Steady state condition and initial-value problem

Question

Assignment:

Q1. Find all the separated solutions of the heat equation u_t = Ku_zz for t > 0,     0< x < π

(satisfying the boundary condition) u_x (0, t) = u_x(π,t) = 0 for t >0.

Q2. Use these separated solutions to write a series solution for the initial value problem posed by the attached pde and the attached boundary conditions, with the initial condition given by u (x, 0) = x for 0< x< π.

Q3. Find the steady state solution for the inital value problem, taking into account the initial condition.

Q4. Show that the series state solution for the initial value problem approaches the steady state solution of the initial value problem as t →∞.

Q5. Give a brief physical interpretation of this limiting behaviour as t →∞.

Provide complete and step by step solution for the question and show calculations and use formulas.