<s>[INST] Wave problem $u_{tt}=u_{xx}-u_t, 0\le x\le 2\pi$ with $u(x,0)=\phi(x)$, $u_t(x,0)=\psi(x)$ and $u(0,t)=u(2\pi,t)=0$ Solve the wave problem $$\begin{cases}u_{tt}=u_{xx}-u_t, 0\le x\le 2\pi\\ u(0,t)=u(2\pi,t)=0\\ u(x,0)=\phi(x)\\ u_t(x,0)=\psi(x)\end{cases}$$ Write the eigenvalues and eigenfunctions explicitly.
Are they orthogonal?  
We propose the separated solution $u(x,t)=X(x)T(t)$.
Then $\frac{(T''+T')}{T}=\frac{X''}{X}=\lambda$. 
If we define $\lambda=\beta^2,$  we have $X(x)=C\cos\beta x+D\sin\beta x$ and $T(t)=Ae^{r_1t}+Be^{r_2t}, r=-1/2\pm\frac{\sqrt{1+4\beta^2}}{2}$. 
Applying boundary condition, we find that $X_n(x)=D_n\sin(\lambda_nx),\lambda=\sqrt{n/2}$
Applying initial conditions we find that $T(0)=A+B=\phi(x)$ and $T'(0)=Ar_1+Br_2=\psi(x)$
My question here is how will I find $A$ and $B$ if we don't know $\phi$ nor $\psi$ ?
How will I find the eigenvalues ?
Help me please
 [/INST] The boundary conditions give
$$ X(x) = \sin\left(\frac{n}{2}x\right) $$
As usual, we drop the remaining constant for now, since it'll absorbed into the constants of $T(t)$
This means $\lambda_n = X''/X = -n^2/4 = -\beta^2$ (you have a sign error). The other equation becomes
$$ T'' + T' + \frac{n^2}{4}T = 0 $$
The two roots of the characteristic are
$$ r = \frac{-1 \pm \sqrt{1-n^2}}{2} $$
Since $n \ge 1$, we rewrite
$$ r = \frac{-1 \pm \sqrt{-(n^2-1)}}{2} = \frac{-1 \pm i\sqrt{n^2-1}}{2} $$
This gives the solution
$$ T(t) = e^{-t/2}\left[A\cos\left(\frac{\sqrt{n^2-1}}{2}t\right) + B\sin\left(\frac{\sqrt{n^2-1}}{2}t\right) \right] $$
For ease of notation, let $\mu_n = \sqrt{n^2-1}$ to make things shorter, so we have
\begin{align} T(t) &= e^{-t/2}\left[A\cos\left(\frac{\mu}{2}t\right) + B\sin\left(\frac{\mu}{2}t\right) \right] \\
T'(t) &= e^{-t/2}\left[\left(-\frac{A}{2} + \frac{\mu B}{2}\right)\cos\left(\frac{\mu}{2}t\right) + \left(-\frac{B}{2} - \frac{\mu A}{2} \right)\sin \left(\frac{\mu}{2}t\right) \right] \end{align}
As usual, the full solution in series form is
$$ u(x,t) = \sum_n X_n(x)T_n(t) = \sum_{n=1}^\infty \sin\left(\frac{n}{2}x\right) e^{-t/2}\left[A\cos\left(\frac{\mu}{2}t\right) + B\sin\left(\frac{\mu}{2}t\right) \right] $$
We can now plug in the given initial conditions
\begin{align} u(x,0) &= \sum_n X_n(x)T_n(0) = \sum_{n=1}^\infty B_n \sin\left(\frac{n}{2}x\right) = \phi(x) \\
u_t(x,0) &= \sum_n X_n(x){T_n}'(0) = \sum_{n=1}^\infty \left(-\frac{A_n}{2} + \frac{\mu B_n}{2}\right) \sin\left(\frac{n}{2}x\right) = \psi(x)
\end{align} 
You'll need to find the Fourier series of these to functions to obtain the coefficients
\begin{align}
B_n &= \frac{1}{\pi}\int_0^{2\pi} \phi(x) \sin\left(\frac{n}{2}x\right) dx \\
\frac{-A_n + \mu B_n}{2} &= \frac{1}{\pi} \int_0^{2\pi} \psi(x) \sin\left(\frac{n}{2}x\right) dx
\end{align}
and solve the resulting system of equations. There's no specific form for the solution here, but you'll know what to do when you are given specific $\phi(x)$ and $\psi(x)$, i.e. just do the integrals.
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