Create a series solution for the initial value problem

Question

Response to the following question:

1.(a) Find all the separated solutions of the heat equation

u_t = Ku_xx for t > 0, 0 < x < π

satisfying the boundary conditions

u_x(0, t) = u_x (π, t) = 0 for t > 0.

(b)    Use these separated solutions to write a series solution for the initial value problem posed by the above pde and the above boundary conditions, with the initial condition given by

 u(x, 0) = x for 0 < x < π.

(c)    Find the steady state solution for the initial value problem, taking into account the initial condition.

(d)    Show that the series solution of the initial value problem approaches the steady state solution of the initial value problem as t  → ∞.

(e)    Give a brief physical interpretation of this limiting behavior as t →∞.