EXERCISES IN SEPARATION OF
VARIABLES
George Baravdish & Bengt Ove Turesson
translated by Peter Basarab-Horwath
2017
c Mathematics Department, Linköping University
1
The heat equation
1. Solve the boundary-value problem

0
00

0 < x < π, t > 0
ut − uxx = 0,
u(0, x) = 1 + cos 3x,
0≤x≤π

 0
0
ux (t, 0) = ux (t, π) = 0, t ≥ 0.
2. Solve the boundary-value problem

00
0

0 < x < π, t > 0
ut − uxx = 0,
2
u(0, x) = cos x,
0≤x≤π

 0
0
ux (t, 0) = ux (t, π) = 0, t ≥ 0.
3. Solve the boundary-value problem

0
00

0 < x < 1, t > 0
ut − uxx = 0,
u(0, x) = 1 + 2x − sin 2πx, 0 ≤ x ≤ 1

 0
u (t, 0) = 1, u(t, 1) = 3,
t ≥ 0.
4. Solve the boundary-value problem

0
00

0 < x < π, t > 0
ut − uxx = 0,
u(0, x) = sin x,
0≤x≤π


0
ux (t, 0) = ux (t, π) = 0, t ≥ 0.
5. Solve the boundary-value problem

0
00

aut − uxx = 0,
(a)
u(0, x) = min{x, L − x},


u(t, 0) = u(t, L) = 0,

0
00

aut − uxx = 0,

πx
u(0, x) = 100 cos
,
(b)
2L


u(t, 0) = 100, u(t, L) = 0,
0 < x < L, t > 0
0≤x≤L
t ≥ 0.
0 < x < L, t > 0
0≤x≤L
t ≥ 0.
6. A thin rod, insulated along all its sides, has its ends at x = 0 and x = L. At
t = 0 the temperature distribution in the rod is given by
u(0, x) = x,
0 ≤ x ≤ L.
The temperature distribution in the rod is governed by the equation
au0t = u0xx .
Find the temperature distribution in the rod for t > 0. Then calculate the limit
lim u(t, x).
t→∞
7. Find a solution to the boundary-value problem

0
00

0 < x < π, t > 1
tut − uxx = 0,
u(1, x) = sin x + 2 sin 3x, 0 ≤ x ≤ π


u(t, 0) = u(t, π) = 0,
t ≥ 1.
8. Find a solution to the boundary-value problem

0
00

0 < x < π, t > 0
ut − uxx + hu = 0,
u(0, x) = 0,
0≤x≤π


u(t, 0) = 0, u(t, π) = 1, t ≥ 0
where h is a positive constant.
2
The wave equation
9. A vibrating string is fixed at its endpoints x = 0 and x = π on the x-axis. The
amplitude of the string at a time t > 0 is given by the curve y = u(t, x), 0 ≤ x ≤
π and u(t, x) satisfies the wave equation u00tt −u00xx = 0. At time t = 0 the string lies
on the x-axis, but its time derivative at t = 0 is given by u0t (0, x) = πx(π − x)/8.
Determine u(t, x) for t > 0.
10. Solve the boundary-value problem
 00
utt − u00xx = 0,



u(0, x) = 0,

u0t (0, x) = sin 3x,



u(t, 0) = u(t, π) = 0,
0 < x < π, t > 0
0≤x≤π
0≤x≤π
t ≥ 0.
11. Solve the boundary-value problem
 00
utt − u00xx = 0,



u(0, x) = 3 sin 2x,

u0t (0, x) = 5 sin 3x,



u(t, 0) = u(t, π) = 0,
0 < x < π, t > 0
0≤x≤π
0≤x≤π
t ≥ 0.
12. Solve the boundary-value problem
 00
utt − u00xx = 0,



u(0, x) = sin3 x,

u0t (0, x) = sin x cos x,



u(t, 0) = u(t, π) = 0,
0 < x < π, t > 0
0≤x≤π
0≤x≤π
t ≥ 0.
13. Solve the boundary-value problem
 00
utt − u00xx = 0,



u(0, x) = x(x − π),

u0t (0, x) = sin3 x,



u(t, 0) = u(t, π) = 0,
0 < x < π, t > 0
0≤x≤π
0≤x≤π
t ≥ 0.
14. Solve the boundary-value problem
 00
utt − u00xx = 0,



u(0, x) = sin x,

u0t (0, x) = π 2 − x2 ,



u(t, −π) = u(t, π) = 0,
−π < x < π, t > 0
−π ≤ x ≤ π
−π ≤ x ≤ π
t ≥ 0.
15. Solve the boundary-value problem
 00
utt − u00xx + 2u0t = 0,



u(0, x) = 0,

u0t (0, x) = sin3 x,



u(t, 0) = u(t, π) = 0,
0 < x < π, t > 0
0≤x≤π
0≤x≤π
t ≥ 0.
3
Laplace’s equation
16. Solve the boundary-value problem
 00
uxx + u00yy = 0,



u(0, x) = sin 2x + sin 3x,

u(x, π) = 0



u(0, y) = u(0, π) = 0,
0 < x < π, 0 < y < π
0≤x≤π
0≤x≤π
0 < y < π.
17. A thin square plate with its corner points at (0, 0), (1, 0), (1, 1) in the xy-plane
and (0, 1) is thermally insulated on its surfaces. Find the stationary temperature
distribution in the plate when three of its sides are at temperature 0 and the
fourth side has temperature distribution given by
(a) u(1, y) = sin πy, 0 < y < 1,
1 (b) u(1, y) = − y −
2
1 , 0 < y < 1.
2
18. Determine the temperature distribution (which is stationary) for a square plate
with its corner points at (0, 0), (1, 0), (1, 1), and (0, 1) in the xy-plane when the
boundary conditions are given by u(x, 0) = 0, u(x, 1) = 1 − 2x for 0 ≤ x ≤ 1.
The plate is thermally insulated on the sides where x = 0 and x = 1.
19. A thin square plate with its corner points at (0, 0), (1, 0), (1, 1) and (0, 1) in the
xy-plane is thermally insulated on its surfaces as well as along that side which
lies on the x-axis. The temperature along the other three sides is given by
(
u(0, y) = 0 0 ≤ y ≤ 1
u(1, y) = 1 0 ≤ y ≤ 1
and


0
1
0≤x≤
2
u(x, 1) =
1

2x − 1
≤ x ≤ 1.
2
Find the stationary temperature distribution u(x, y) in the plate.
20. Solve the boundary-value problem

00
00

0 < x < 1, 0 < y < 1
uxx + uyy = x
u(x, 0) = u(x, 1) = 0 0 ≤ x ≤ 1


u(0, y) = u(1, y) = 0 0 ≤ y ≤ 1.
Answers
1. u(t, x) = 1 + e−9t cos 3x
1 1
2. u(t, x) = + e−4t cos 2x
2 2
2
3. u(t, x) = 1 + 2x − e−4π t sin 2πx
∞
2
4X
1
2
4. u(t, x) = −
e−4k t cos 2kx
2
π π k=1 4k − 1
∞
4L X (−1)k −(2k+1)2 π2 t/aL2
(2k + 1)π
5. (a) u(t, x) = 2
x
e
sin
2
π k=0 (2k + 1)
L
∞
x 200 X
1
kπ
k2 π 2 t/aL2
(b) u(t, x) = 100 1 −
+
e
sin
x
L
π k=1 k(4k 2 − 1)
L
∞
L 4L X
1
(2k + 1)πx
2 2
2
6. u(t, x) = − 2
e−(2k+1) π t/aL cos
,
2
2
π k=0 (2k + 1)
L
lim u(t, x) =
t→∞
L
2
sin x 2 sin 3x
+
t
t9
√
∞
sinh hx
2 X (−1)k k −(k2 +h)t
√
e
sin kx +
8. u(t, x) =
π k=1 k 2 + h
sinh hπ
∞
X
1
sin(2k + 1)t sin(2k + 1)x
9. u(t, x) =
4
(2k
+
1)
k=0
7. u(t, x) =
10. u(t, x) =
1
sin 3t sin 3x
3
11. u(t, x) = 3 cos 2t sin 2x +
5
sin 3t sin 3x
3
3
1
1
cos t sin x + sin 2t sin 2x − cos 3t sin 3x
4
4
4
∞
3
1
8X
1
13. u(t, x) = sin t sin x −
sin 3t sin 3x −
cos(2k + 1)t sin(2k + 1)x
4
12
π k=0 (2k + 1)3
12. u(t, x) =
∞
14.
15.
16.
17.
64 X (−1)k
(2k + 1)t
(2k + 1)x
u(t, x) = cos t sin x +
sin
cos
4
π k=0 (2k + 1)
2
2
√
3
1
u(t, x) = te−t sin x − √ e−t sin( 8t) sin 3x
4
8 2
u(x, y) = (cosh 2y − coth 2π sinh 2y) sin 2x + (cosh 3y − coth 3π sinh 3y) sin 3x
∞
sinh πx
2 X 1 − (−1)k
(a) u(x, y) =
sin πy
(b) u(x, y) = 2
sinh kπx sin kπy
sinh π
π k=0 k 2 sinh kπ
∞
8 X
1
18. u(x, y) = 2
sinh(2k + 1)πy cos(2k + 1)πx
2
π k=0 (2k + 1) sinh(2k + 1)π
∞
4 X
(−1)k+1
19. u(x, y) = x + 2
cosh(2k + 1)πy sin(2k + 1)πx
π k=0 (2k + 1)2 cosh(2k + 1)π
∞
1
2 X (−1)k+1
(ekπy + ekπ−kπy ) sin kπx
20. u(x, y) = (x3 − x) + 3
6
π k=1 k 3 (ekπ + 1)