MA2332 Tutorial Sheet: due on 1st April, 2016
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1. Use separation of variables to solve the two-dimensional Laplace equation for φ(x, y)
with boundary conditions given by
φ(0, y) = 0,
∂
1
φ(0, y) =
sin(ny), (n ∈ Z).
∂x
n
called the Hadamard conditions.
2. Using separation of variables to solve the two-dimensional Laplace equation
∇2 V (x, y) = 0,
subject to the boundary conditions: V (0, y) = V (d, y) = 0 and V (x, 0) = V0 sin
2πx
,
d
with d and V0 ∈ R .
3. Use separation of variables to find a solution to the 1-dimensional wave equation
2
∂ 2φ
2∂ φ
=
c
∂t2
∂x2
∂φ
(x, 0) = g(x)
∂t
Note: you may assume that the formula for the coefficients of a Fourier sine series
for a function h(x) defined on 0 ≤ x ≤ L is
with the boundary conditions: φ(x, 0) = f (x), φ(0, t) = 0, φ(L, t) = 0,
2
bn =
L
1
Z
L
h(x) sin
nπx 0
L
dx, n = 1, 2, 3, . . .
Sinéad Ryan, see http://www.maths.tcd.ie/˜ryan/231.html
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