I'm trying solve this wave equation using Fourier method, but I am stuck...
$${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ u }( x,0) = f(x), { u }_{ t }( x,0) = g(x) $$ I know that first I have to use separation of variables: $${ u }( x,t) = T(t)X(x). $$ Making the calculations $$\frac{T''+ \alpha T}{c^{2}T} = \frac{X''}{X} = -\lambda $$ I guess I'm right at this point? Okay? Now I have to solve: $$X'' + \lambda X = 0,$$ and $$\frac{T'' + \alpha T}{c^{2}T} = -\lambda ,$$ $$ T'' + (\alpha + \lambda c^2)T = 0.$$ I don't now how to solve the second equation and how I add the two equation to solve the first problem.
I will be very grateful for the help!!!!
When you solve the equation for $X$ for $\lambda =\alpha^2>0$, this gives you $$ X=A\sin(\sqrt{\alpha}x)+B\cos(\sqrt{\alpha}x). $$ Applying the initial conditions gives $$ 0=A\sin(\sqrt{\alpha}L) $$ which implies that either $A=0$ or $\sin(\sqrt{\alpha}L)=0$. Note that $A=0$ gives a trivial result so we must have $\sin(\sqrt{\alpha}L)=0,$ ie, $\alpha_nL=n\pi,~n=1,2,3,\cdots$. Corresponding to each $n\in\mathbb{Z}^+,~\exists$ eigenvalue $\lambda_n=\alpha^2=\frac{n^2\pi^2}{L^2},~n=1,2,3,\cdots$. This implies that the solution is $$ X_n(x)=A\sin\left(\frac{n\pi x}{L}\right) $$ were $\lambda_n$ are eigenvalues and the function $B_n\sin(\frac{n\pi x}{L})$ is the corresponding eigenfunction.
With this information, we can go after the equation for $T.$ For given $\lambda_n,$ we have $$ T''_n+\left( \alpha+\frac{n^2\pi^2c^2}{L^2} \right)T=0 $$ which has Auxiliary equation $$ r^2+\alpha+\frac{n^2\pi^2c^2}{L^2}=0 \\ \Rightarrow r^2=\pm i\sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}. $$ Solving this ODE gives $$ T_n(t)=a_n\cos\left( \sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}t \right)+b_n\sin\left( \sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}t \right),~n=1,2,3,\cdots $$ After applying the final initial conditions, we can write the solution $$ u_n(x,t)=X_n(x)T_n(t) $$ and take ay linear combination of the functions $u_n$. Is that what you were looking for?