APM3701
Assignment 2
Unique No: 700123
Due 8 August 2025
,Student Name: APM3701 Assignment 2
APM3701 Assignment 2
Due Date: 8 August 2025
Heat Flow Problem
Question 1(a)
Formulate the initial-boundary value problem (IBVP) governing the temperature u(x, t)
in a rod of length L, with heat conductivity 1, initial temperature f (x), and time-
dependent boundary temperatures g1 (t) and g2 (t).
IBVP:
∂u ∂ 2u
= , 0 < x < L, t > 0
∂t
∂x2
u(x, 0) = f (x), 0≤x≤L
u(0, t) = g1 (t), u(L, t) = g2 (t), t ≥ 0
Variables and Parameters:
• u(x, t): Temperature at position x and time t
• x: Spatial coordinate, 0 ≤ x ≤ L
• t: Time, t ≥ 0
• L: Length of the rod
• f (x): Initial temperature distribution
• g1 (t), g2 (t): Boundary temperatures at x = 0 and x = L
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, Student Name: APM3701 Assignment 2
Question 1(b)
Assume f (x), g1 (t), and g2 (t) are bounded:
m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M
and the solution u(x, t) satisfies:
m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], t ≥ 0
Show that the solution is unique.
Let u1 (x, t) and u2 (x, t) be two solutions. Define:
w(x, t) = u1 (x, t) − u2 (x, t)
Then w satisfies:
∂w ∂ 2w
= , 0 < x < L, t > 0
∂t
∂x2
w(x, 0) = 0, 0≤x≤L
w(0, t) = 0, w(L, t) = 0, t ≥ 0
Define the energy functional:
Z L
E(t) = w2 (x, t) dx
0
Differentiate:
L L
∂ 2w
Z Z
dE ∂w
=2 w dx = 2 w 2 dx
dt 0 ∂t 0 ∂x
Integrating by parts and using boundary conditions:
Z L 2
dE ∂w
= −2 dx ≤ 0
dt 0 ∂x
Since E(0) = 0, we conclude E(t) = 0 ⇒ w(x, t) = 0 for all x, t.
Page 2
Assignment 2
Unique No: 700123
Due 8 August 2025
,Student Name: APM3701 Assignment 2
APM3701 Assignment 2
Due Date: 8 August 2025
Heat Flow Problem
Question 1(a)
Formulate the initial-boundary value problem (IBVP) governing the temperature u(x, t)
in a rod of length L, with heat conductivity 1, initial temperature f (x), and time-
dependent boundary temperatures g1 (t) and g2 (t).
IBVP:
∂u ∂ 2u
= , 0 < x < L, t > 0
∂t
∂x2
u(x, 0) = f (x), 0≤x≤L
u(0, t) = g1 (t), u(L, t) = g2 (t), t ≥ 0
Variables and Parameters:
• u(x, t): Temperature at position x and time t
• x: Spatial coordinate, 0 ≤ x ≤ L
• t: Time, t ≥ 0
• L: Length of the rod
• f (x): Initial temperature distribution
• g1 (t), g2 (t): Boundary temperatures at x = 0 and x = L
Page 1
, Student Name: APM3701 Assignment 2
Question 1(b)
Assume f (x), g1 (t), and g2 (t) are bounded:
m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M
and the solution u(x, t) satisfies:
m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], t ≥ 0
Show that the solution is unique.
Let u1 (x, t) and u2 (x, t) be two solutions. Define:
w(x, t) = u1 (x, t) − u2 (x, t)
Then w satisfies:
∂w ∂ 2w
= , 0 < x < L, t > 0
∂t
∂x2
w(x, 0) = 0, 0≤x≤L
w(0, t) = 0, w(L, t) = 0, t ≥ 0
Define the energy functional:
Z L
E(t) = w2 (x, t) dx
0
Differentiate:
L L
∂ 2w
Z Z
dE ∂w
=2 w dx = 2 w 2 dx
dt 0 ∂t 0 ∂x
Integrating by parts and using boundary conditions:
Z L 2
dE ∂w
= −2 dx ≤ 0
dt 0 ∂x
Since E(0) = 0, we conclude E(t) = 0 ⇒ w(x, t) = 0 for all x, t.
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