The temperature T of a rod of finite length L=1 is governed by the heat equation aT (H) at Oz2 ' subject to the boundary conditions T(0,t)=12, T(1,t)=1 and initial condition T(x,0)=x^2+1. (a) Discuss the physical meaning of the two boundary conditions. (b) Assume T(x,t)=v(x.t)+w(x) and convert (H) into a partial differential equation for v such that v(0,t)=v(1,t)=0 and an ODE for w. What are the boundary conditions for w? (c) Solve the ODE for w. (d) The general solution for v is given as v(x, t) = Bn sin(nux)e (nx)'t n=1 for arbitrary constants Bn. (i) Determine the arbitrary constants Bn. (ii) Write down the unique solution for T to the heat equation (H). (iii) What is the equilibrium temperature T of the rod as too