PDE Homework #4: Fourier Transform and Heat Equation on Infinite Domains
1. Find the Fourier transform of the following. Show all work.
a)
f x H x H x XXXXXXXXXX) ? ? ? ?
where H(x) denotes the Heaviside function
b)
234
234 , , ,
df d f d f d f
dx dx dx dx
. Hint: Integrate by parts as shown in class.
2. Let the “convolution” of f(x) and g(x) be given by
f g f x g d ( ) ( ) ? ? ?
?
??
? ? ? ?
.
Show that the Fourier transform of the convolution is equal to the product of the
Fourier transforms,
ˆ
F f g f g [ ] ( ) ( ) ? ? ? ?
ˆ .
3. Use Fourier transforms to solve the ODE:
2
? ? ? ? ? ? ? ? y a y g x x '' ( ),
Assume that
y x ? ? ?? 0 as .
4. Verify that the heat kernel (aka fundamental solution)
2
4
1
( , ) , , 0
4
x
at k x t e x t
?at
?
? ? ? ? ? ? ?
satisfies the heat equation.
5. Look up the definition of the complementary error function, erfc x( )
. Express the
solution of the following heat equation problem in terms of erfc.
0
1
, , 0
0
( ,0)
0
t xx u u x t
T x
u x
T x
? ? ? ? ? ? ?
? ?
? ?
? ?
6. Solve
, 0 , 0
(0, ) cos
0 as
t xx u u x t
u t t
u x
? ? ? ? ?
?
? ? ?
Document Preview: PDE Homework #4: Fourier Transform and Heat Equation on Infinite Domains
1. Find the Fourier transform of the following. Show all work.
a) f (x)?H(x?1)?H(x?1) where H(x) denotes the Heaviside function
234
df d f d f d f
b) , , , . Hint: Integrate by parts as shown in class.
234
dx dx dx dx
?
2. Let the “convolution” of f(x) and g(x) be given by f?g? f (x??)g(?)d? .
?
??
Show that the Fourier transform of the convolution is equal to the product of the
ˆ
Fourier transforms, F[f?? g] f (?? )gˆ( ) .
2
3. Use Fourier transforms to solve the ODE: ?y''?a y? g(x), ??? x??
Assume that yx? 0 as ??? .
4. Verify that the heat kernel (aka fundamental solution)
2
x
?
1
4at
k(x,t)? e , ??? x??, t? 0
4?at
satisfies the heat equation.
5. Look up the definition of the complementary error function, erfc() x . Express the
solution of the following heat equation problem in terms of erfc.
u?u , ??? x??, t? 0
t xx
Tx? 0
?
0
ux ( ,0)?
?
Tx? 0
? 1
6. Solve
u?u , 0? x??, t? 0
t xx
u(0,t)? cost
ux? 0 as ??