4. (a) Write down the fundamental solution of the diffusion equation ut kur, on Rx (0, ). U = You do not need to give any justification for your answer. (2 points) (b) Use the method of reflection to find a solution for the diffusion equation on the half line [0, ) with a Neumann boundary condition in 0: U t = kux, on (0, ) x (0, ) u(,0) = 6 on [0, ) (1) ux (0,.) = 0 on (0, ). (4 points) (c) Use the energy method to show that there is at most one solution of (1) with u(x, t) 0 for x. (4 points)