The solution of $$T' \lambda D T = 0$$ with eigenvalue $$\displaystyle{\lambda_n = \left(\frac{n \pi}{L}\right)^2}$$ results in the eigenfunctions

If $$\displaystyle{u(x,t) = {a_0}/{2} \sum_{n=1}^\infty a_n \cos{(n\pi x/L)} \exp{(-n^2\pi^2 D t/L^2)}}$$ and $$u(x,0) = f(x)$$, the general formula for $$a_n$$ is given by

Suppose that a pipe with open ends has an initial dye concentration centered at one-quarter of the pipe's length. The long-time $$t>>L^2/D$$ solution for the concentration is given by