Question

I've been trying to solve this for ages, but get only long and ugly solutions :(
\[f(x) = x^2,\ x \in [0,1]\] \[f(x) = (x-2)^2,\ x \in [1,2]\]
Need fourier series of this piecewise function (how do I write them with one big bracket?) and the solution to the PDE (heat equation)
\[\frac{\partial^2 u}{\partial x^2} (x,t)= \frac{\partial u}{\partial x} (x,t)\]
with \[\frac{\partial u}{\partial x} (0,t)= \frac{\partial u}{\partial x} (2,t) = 0\] and \[u(x,0) = f(x),\ x \in [0,2]\]

Answer 1

I wouldn't be surprised if the solutions here aren't pretty, as the boundary condition function f isn't so elementary in Fourier series terms.