Applied Math

Applied Math

1. Find the Fourier transform of f (x ) = 
Ae -r1 x
, x = 0
Ae r2 x
, x < 0
where A, r1 > 0, r2 > 0 are constant. Write the answer
as a single fraction.
2. Show if f is an even/odd function then so is fb.
3. In class we showed the heat conduction problem
?u
? t
= a
?
2u
? x
2
, u(x ,0) = f (x ), -8 < x <8, t > 0
has solution
u(x ,t ) = 1
2p
? 8
-8
fb(k)e
-a k2
t
e
i k x dk.
(a) Write this as a real integral in the case of even f .
(b) Find the solution (as an integral) if f (x ) = 1
1 + x
2
.
(c) Verify directly that your solution satisfies the original heat conduction problem.
(d) Show that, at any time, the temperature is greatest at x = 0.
Bonus. Show u(0,t ) ~
p
p
2
p
a t
for large t . Hint. Substitute u = a k2
t .
4. Let d be the delta function. Evaluate
(a) ? 8
-8
(x – 2)
3d(x ) dx
(b) ? 8
-8
(x – 2)
3d(x – 6) dx
(c) ? 8
-8
(x – 2)
3d(2x ) dx
(d) ? 8
-8
(x – 2)
3d(2x – 6) dx
Bonus. ? 8
-8
(x – 2)
3d
‘
(x ) dx Hint. Integrate by parts.
Bonus. For the function f in Hw7 Q1, find fc”(k) and its Fourier inversion.
Bonus. In class I said i kHÒ(k) = HÓ’
(k) = db(k) = 1 so HÒ(k) = 1
i k
but this is incorrect — in fact it contradicts Q2.
What did I do wrong and what is the correct result?

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