An insulated composite rod is formed of two parts arranged end to end, and both halves are of equal length. Part α has thermal conductivity k_α, for 0 ≤ x ≤ 1/2, and part b has thermal conductivity k_b, for 1/2 ≤ x ≤ 1. The nondimensional transient heat conduction equations that describe the temperature u over the length x of the composite rod are

∂²u/∂x² = ∂u/∂t             0 ≤ x ≤ 1/2

r ∂²u/∂x² = ∂u/∂t          1/2 ≤ x ≤ 1

Where u = temperature, x = axial coordinate, t = time, and r = k_α/k_b. The boundary and initial conditions are

Boundary conditions              u(0, l) = 1                                u(1, l) = 1

                                                (∂u/∂x)_α = (∂u/∂x)b                  x = 1/2

Initial conditions                     u(x, 0) = 0                               0 < x < 1

Solve this set of equations for the temperature distribution as a function of time. Use second-order accurate finite-difference analogues for the derivatives with a Crank-Nicolson formulation to integrate in time. Write a computer program for the solution, and select values of ∆x and ∆t for good accuracy. Plot the temperature u versus length x for various values of time t. Generate a separate curve for the following values of the parameter r = 1, 0.1, 0.01, 0.001, and 0.