 Solution

A Fourier sine series automatically meets the zero-temperature boundary conditions because $\sin 0=\sin (n\pi )=0$. Equivalently, extend ${\Theta }_0$ oddly from $(0,L)$ to $(-L,L)$ and then periodically with period $2L$. Its Fourier series has only sine terms, withThe coefficient formula follows from the orthogonality identityFor piecewise smooth initial data the series equals ${\Theta }_0$ at interior continuity points and the average of its one-sided values at interior jumps. More general square-integrable data have convergence in L2 norm convergence. A classical solution continuous at the initial corners additionally requires ${\Theta }_0(0)={\Theta }_0(L)=0$, in agreement with the prescribed boundary conditions.