Summary

$$\boxed{ u(x,t) = \sum_{n=1}^{\infty} \Bigg[ \frac{2}{L} \int_0^L ... ...\sin\Big(\frac{n\pi x}{L}\Big) \, \exp\Big[-\Big(\frac{n\pi}{L}\Big)^2 t\Big] }$$ (10)

This is the classical solution of the 1D heat equation on a finite rod with zero Dirichlet boundaries.