7.4 Potential Equation 419
Figure 5 Regions and mesh numbering for Exercises 5–9.EXERCISES
Set up and solve replacement equations for each of the following problems.
Use symmetry to reduce the number of unknowns.
- ∇^2 u=−1, 0<x<1, 0<y<1,u=0 on the boundary. x= y= 1 /4.
- Same as Exercise 1 with x= y= 1 /8. Compare the solutions.
- ∇^2 u=0, 0<x<1, 0<y<1,u( 0 ,y)=0,u(x, 0 )=0,u( 1 ,y)=y,
u(x, 1 )=x. x= y= 1 /4. - Same as Exercise 3 with x= y= 1 /8.
- The regionRis a square of side 1 from the center of which a similar square
of side 1/7hasbeenremoved;∇^2 u=0inR,u=0 on the outside bound-
ary, andu=1 on the inside boundary; x= y= 1 /7. See Fig. 5. - Same as Exercise 5, but the partial differential equation is∇^2 u=−1, and
the boundary condition isu=0 on all boundaries. See Fig. 5. - The regionRhas the shape of aT, made by removing strips from the cor-
ners of a 1×1 square. The partial differential equation is∇^2 u=−25 inR,
andu=0 on the boundary. Take x= y= 1 /5. See Fig. 5 for numbering
of mesh points. - The region is a rectangle, 2 units wide and 1 unit high. The potential equa-
tionholdsintheinterior;u=1 on the upper half of the boundary (the top