11.2 Heat equation Let u(t, x) satisfy the equation ut(t, x) = 4uxx(t, x) 1, 0 < x < 1, t > 0 with initial condition u(0, x) = 0, 0 < x < 1, and boundary conditions u(t, 0) = 0, u(t, 1) = 0, t 0. This equation describes the temperature in a rod. The rod initially has a temperature of 0c (zero degree Celsius), and is then heated at a uniform rate 1. However, its two endpoints are kept at the temperature of 0c at all times. The unknown function u(t, x) describes the temperature in the rod at time t 0 at the point x [0, 1]. (a). Set up the forward-Euler method. (b). Set up the backward-Euler method. Write out the tri-diagonal system one needs to solve at every time step. 1