PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
1. The temperature u(x, t) on a rod of length L satisfies the heat equation
∂u
∂ 2u
= c2 2
∂t
∂x
t > 0, 0 < x < L
(a) Using separation of variables with u(x, t) = V (x)W (t), show that the ODEs satisfied by
V (x) and W (t) are
and
V ′′ − kV = 0
(1)
W ′ − kc 2 W = 0
(2)
for any constant k.
(b) If the ends of the rod are kept at zero temperature, i.e.,
u=0
on
x = 0 and
x = L,
(3)
what are the boundary conditions that should be satisfied by V ?
(c) Find the solutions to V which satisfy the equation (1) and the boundary conditions obtained
in (b). Hence find the corresponding solutions W that satisfy the equation (2). Conclude
that the most general solution for u satisfying the heat equation and the boundary conditions
(3) is given by
∞
∑
πnx
π2 n2 c 2 t
cn sin
u(x, t) =
exp −
(4)
L
L2
n=1
where cn are constants.
(d) If u(x, 0) = f (x), express cn as an integral.
(e) Find the solution u to the heat equation if u satisfies the boundary conditions (3) and
(i) c 2 = 5, L = π, u(x, 0) = sin 3x
2πx
(ii) c 2 = 6, L = 5, u(x, 0) = 3 sin
− 5 sin(4πx)
5
For each case, determine the steady state temperature lim u(x, t).
t→∞
1
2
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
2. The temperature u(x, t) on a rod of length L satisfies the heat equation
∂ 2u
∂u
= c2 2
∂t
∂x
t > 0, 0 < x < L
(a) Using separation of variables with u(x, t) = V (x)W (t), show that the ODEs satisfied by
V (x) and W (t) are
and
V ′′ − kV = 0
(5)
W ′ − kc 2 W = 0
(6)
for any constant k.
(b) If the ends of the rod are insulated, i.e.,
∂u
= 0 on x = 0 and x = L,
∂x
what are the boundary conditions that should be satisfied by V ?
(7)
(c) Find the solutions to V which satisfy the equation (5) and the boundary conditions obtained
in (b). Hence find the corresponding solutions W that satisfy the equation (6). Conclude
that the most general solution for u satisfying the heat equation and the boundary conditions
(7) is given by
∞
∑
πnx
π2 n2 c 2 t
u(x, t) =
cn cos
exp −
(8)
2
L
L
n=0
where cn are constants.
(d) If u(x, 0) = f (x), express cn as an integral.
(e) Find the solution u to the heat equation if u satisfies the boundary conditions (7) and
c 2 = 4, L = 4, u(x, 0) = 5 − cos(πx) + 3 cos(2πx). Also, determine the steady state
temperature lim u(x, t).
t→∞
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
3
3. The function u(x, t) satisfies the wave equation
2
∂ 2u
2∂ u
=
c
,
t ≥ 0, 0 ≤ x ≤ L
(9)
∂ t2
∂ x2
(a) Using separation of variables with u(x, t) = V (x)W (t), show that the ODEs satisfied by
V (x) and W (t) are
and
V ′′ − kV = 0
(10)
W ′′ − kc 2 W = 0
(11)
for any constant k.
(b) If u satisfies the boundary conditions
u=0
on
x = 0 and
x = L,
(12)
what are the boundary conditions that should be satisfied by V ?
(c) Find the solutions to V which satisfy the equation (10) and the boundary conditions obtained
in (b). Hence find the corresponding solutions W that satisfy the equation (11). Conclude
that the most general solution for u satisfying the wave equation and the boundary conditions (12) is given by
∞
∑
πnx
πnc t
πnc t
u(x, t) =
an cos
+ bn sin
sin
L
L
L
n=1
where an and bn are constants.
(d) If u(x, 0) = f (x) and u t (x, 0) = g(x), express an and bn as integrals.
(e) Find the solution u to the wave equation if u satisfies the boundary conditions (12) and
c = 3, L = 4, u(x, 0) = 3 sin(2πx), u t (x, 0) = −5 sin(3πx)
4
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
4. Consider the wave equation
2
∂ 2u
2∂ u
=
c
∂ t2
∂ x2
(a) By introducing the variables y = x + c t, y = x − c t, show that for any functions ϕ( y) and
ψ(z)
u(x, t) = ϕ( y) + ψ(z)
is a solution of the wave equation.
(b) Use the result of (a) to find the D’Alembert’s solution of the wave equation if
(i) c = 2, u(x, 0) = sin 3x, u t (x, 0) = 0
(ii) c = 2, u(x, 0) = 0, u t (x, 0) = 3 sin 5x
(iii) c = 2, u(x, 0) = sin
3x, u t (x, 0) = 3 sin 5x
sin πx,
0 ≤ x ≤ 1,
(iv) c = 2, u(x, 0) =
, u t (x, 0) = 0
0,
otherwise
5. The function u(x, y) satisfies the Laplace equation
∂ 2u ∂ 2u
+
=0
∂ x2 ∂ y2
(13)
in a region R.
(a) Using separation of variables with u(x, y) = V (x)W ( y), show that the ODEs satisfied by
V (x) and W ( y) are
V ′′ − kV = 0
and
(14)
′′
W + kW = 0
(15)
for any constant k.
(b) If u satisfies the boundary conditions
u=0
on
x = 0 and
x = a,
(16)
what are the boundary conditions that should be satisfied by V ?
(c) Find the solutions to V which satisfy the equation (14) and the boundary conditions obtained
in (b). Hence find the corresponding solutions W that satisfy the equation (15). Conclude
that the most general solution for u satisfying the Laplace equation and the boundary conditions (16) is given by
∞
πn y
πn y
∑
πnx
u(x, t) =
sin
an exp
+ bn exp −
a
a
a
n=1
where an and bn are constants.
(d) Find the solution u to the Laplace equation if u satisfies the boundary conditions (16) and
(i) a = 3, u(x, 0) = 4 sin πx, u → 0 as y→ ∞
7πx
(ii) a = 4, u(x, 0) = 0, u(x, 3) = 2 sin
2
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
5
6. The function u(x, y) satisfies the Laplace equation
∂ 2u ∂ 2u
+
=0
∂ x2 ∂ y2
(17)
in a region R.
(a) Show that for any constants A and B and any positive integer n,
πnx
πnx
πn y
u(x, y) = sin
Ae b + Be− b
b
satisfies (17) and the boundary conditions
u = 0 on
y = 0 and
y=b
(18)
(b) Find A, B and n such that u(x, y) satisfies (17), (18) and
(i) b = 3, u(0, y) = sin π y, u(2, y) = 2 sin π y
(ii) b = 3, u(0, y) = sin π y, u → 0 as x → ∞
Write down the solution of u explicitly.
(c) Show that for any constants A, B and C and any positive integer n,
πnx
πnx
πn y
u(x, y) = A + cos
Be b + C e− b
b
satisfies (17) and the boundary conditions
uy = 0
on
y =0
and
y=b
(d) Find A, B, C and n such that u(x, y) satisfies (17), (19) and
(i) b = 3, u(0, y) = cos π y, u(2, y) = 0
(ii) b = 3, u(0, y) = 3 + cos π y, u(2, y) = 3
Write down the solution of u explicitly.
(19)
6
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
Answers:
1. (b) V (0) = V (L) = 0
πnx
π2 n2 c 2 t
, Wn (t) = exp −
, n = 1, 2, . . .
(c) Vn (x) = cn sin
L
L2
∫ L
2
πnx
(d) cn =
f (x) sin
dx
L 0
L
(e)
(i) e−45t sin 3x; 0
24π2 t
2πx
(ii) 3 exp −
sin
− 5 exp −96π2 t sin 4πx; 0
25
5
2. (b) V ′ (0) = V ′ (L) = 0
πnx
π2 n2 c 2 t
, Wn (t) = exp −
, n = 0, 1, 2, . . .
L
L2
∫ L
2
πnx
d x, n = 1, 2, 3, . . .
f (x)d x, cn =
f (x) cos
L 0
L
(c) Vn (x) = cn cos
1
(d) c0 =
L
∫
L
0
2
2
(e) 5 − e−4π t cos πx + 3e−16π t cos 2πx; 5
3. (b) V (0) = V (L) = 0
πnx
πnc t
πnc t
, Wn (t) = an cos
+ bn sin
, n = 1, 2, . . .
L
L
L
∫ L
πnx
πnx
2
f (x) sin
g(x) sin
d x, bn =
dx
L
πnc 0
L
(c) Vn (x) = cn sin
2
(d) an =
L
∫
0
L
(e) 2 sin(2πx) cos(6πt) −
4. (b)
(i)
5
sin(3πx) sin(9πt)
9π
1
(sin 3(x + 2t) + sin 3(x − 2t))
2
(ii) −
3
(cos 5(x + 2t) − cos 5(x − 2t))
20
3
1
(sin 3(x + 2t) + sin 3(x − 2t)) −
(cos 5(x + 2t) − cos 5(x − 2t))
2
20
1
0 ≤ x + 2t ≤ 1, 0 ≤ x − 2t ≤ 1
2 (sin π(x + 2t) + sin π(x − 2t)) ,
1
sin π(x + 2t),
0 ≤ x + 2t ≤ 1, x − 2t < 0
(iv) 12
sin π(x − 2t),
0 ≤ x − 2t ≤ 1, x + 2t > 1
2
0,
x − 2t < 1, x + 2t > 1
(iii)
PROBLEM SET 5 PARTIAL DIFFERENTIAL EQUATIONS
5. (b) V (0) = V (a) = 0
(c) Vn (x) = cn sin
(d)
(i) 4e−π y sin πx
(ii)
6. (b)
πn y
πn y
πnx
, Wn (t) = an exp
+ bn exp −
, n = 1, 2, . . .
a
a
a
(i)
2
sinh
21π
2
sin
7π y
7πx
sinh
2
2
sin π y sinh π(2 − x) 2 sin π y sinh πx
+
sinh 2π
sinh 2π