Chapter 32, Problem 16P

Solve the nondimensional transient heat conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate. The governing equation is

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=\frac{\partial u}{\partial t}$

where $u=$ temperature, x and y are spatial coordinates, and $t=$ time. The boundary and initial conditions are

Boundary conditions	$u\left|x,0,t\right|=0$ $u\left|0,y,t\right|=0$	$u\left|x,0,t\right|=1$ $u\left|1,y,t\right|=1$
Initial conditions	$u\left|x,y,0\right|=0$	$0\le x<10\le y<1$

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.