APM3701
Assignment 02
Unique No: 700123
Closing Date: 8 August 2025
,APM3701 Assignment 02 Due 8 August 2025 Page 1
Heat Equation Problem
Step-by-Step Mathematical Solution
Question 1
Problem Statement
Consider the heat flow in a horizontal rod of length L units and heat conductivity 1. We
assume that initially the rod was submerged in a medium where the temperature at each
point x of the rod is described by the function f (x). We also suppose that the left and
the right ends of the rod are in contact with media whose temperatures change with time
and are described by the functions g1 (t) and g2 (t), respectively.
(a) Write down the initial-boundary value problem satisfied by the temperature distri-
bution u(x, t) in the rod at any point x and time t. Explain all the variables and
parameters used.
(b) Suppose that f, g1 , g2 are bounded. There exist constants m and M such that for
all x in the domain of f , and for all t ≥ 0, we have:
m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M
and the solution u(x, t) of the IBVP described above satisfies:
m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], t ≥ 0.
Show that the solution u(x, t) is unique.
(c) Suppose u1 (x, t) and u2 (x, t) are solutions of the heat problem (with different initial
and boundary conditions), such that:
u1 (0, t) ≤ u2 (0, t), u1 (L, t) ≤ u2 (L, t), u1 (x, 0) ≤ u2 (x, 0)
Show that:
u1 (x, t) ≤ u2 (x, t), ∀x ∈ [0, L], t ≥ 0
, APM3701 Assignment 02 Due 8 August 2025 Page 2
1 Solution
(a) Initial-Boundary Value Problem
Variables and parameters:
• u(x, t): Temperature at position x ∈ [0, L] and time t ≥ 0
• f (x): Initial temperature distribution, i.e., u(x, 0) = f (x)
• g1 (t): Boundary temperature at x = 0, i.e., u(0, t) = g1 (t)
• g2 (t): Boundary temperature at x = L, i.e., u(L, t) = g2 (t)
The heat equation (with conductivity = 1):
∂u ∂ 2u
= , 0 < x < L, t > 0
∂t ∂x2
Initial condition:
u(x, 0) = f (x), 0≤x≤L
Boundary conditions:
u(0, t) = g1 (t), u(L, t) = g2 (t), t≥0
Assignment 02
Unique No: 700123
Closing Date: 8 August 2025
,APM3701 Assignment 02 Due 8 August 2025 Page 1
Heat Equation Problem
Step-by-Step Mathematical Solution
Question 1
Problem Statement
Consider the heat flow in a horizontal rod of length L units and heat conductivity 1. We
assume that initially the rod was submerged in a medium where the temperature at each
point x of the rod is described by the function f (x). We also suppose that the left and
the right ends of the rod are in contact with media whose temperatures change with time
and are described by the functions g1 (t) and g2 (t), respectively.
(a) Write down the initial-boundary value problem satisfied by the temperature distri-
bution u(x, t) in the rod at any point x and time t. Explain all the variables and
parameters used.
(b) Suppose that f, g1 , g2 are bounded. There exist constants m and M such that for
all x in the domain of f , and for all t ≥ 0, we have:
m ≤ f (x) ≤ M, m ≤ g1 (t) ≤ M, m ≤ g2 (t) ≤ M
and the solution u(x, t) of the IBVP described above satisfies:
m ≤ u(x, t) ≤ M, ∀x ∈ [0, L], t ≥ 0.
Show that the solution u(x, t) is unique.
(c) Suppose u1 (x, t) and u2 (x, t) are solutions of the heat problem (with different initial
and boundary conditions), such that:
u1 (0, t) ≤ u2 (0, t), u1 (L, t) ≤ u2 (L, t), u1 (x, 0) ≤ u2 (x, 0)
Show that:
u1 (x, t) ≤ u2 (x, t), ∀x ∈ [0, L], t ≥ 0
, APM3701 Assignment 02 Due 8 August 2025 Page 2
1 Solution
(a) Initial-Boundary Value Problem
Variables and parameters:
• u(x, t): Temperature at position x ∈ [0, L] and time t ≥ 0
• f (x): Initial temperature distribution, i.e., u(x, 0) = f (x)
• g1 (t): Boundary temperature at x = 0, i.e., u(0, t) = g1 (t)
• g2 (t): Boundary temperature at x = L, i.e., u(L, t) = g2 (t)
The heat equation (with conductivity = 1):
∂u ∂ 2u
= , 0 < x < L, t > 0
∂t ∂x2
Initial condition:
u(x, 0) = f (x), 0≤x≤L
Boundary conditions:
u(0, t) = g1 (t), u(L, t) = g2 (t), t≥0
