Please solve this question

5. Consider heat flow along a thin, insulated bar of length L, density 4 and specific heat 0.01, modelled by the 1D heat equation OT at for 0 0, (6) where T(x, t) is the temperature. Let the boundary conditions be T. (0, t) = T(L, t) = 0 for t > 0. (7) (a) Find the thermal conductivity of the bar. (b) Substituting the separation solution T(x, t) = F(x) G(t) into the heat equa- tion (6), and rearranging, gives F" F = k, (8) where k is a separation constant. Determine a Sturm-Liouville problem for F(x). (c) Find the eigenvalues k = kn and eigenfunctions F(x) = Fn(x) (n = 1, 2, ...) of the Sturm-Liouville problem. You should consider all possible values of k. (d) Find corresponding functions G(t) = Gn(t) (n = 1, 2, ...). (e) Find a series representation for T(x, y) that satisfies the heat equation and the homogeneous BC's. (f) Find the temperature T in the limit t -+ co