \end{aligned} where $$\varepsilon = u_x$$, but this might not be straightforward due to the right-hand side. You may have a look at (1), chap. 12.