1. a. Find the solution u(x, y) of Laplace's equation in the

rectangle 0 < x < a, 0 ≤ y ≤ b, that satisfies the boundary

conditions

u(0, y) = 0,

u(x,0) = 0,

b. Find the solution if

u(a, y) = 0,

u(x, b) = g(x),

g(x) = = {*_.

0 < y

0 < x

0≤x≤a/2,

a-x, a/2 ≤ x ≤a.

G c. For a = 3 and b = 1, plot u versus x for several values

of y and also plot u versus y for several values of x. (Use

enough terms in the Fourier series to accurately approximate the

nonhomogeneous boundary condition.)

Gd. Plot u versus both x and y in three dimensions. Also draw

a contour plot showing several level curves of u(x, y) in the xy-

plane.

2. Find the solution u(x, y) of Laplace's equation in the rectangle

0

u(a, y) = 0,

u(x, b) = 0,

3. a. Find the solution u(x, y) of Laplace's equation in the

rectangle 0 < x < a, 0 ≤ y ≤ b, that satisfies the boundary

u(0, y) = 0,

u(x,0) = h(x),

0

0 < x

10.8 Laplace's Equation 521

Nc. Let a = 5. Find the smallest value of yo for which

u(x, y) ≤ 0.1 for all y ≥ yo-/n5. Find the solution u(r, 0) of Laplace's equation

1

7400

outside the circle r = a, that satisfies the boundary condition

u(a,0) = f(0), 0≤0 < 2π,

1

Urr + ₂ +

= 0,

0, r>a, 0<0 < 2π,

on the circle. Assume that u(r, 0) is single-valued and bounded for

r> a./n14. a. Find the solution u(x, y) of Laplace's equation in the

rectangle 0 < x < a, 0 < y < b, that satisfies the boundary

conditions

u₂(0, y) = 0,

u(x,0) = 0,

u,(a, y) = 0,

u(x, b) = g(x),

0 < y

0

b. Find the solution if g(x) = 1 + x²(x -a)².

G c. Let a = 3 and b = 2; plot several different views of the

solution.