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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST
Name:
Work out everything as far as you can before making decimal approximations.
- Consider d’Alembert’s solution
u(x, t) =
(f (x − ct) + f (x + ct)) +
2 c
∫ (^) x+ct
x−ct
g(s) ds
of the wave equation for a vibrating string, where f (x) is the odd function with period 2L which on the interval 0 ≤ x ≤ L gives the initial position of the string, and similarly g(x) gives the initial velocity on 0 ≤ x ≤ L and is odd and 2L periodic. Suppose that f (x) = sin πx g(x) = 0
with L = 1 and c = (^) π^1. What is the first positive time t that the string returns to its original shape?
Date: October 26, 2000. 1
- Use separation of variables to find the general solution of the heat equation in a rectangular plate with edges of lengths a and b, and with all edges insulated.
- Show that the function
un(r, θ) =
( (^) r R
)n (an cos(nθ) + bn sin(nθ))
satisfies the Laplace equation ∇^2 u = 0 on the disk of radius R, where in polar coordinates
∇^2 u = ∂^2 u ∂r^2
r
∂u ∂r
r^2
∂^2 u ∂θ^2
Now adding these together, why does the function
u(r, θ) = a 0 +
∑^ ∞
n=
( (^) r R
)n (an cos (nθ) + bn sin (nθ))
satisfy the Laplace equation? Moreover show that if we pick these a 0 , am and bm according to
a 0 =
2 π
∫ (^2) π
0
f (θ) dθ
am =
π
∫ (^2) π
0
f (θ) cos(nθ) dθ
bm =^1 π
∫ (^2) π
0
f (θ) sin(nθ) dθ
(as Fourier amplitudes of f (θ)) then
u(R, θ) = f (θ)
so that this is the steady state of the heat equation in the disk with boundary held fixed at temperature u(R, θ) = f (θ).
0 –0.5 – 1 0.5 x
–0. 0 0. 1
y
1
Figure 1. A round peak
- Looking at the wave equation ∂^2 u ∂t^2 =^ c
2
∂^2 u ∂x^2 +^
∂^2 u ∂y^2
and recalling second derivatives from calculus, explain why a round peak in the graph of u (as in figure 1) will be accelerated downward.
