7.5 Two-Dimensional Problems 427
Figure 9 Regions for Exercises 1–3.- Same as Exercise 2, except that the region is a cross. (See Fig. 9c.)
- Same as Eqs. (9)–(11), except that the boundary condition isu=1onthe
bottom(y= 0 )andu=0 elsewhere. (See Fig. 7.) - ∇^2 u=
∂u
∂t,0<x<1,^0 <y<1,^0 <t,
u( 0 ,y,t)=0, u( 1 ,y,t)=1, 0 <y<1, 0 <t,
u(x, 0 ,t)=0, u(x, 1 ,t)=1, 0 <x<1, 0 <t,
u(x,y, 0 )=0, 0 <x<1, 0 <y<1,
x= y= 1 /4. (See Fig. 2.)- Find a numerical solution of the heat problem on a 1×1 square with x=
y= 1 /4. Initiallyu=0 and on the outside boundaryu=0. There is a
tiny hole in the center of the square, sou( 1 / 2 , 1 / 2 ,t)=1,t>0. (Actually,
the region is a punctured square.) - Solve numerically Eqs. (14)–(18) with x= y= 1 /4,ρ^2 = 1 /2. Take
f(x,y)≡0and
g(x,y)={
4
√
2at( 1
2 ,
1
2)
,
0elsewhere.
Physically,udescribes the vibrations of a square membrane struck in the
middle.- Obtain an approximate solution of Eqs. (14)–(18) withf(x,y)≡0and
g(x,y)= 4
√
- Take x= y= 1 /4andρ^2 = 1 /2.
- Same as Exercise 8, butf(x,y)≡1andg(x,y)≡0inthesquare.
10.Obtain an approximate numerical solution to the wave equation on anL-
shaped region (a 1×1 square with a 1/ 4 × 1 /4squareremovedfromthe
upper right corner). Assume initial displacement=1 in the lower right
corner, initial velocity equal to 0, and zero displacement on the boundary.
Ta k e x= y= 1 /4andρ^2 = 1 /2.