4.1.8. (a) Find the series solution to the heat equation $u_t = u_{xx}$ on $-2 < x < 2$, $t > 0$, when subject to the Dirichlet boundary conditions $u(t, -2) = u(t, 2) = 0$ and the initial condition
$u(0, x) = \begin{cases} x, & |x| < 1, \\ 0, & \text{otherwise}. \end{cases}$ (b) Sketch a graph of the solution at some representative times. (c) At what rate does the temperature approach thermal equilibrium?