,APM3701 Assignment 2 (COMPLETE ANSWERS) 2025
(700123) - DUE 8 August 2025; 100% TRUSTED Complete,
trusted solutions and explanations.
QUESTION 1 Consider the heat flow in an horizontal rod of
length L units and heat conductivity 1. We assume that initially
the rod was submerged in a meduim 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 which temperatures change with time and
are described by the functions g1 (t) and g2 (t) respectively. (a)
Write down the initial-boundary problem satisfied by the
temperature distribution u (x, t) in the rod at any point x and
time t (Explain all the meaning of the variables and parameters
used). (5 Marks) (b) Suppose that f, g1, g2 are bounded, there
exist constants m and M such that for all x in the domain of g1
and g2, and all t 0, we have m f (x) M;m g1 (x)
M;m g2 (x) M; and the temperature u (x, t) solution of the
IBVP described above satisfies the inequalities m u (x, t)
M; for all x and t 0. Show that the solution u (x, t) of the heat
problem described above is unique. (Explain clearly all the steps
(10 Marks) (c) Suppose that u1 (x, t) and u2 (x, t) are solutions
of the heat problem above (with different initial and boundary
conditions) are such that u1 (0, t) u2 (0, t) , u1 (L, t) u2 (L,
t) , and u1 (x, 0) u2 (x, 0) . Show that u1 (x, t) u2 (x, t) for
all 0 x L and all t 0. (10 Marks) [25 Marks] 7
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, (a) Initial-Boundary Value Problem (IBVP)
The heat equation governs the temperature distribution
u(x,t)u(x,t)u(x,t) in a rod over time. This is a parabolic partial
differential equation (PDE), which can be written as:
∂u∂t=∂2u∂x2for0<x<Landt>0.\frac{\partial u}{\partial t} =
\frac{\partial^2 u}{\partial x^2} \quad \text{for} \quad 0 < x < L
\quad \text{and} \quad t > 0.∂t∂u=∂x2∂2ufor0<x<Landt>0.
Here:
u(x,t)u(x,t)u(x,t) is the temperature at position xxx and time
ttt.
LLL is the length of the rod.
The heat conductivity of the rod is assumed to be 111,
simplifying the PDE.
Initial Condition: At t=0t = 0t=0, the temperature distribution
along the rod is given by f(x)f(x)f(x):
u(x,0)=f(x)for0≤x≤L.u(x, 0) = f(x) \quad \text{for} \quad 0 \leq
x \leq L.u(x,0)=f(x)for0≤x≤L.
Boundary Conditions: The temperature at the left end (at x=0x
= 0x=0) and the right end (at x=Lx = Lx=L) is governed by the
functions g1(t)g_1(t)g1(t) and g2(t)g_2(t)g2(t), respectively.
These describe how the temperatures at the ends of the rod
change over time:
u(0,t)=g1(t)fort≥0,u(0, t) = g_1(t) \quad \text{for} \quad t \geq
0,u(0,t)=g1(t)fort≥0, u(L,t)=g2(t)fort≥0.u(L, t) = g_2(t) \quad
\text{for} \quad t \geq 0.u(L,t)=g2(t)fort≥0.
(700123) - DUE 8 August 2025; 100% TRUSTED Complete,
trusted solutions and explanations.
QUESTION 1 Consider the heat flow in an horizontal rod of
length L units and heat conductivity 1. We assume that initially
the rod was submerged in a meduim 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 which temperatures change with time and
are described by the functions g1 (t) and g2 (t) respectively. (a)
Write down the initial-boundary problem satisfied by the
temperature distribution u (x, t) in the rod at any point x and
time t (Explain all the meaning of the variables and parameters
used). (5 Marks) (b) Suppose that f, g1, g2 are bounded, there
exist constants m and M such that for all x in the domain of g1
and g2, and all t 0, we have m f (x) M;m g1 (x)
M;m g2 (x) M; and the temperature u (x, t) solution of the
IBVP described above satisfies the inequalities m u (x, t)
M; for all x and t 0. Show that the solution u (x, t) of the heat
problem described above is unique. (Explain clearly all the steps
(10 Marks) (c) Suppose that u1 (x, t) and u2 (x, t) are solutions
of the heat problem above (with different initial and boundary
conditions) are such that u1 (0, t) u2 (0, t) , u1 (L, t) u2 (L,
t) , and u1 (x, 0) u2 (x, 0) . Show that u1 (x, t) u2 (x, t) for
all 0 x L and all t 0. (10 Marks) [25 Marks] 7
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, (a) Initial-Boundary Value Problem (IBVP)
The heat equation governs the temperature distribution
u(x,t)u(x,t)u(x,t) in a rod over time. This is a parabolic partial
differential equation (PDE), which can be written as:
∂u∂t=∂2u∂x2for0<x<Landt>0.\frac{\partial u}{\partial t} =
\frac{\partial^2 u}{\partial x^2} \quad \text{for} \quad 0 < x < L
\quad \text{and} \quad t > 0.∂t∂u=∂x2∂2ufor0<x<Landt>0.
Here:
u(x,t)u(x,t)u(x,t) is the temperature at position xxx and time
ttt.
LLL is the length of the rod.
The heat conductivity of the rod is assumed to be 111,
simplifying the PDE.
Initial Condition: At t=0t = 0t=0, the temperature distribution
along the rod is given by f(x)f(x)f(x):
u(x,0)=f(x)for0≤x≤L.u(x, 0) = f(x) \quad \text{for} \quad 0 \leq
x \leq L.u(x,0)=f(x)for0≤x≤L.
Boundary Conditions: The temperature at the left end (at x=0x
= 0x=0) and the right end (at x=Lx = Lx=L) is governed by the
functions g1(t)g_1(t)g1(t) and g2(t)g_2(t)g2(t), respectively.
These describe how the temperatures at the ends of the rod
change over time:
u(0,t)=g1(t)fort≥0,u(0, t) = g_1(t) \quad \text{for} \quad t \geq
0,u(0,t)=g1(t)fort≥0, u(L,t)=g2(t)fort≥0.u(L, t) = g_2(t) \quad
\text{for} \quad t \geq 0.u(L,t)=g2(t)fort≥0.
