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Supplementary Problems

2
2
2
Exercise 225. Solve u xx + u yy = 0 in the quarter-disk (x + y < a , x > 0,
y > 0) with the following BCs: u = 0 on x = 0 and on y = 0 and ∂u = 1 on
∂r
r = a. Write the answer as an infinite series and write the first two nonzero terms
explicitly.

Exercise 226. Solve u xx + u yy = 0 in the region (α < θ < β, a < r < b) with
the boundary conditions u = 0 on the two sides θ = a and θ = β, u = g(θ) on the
arc r = a, and u = h(θ) on the arc r = b.

Exercise 227. Find the solution u(x, y) of the reduced Helmholtz equation ∆u−
ku = 0 (k is a positive parameter) in the square 0 < x, y < π, where u satisfies
the boundary condition u(0, y) = 1, u(π, y) = u(x, 0) = u(x, π) = 0.

Exercise 228. Solve the Laplace equation ∆u = 0 in the square 0 < x, y < π,
subject to the boundary condition u(x, 0) = u(x, π) = 1, u(0, y) = u(π, y) = 0.

Exercise 229.
(a) Solve the problem
∆u = 0 0 < x < π, ) < y < π,

u(x, 0) = u(x, π) = 0 0 ≤ x ≤ π,
u(0, y) = 0 0 ≤ y ≤ π,
u(π, y) = sin y 0 < y < π.
(b) Is there a point (x, y) ∈ {(x, y)| 0 < x < π, 0 < y < π} such that u(x, y) =
0?

Exercise 230. Consider the Laplace equation ∆u = 0 in the domain 0 < x, y < π
2
with the boundary condition u (x, π) = x −a, u (0, y) = u (π, y) = u (x, 0) = 0.
y
y
x
x
Find all the values of the parameter a for which the problem is solvable. Solve the
problem for these values of a.
2
2
Exercise 231. Solve the Laplace equation ∆u = 0 in the domain x + y > 4,
2
2
subject to the boundary condition u(x, y) = y on x + y = 4, and the decay
condition lim u(x, y) = 0.
|x|+|y|→∞

Exercise 232. Solve the problem
u xx + u yy = 0 0 < x < 2π, −1 < y < 1,
u(x, −1) = 0, u(x, 1) = 1 + sin 2x 0 ≤ x ≤ 2π,
u (0, y) = u (2π, y) = 0 −1 < y < 1.
x
x
Exercise 233. Solve the equation u = 2u xx in the domain 0 < x < π, t > 0
t
under the initial boundary value conditions u(0, t) = u(π, t) = 0, u(x, 0) = f(x) =
2
2
x(x − π ).

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