Page 447 - Applied Numerical Methods Using MATLAB

436 PARTIAL DIFFERENTIAL EQUATIONS
the y-axis to [0 5] and [0 5], respectively, in the dialog box opened by
clicking ‘Axes Limits’ in the Options pull-down menu. You can also click
‘Axes Equal’ in the Options pull-down menu to have the x axis and
the y axis of equal scale so that a circle/square may not look like an
ellipse/rectangle.
1. Click the button in the tool-bar and click-and-drag on the graphic
region to create a rectangle of domain. Then, in the Object dialog box
opened by double-clicking the rectangle, set the Left/Bottom/Width/Height
to 0/0/4/4. In this case, you don’t have to construct a CSG model by
editing the set formula, because the domain consists of a single object:
a rectangle.
2. Click the button in the tool-bar and double-click each boundary seg-
ment to specify the boundary condition as Eqs. (E9.7.2,3) in the boundary
condition dialog box (see Fig. 9.13a).
3. Open the PDE specification dialog box by clicking the PDE button in the
tool-bar, check the box on the left of Elliptic as the type of PDE, and set
its parameters in Eq. (E9.7.1) as depicted in Fig. 9.13b.
4. Click the button in the tool-bar to divide the domain into a number of
triangular subdomains to get the triangular mesh as depicted in Fig. 9.13c.
You can click the button in the tool-bar to refine the mesh successively
for better accuracy.
5. Click the = button in the tool-bar to plot the solution in the form of two-
dimensional graph with the value of u(x, y) shown in color.
6. If you want to plot the solution in the form of a three-dimensional graph
with the value of u(x, y) shown in height as well as color, check the box
before Height on the far-left side of the Plot selection dialog box opened
by clicking the button in the tool-bar. If you want the mesh shown in
the solution plot as Fig. 9.13d, check the box before Show mesh on the
far-left and low side and click the Plot button at the bottom of the Plot
selection dialog box (Fig. 9.12h). You can compare the result with that of
Example 9.3 depicted in Fig. 9.4.
7. If you have the true analytical solution

y
x
u(x, y) = e cos x − e cos y (E9.7.4)
and you want to plot the difference between the PDEtool (FEM) solution
and the true analytical solution, change the entry ‘u’ into ‘user entry’ in the
Color/Contour row and the Height row of the Property column and write
‘u-(exp(y).*cos(x)-exp(x).*cos(y))’ into the corresponding fields in
the User entry column of the Plot selection dialog box opened by clicking
the button in the tool-bar and click the Plot button at the bottom of the
dialog box.