Solve the nondimensional transient heal conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate. The governing equation is
∂2u/∂x2 + ∂2u/∂y2 = ∂u/∂t
Where u = temperature, x and y are spatial coordinates, and t = time. The boundary and initial conditions are
Boundary conditions u(x, 0, l) = 0 u(x, 1, l) = 1
u(0, y, l) = 0 u(1, y, l) = 1
Initial conditions u(x, y, 0) = 0 0 ≤ x < 1 0 ≤ y < 1
Solve using the alternating direction-implicit technique. Write a computer program to implement the solution. Plot the results using a three-dimensional plotting routine where the horizontal plan contains the x and y axes and the z axis is the dependent variable u. Construct several plots at various times, including the following:
(a) The initial conditions;
(b) One intermediate time, approximately halfway to steady state; and
(c) The steady-state condition.