How can I get to the solution of the following IBVP in PDE? I have followed the solution in this document: http://www.math.fsu.edu/~bellenot/class/f09/fun/ft.pdf
But we have gotten a different solution in class using boundary conditions. I would like to bridge the gap. Thanks.
Problem: $$ku_{xx}=u_t, \ \ 0<x<L, \ \ t>0 \\ u(0,t)=0, u(L,t)=0 \\ u(x,0)=f(x)$$
Solution in class: $$u(x,t) = \sum_{n=1}^{\infty}c_n\exp\left(-\left(\frac{n\pi}{L}\right)^2kt\right)\sin\left(\frac{n\pi x}{L}\right), \\ c_n=\frac{2}{L}\int_{0}^{L}f(x)\sin\frac{n\pi x}{L}dx$$
Never mind, I found the answer here: http://www.acad.polyu.edu.hk/~mackchan/Teaching0405/ama202/PDE.pdf