Assignment 5
Due date: August 11, 2015
(
1. Let φ(x) =
x, if |x| ≤ 1,
0, if |x| > 1,
2. Solve the following problem:





and u(x, t) =
φ(x − t) + 2φ(x − 2t)
. Find the expression of u(x, 1).
2
uxx + uyy = 0,
0 < x < a, 0 < y < b,
BC : u(x, 0) = uy (x, b) = 0,
0 ≤ x ≤ a,
u(0, y) = 0, ux (a, y) = f (y),
(1)
0 ≤ y ≤ b.
3. Use the method of eigenfunction expansions to find the eigenvalues and corresponding eigenfunctions of the
following problem:

uxx + uyy = λu, 0 < x < a, 0 < y < b,


(2)
BC : u(x, 0) = u(x, b) = 0, 0 ≤ x ≤ a,


u(0, y) = u(a, y) = 0, 0 ≤ y ≤ b.
4. Find the solution u(r, θ) of Laplace’s equation:

1
1


urr + ur + 2 uθθ = 0, r > a, 0 < θ < 2π,


r
r



BC : u(r, 0) = u(r, 2π), uθ (r, 0) = uθ (r, 2π), r ≥ a,



u(a, θ) = f (θ), 0 < θ < 2π,




u(r, θ) is bounded for all r ≥ a and 0 ≤ θ ≤ 2π.
5. Guess a solution to the following Poisson’s equation:
(
uxx + uyy = 1,
BC : u(x, y) = 0,
x2 + y 2 < a2 ,
x2 + y 2 = a2 .
6. Let α be a constant, consider the following problem:

uxx + uyy = 0, 0 < x < 1, 0 < y < 2,


BC : uy (x, 0) = uy (x, 2) = 0, 0 ≤ x ≤ 1,


ux (0, y) = y, ux (1, y) = α, 0 ≤ y ≤ 2.
(a) Find α such that (5) has a solution.
(b) Use the method of eigenfunction expansions to solve (5) when taking α in part (a).
(3)
(4)
(5)
7. For the following problem:
ut = uxx + ex , 0 < x < 1, t > 0,
BC : u(0, t) = 0, u(1, t) = 1, t > 0,

IC : u(x, 0) = f (x), 0 ≤ x ≤ 1.


Briefly describe how you would use the method of finite differences to find an approximate solution to this
problem. Use the notation ukn = u(xn , tk ) to denote the values of u on the finite difference mesh, and include
how you propose to incorporate the boundary and initial conditions.
8. For the following problem:







where
utt = uxx ,
−1 < x < 1, t > 0,
BC : ux (−1, t) = ux (1, t) = 0,
IC :
t > 0,
u(x, 0) = f (x), ut (x, 0) = 0,
−1 ≤ x ≤ 1.

 2(1 + x), if −1 < x < − 21 ,
1,
if − 12 ≤ x ≤ 12 ,
f (x) =

2(1 − x), if 21 < x < 1.
Briefly describe how you would use the method of finite differences to find an approximate solution to this
problem. Use the notation ukn = u(xn , tk ) to denote the values of u on the finite difference mesh, and include
how you propose to incorporate the boundary and initial conditions.
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