Page 446 - Applied Numerical Methods Using MATLAB

GUI OF MATLAB FOR SOLVING PDES: PDETOOL 435
6. In Plot mode, you can change the plot option in the Plot selection dialog
box opened by clicking the button in the tool-bar or ‘Parameters’ in
the Plot pull-down menu (Fig. 9.12g). In the Plot selection dialog box
(Fig. 9.12h), you can set the plot type to, say, Color/Height(3-D) and set
the plot style to, say, interpolated shading and continuous (interpolated)
height. If you want the mesh to be shown in the solution graph, check
the box of Show mesh. In case you want to plot the graph of a known
function, change the option(s) of the Property into ‘user entry’, type in the
MATLAB expression describing the function and click the Plot button. You
can save the plot parameters as the current default by clicking the Done
button. You can also change the color map in the second line from the
bottom of the dialog box.

(cf) We can extract the parameters involved in the domain geometry by clicking ‘Export..’
in the Draw pull-down menu, the parameters specifying the boundary by clicking
‘Export..’ in the Boundary pull-down menu, the parameters specifying the PDE by
clicking ‘Export..’ in the PDE pull-down menu, the parameters specifying the mesh
by clicking ‘Export..’ in the Mesh pull-down menu, the parameters related to the
solution by clicking ‘Export..’ in the Solve pull-down menu, and the parameters
related to the graph by clicking ‘Export..’ in the Plot pull-down menu. Whenever
you want to save what you have worked in PDEtool, you may select File/Save in
the top menu-bar.
(cf) Visit the website “http://www.mathworks.com/access/helpdesk/help/helpdesk.
html” for more details.

9.5.3 Examples of Using PDETOOL to Solve PDEs
In this section, we will make use of PDEtool to solve some PDE problems that
were dealt with in the previous sections.

Example 9.7. Laplace’s Equation: Steady-State Temperature Distribution Over
a Plate. Consider the Laplace’s equation (Example 9.1)

2
2
∂ u(x, y) ∂ u(x, y)
2
∇ u(x, y) = + = 0 for 0 ≤ x ≤ 4, 0 ≤ y ≤ 4
∂x 2 ∂y 2
(E9.7.1)
with the following boundary conditions.
y
y
4
u(0,y) = e − cos y, u(4,y) = e cos 4 − e cos y (E9.7.2)
x
x
4
u(x, 0) = cos x − e , u(x, 4) = e cos x − e cos 4 (E9.7.3)
The procedure for using PDEtool to solve this problem is as follows:
0. Type ‘pdetool’ into the MATLAB command window to have the PDE
toolbox window on the screen. Then, adjust the ranges of the x-axis and