1 Partial di↵erential equations: Questions
1. Solve the di↵usion equation
ut = uxx ,
subject to the initial and boundary conditions
u(x, 0) = sin(⇡x) + 3 sin(2⇡x) 2 sin(3⇡x), u(0, t) = 0, u(1, t) = 0.
2. Find the solution to the wave equation
@ 2u @ 2u
= ,
@t2 @x2
subject to the boundary conditions
u(0, t) = 0 u(2, t) = 0,
and initial conditions
u(x, 0) = 0 ut (x, 0) = 1 for 0 < x < 2.
3. Solve the di↵usion equation
ut = uxx ,
subject to the initial conditions
⇢
2x 0 < x < 1/2
u(x, 0) = ,
2 2x 1/2 < x < 1
and boundary conditions
@u @u
= 0, = 0.
@x x=0 @x x=1
4. Find the solution to the wave equation
@ 2u @ 2u
= ,
@t2 @x2
subject to the boundary conditions
ux (0, t) = 0 ux (1, t) = 0,
and initial conditions
u(x, 0) = F (x) ut (x, 0) = G(x).
Hence write down the solution to the special case
u(x, 0) = x for 0 < x < 1, ut (x, 0) = 0.
1
,5. By using the method of separation by variables find the electric potential inside, and outside,
a sphere of radius r = 1 which is held at potential = 10 cos3 ✓ 3 cos2 ✓ 5 cos ✓ 1. Here
(r, ✓, ) are spherical polar coordinates and the potential obeys Laplace’s equation
✓ ◆ ✓ ◆
2 1 @ 2@ 1 @ @ 1 @
r = 2 r + sin ✓ + = 0.
r @r @r sin ✓ @✓ @✓ sin2 ✓ @ 2
and ! 0 as r ! 1. [You are given that P0 (x) = 1, P1 (x) = x, P2 (x) = (1/2)(3x2 1),
P3 (x) = (1/2)(5x3 3x).]
6. A stretched square membrane has height . Given the height of the membrane along the
edges of the square it is possible to find its height inside the square by solving
@2 @2
+ = 0, (1)
@x2 @y 2
In the instance where
(a) (0, y) = 0, (b) (⇡, y) = 0, (c) (x, 0) = 0, (2)
(x, ⇡) = G(x), (3)
find an expression for the membrane displacement (x, y). Write down this expression for the
special case G(x) = x(⇡ x).
2
, 1stpower 7Mt
9 K Heat equation
variables
Separation of
subject to ulat o and Ullitt o
Xt flag t
o
u at fColglt flat o
solution
o trivial
Jlt give
o
UCitt forget
t o fan O
g
Faget flag'll
f x 9 t h h
ya gets get
f at Afa o I I got
ya fog f dat
c
Mm hey
me Ing It to
Acacia t Bin del go Aed
g x
1. Solve the di↵usion equation
ut = uxx ,
subject to the initial and boundary conditions
u(x, 0) = sin(⇡x) + 3 sin(2⇡x) 2 sin(3⇡x), u(0, t) = 0, u(1, t) = 0.
2. Find the solution to the wave equation
@ 2u @ 2u
= ,
@t2 @x2
subject to the boundary conditions
u(0, t) = 0 u(2, t) = 0,
and initial conditions
u(x, 0) = 0 ut (x, 0) = 1 for 0 < x < 2.
3. Solve the di↵usion equation
ut = uxx ,
subject to the initial conditions
⇢
2x 0 < x < 1/2
u(x, 0) = ,
2 2x 1/2 < x < 1
and boundary conditions
@u @u
= 0, = 0.
@x x=0 @x x=1
4. Find the solution to the wave equation
@ 2u @ 2u
= ,
@t2 @x2
subject to the boundary conditions
ux (0, t) = 0 ux (1, t) = 0,
and initial conditions
u(x, 0) = F (x) ut (x, 0) = G(x).
Hence write down the solution to the special case
u(x, 0) = x for 0 < x < 1, ut (x, 0) = 0.
1
,5. By using the method of separation by variables find the electric potential inside, and outside,
a sphere of radius r = 1 which is held at potential = 10 cos3 ✓ 3 cos2 ✓ 5 cos ✓ 1. Here
(r, ✓, ) are spherical polar coordinates and the potential obeys Laplace’s equation
✓ ◆ ✓ ◆
2 1 @ 2@ 1 @ @ 1 @
r = 2 r + sin ✓ + = 0.
r @r @r sin ✓ @✓ @✓ sin2 ✓ @ 2
and ! 0 as r ! 1. [You are given that P0 (x) = 1, P1 (x) = x, P2 (x) = (1/2)(3x2 1),
P3 (x) = (1/2)(5x3 3x).]
6. A stretched square membrane has height . Given the height of the membrane along the
edges of the square it is possible to find its height inside the square by solving
@2 @2
+ = 0, (1)
@x2 @y 2
In the instance where
(a) (0, y) = 0, (b) (⇡, y) = 0, (c) (x, 0) = 0, (2)
(x, ⇡) = G(x), (3)
find an expression for the membrane displacement (x, y). Write down this expression for the
special case G(x) = x(⇡ x).
2
, 1stpower 7Mt
9 K Heat equation
variables
Separation of
subject to ulat o and Ullitt o
Xt flag t
o
u at fColglt flat o
solution
o trivial
Jlt give
o
UCitt forget
t o fan O
g
Faget flag'll
f x 9 t h h
ya gets get
f at Afa o I I got
ya fog f dat
c
Mm hey
me Ing It to
Acacia t Bin del go Aed
g x
