A rod of length L is heated at a rate that increases linearly with distance along the rod from its insulated end. The other end of the rod is maintained at temperature T₀. The initial rod temperature in T₀. The relevant equations determining the temperature distribution along the rod are thus given by mcT_t = KT_xx + q, where q = q₀ X/L, on 0

and with T_x(0,t) = 0, T(L,t) = T₀ and T(x,0) = T₀

Here c the specific heat, m is the mass per unit rod length, q the heat input rate per unit rod length at location x, and K is the effective rod conductivity,

a) scale the equations appropriately. Comment on the dependence of these scales on the parameters of the problem. Determine the (scaled) steady state solution. where along the rod, is the maximum temperature reached?

b) determine (the scaled) temperature distribution. how long (in unscaled time) does it take for steady state to be reached?

Comment on "diffusive smoothing" in context.