wave equation PDE

Question

Solve the initial boundary value problem

\begin{eqnarray} u_{tt} &=& u_{xx} + u_t ~~\mbox{in}~~ (0, \ell) \times (0, \infty), \\ u(0, t) &=& u(\ell, t) ~~\mbox{on}~~ (0, \infty), \\ u(x, 0) &=& \sin x, ~ u_t(x, 0) = \cos x ~~\mbox{on}~~ (0, \ell). \end{eqnarray}

My try :

I directly applied separation of variables, I wasnt able to get the transformation so tat it can be converted to $$w_{tt} = w_{xx}$$ and i solved Time second degree ODE using characteristic equation. I used Boundary conditions , but when i try to implement intital i am not sure how to implement the last condition.

My form so far after applying BCs : $$u(x,t)=\sum_{n=1}^{\infty}\sin(n\pi x/l)[c_1 e^{\frac t2(1+\sqrt{1-4(\frac{n\pi}{l})^2})} + c_2 e^{\frac t2(1-\sqrt{1-4(\frac{n\pi}{l})^2})}]$$

Answer 1

So far you have

$$ u(x, t) = \sum_n \sin\left(\frac{n\pi x}{l}\right)\left[ A_n e^{\lambda_+ t} + B_n e^{\lambda_- t} \right] \tag{1} $$

where

$$ \lambda_\pm = \frac{1}{2}\left[1 \pm \sqrt{1 - \left(\frac{2n\pi}{l}\right)^2}\right] $$

Now consider the initial conditions

$u(x, 0)$

Replacing that in (1) you get

$$ \sin x = \sum_n \sin\left(\frac{n\pi x}{l}\right) [A_n + B_n] $$

and from here

\begin{eqnarray} \int_0^l {\rm d}x~ \sin\left(\frac{m\pi x}{l}\right) \sin x &=& \sum_n [A_n + B_n] \int_0^l {\rm d}x~\sin\left(\frac{n\pi x}{l}\right) \sin\left(\frac{m\pi x}{l}\right) \\ &=& \sum_{n}[A_n + B_n] \left(\frac{l}{2}\delta_{mn}\right) \\ &=& \frac{l}{2}(A_m + B_m) \\ &=& \frac{l m \pi}{l^2 - m^2\pi^2} \cos(m\pi)\sin l \\ &=& (-1)^m\frac{l m \pi}{l^2 - m^2\pi^2} \sin l \end{eqnarray}

So in summary

$$ A_m + B_m = (-1)^m\frac{2 m \pi}{l^2 - m^2\pi^2} \sin l \tag{2} $$

$u_t(x, 0)$

I will leave this one to you, you should get something like

$$ \lambda_+ A_m + \lambda_- B_m = \cdots \tag{3} $$

Equations (2) and (3) can be solved to find $A_n$ and $B_n$