Separation of variable of PDE

Question

For any $u_0,u_1\in L^2(0,\pi)$ and $f\in L^2((0,\pi)\times(0,+\infty))$ find using separation of variables and Fourier series a formal explicit expression of the solution of the problem

$u_{tt}-u_{xx}=f(x,t)$ for $(x,t)\in(0,\pi)\times(0,+\infty),$

$u(x,0)=u_0(x),u_t(x,0)=u_1(x)$ for any $x\in(0,\pi)$

$u(0,t)=u(\pi,t)=0,$ for any $t\in(0,+\infty)$

I do easily when it deals with first derivative, this problem bothering me but this is very important for me for qualifying so please could you solve it for me...

Answer 1

1. Write down eigenvalues and eigenfunctions solving the Sturm-Liouville problem $\{LX=\lambda X, \;X\in D_L\}$ for a self-adjoint ordinary differential operator $$ \begin{align*} L=\frac{d\,}{dx^2}\colon D_L=\{v\in H^2(0,\pi)\colon v(0)=v(\pi)=0\}\subset L^2(0,\pi)\to L^2(0,\pi)\}\\ X_n=\sin{(nx)},\;\;\lambda_n=-n^2,\;\;n\geqslant 1. \end{align*} $$

2. Expanding the desired solution $u\,$ w.r.t. the orthogonal basis $\{X_n\}$ in $L^2(0,\pi)$ construct solution in the form of Fourier series $$ u(x,t)=\sum_{n=1}^{\infty}T_n(t)\sin{(nx)},\quad T_n(t)=\frac{(u,X_n)}{\|X_n\|^2}= \frac{2}{\pi}\int\limits_0^{\pi}u(x,t)\sin{(nx)}\,dx,\;\;n\geqslant 1, $$ with notation $(\cdot,\cdot)$ standing for the inner product in $L^2(0,\pi)$. Fourier coefficients $T_n=T_n(t)$ are to be found by solving Cauchy problems $$ \begin{align*} T_n''-\lambda_n T_n=f_n(t)=\frac{(f,X_n)}{\|X_n\|^2}= \frac{2}{\pi}\int\limits_0^{\pi}f(x,t)\sin{(nx)}\,dx,\\ T_n(0)=\alpha_n\overset{\rm def}{=}\frac{(u_0,X_n)}{\|X_n\|^2}= \frac{2}{\pi}\int\limits_0^{\pi}u_0(x)\sin{(nx)}\,dx,\\ T'_n(0)=\beta_n\overset{\rm def}{=}\frac{(u_1,X_n)}{\|X_n\|^2}= \frac{2}{\pi}\int\limits_0^{\pi}u_1(x)\sin{(nx)}\,dx.\end{align*} $$ To obtain ODE $\,T''_n+n^2T_n=f_n$, it suffices to multiply PDE $\,u_{tt}-u_{xx}=f$ by an eigenfunction $X_n\,$ w.r.t. the inner product $(\cdot,\cdot)$, and employ the fact that operator $L$ is self-adjoint, i.e., $\,(Lu,X_n)=(u,LX_n)=\lambda_n(u,X_n)=-n^2(u,X_n)$.

3. Solving the Cauchy problems for Fourier coefficients $T_n$ we find $$ T_n(t)=\alpha_n \cos{(nt)}+\frac{\beta_n}{n}\sin{(nt)}+ \frac{1}{n}\int\limits_0^t f_n(s)\sin{[n(t-s)]}\,ds,\;\;n\geqslant 1, $$ where the first two terms represent a solution of homogeneous equation with Cauchy data $\alpha_n,\beta_n\,$, while the last term represents a solution of equation corresponding to the RHS $\,f_n=f_n(t)\,$ with homogeneous Cauchy data.