,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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, Part (a): Initial-Boundary Value Problem (IBVP)
We are tasked with describing the temperature distribution
u(x,t)u(x, t)u(x,t) in the rod at any point xxx and time ttt, given
the conditions.
Variables and Parameters:
u(x,t)u(x,t)u(x,t) is the temperature at position xxx and time
ttt.
f(x)f(x)f(x) is the initial temperature distribution in the rod
at t=0t = 0t=0, with f(x)f(x)f(x) defined for 0≤x≤L0 \leq x
\leq L0≤x≤L.
g1(t)g_1(t)g1(t) is the temperature at the left end of the rod
at time ttt, i.e., at x=0x = 0x=0.
g2(t)g_2(t)g2(t) is the temperature at the right end of the
rod at time ttt, i.e., at x=Lx = Lx=L.
LLL is the length of the rod.
Heat Equation:
The temperature distribution u(x,t)u(x,t)u(x,t) in the rod satisfies
the heat equation, which describes the flow of heat:
∂u(x,t)∂t=∂2u(x,t)∂x2\frac{\partial u(x,t)}{\partial t} =
\frac{\partial^2 u(x,t)}{\partial x^2}∂t∂u(x,t)=∂x2∂2u(x,t)
This equation models the temperature evolution in a material
with constant thermal conductivity.
Initial Condition:
At t=0t = 0t=0, the temperature distribution is given by
f(x)f(x)f(x):
(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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, Part (a): Initial-Boundary Value Problem (IBVP)
We are tasked with describing the temperature distribution
u(x,t)u(x, t)u(x,t) in the rod at any point xxx and time ttt, given
the conditions.
Variables and Parameters:
u(x,t)u(x,t)u(x,t) is the temperature at position xxx and time
ttt.
f(x)f(x)f(x) is the initial temperature distribution in the rod
at t=0t = 0t=0, with f(x)f(x)f(x) defined for 0≤x≤L0 \leq x
\leq L0≤x≤L.
g1(t)g_1(t)g1(t) is the temperature at the left end of the rod
at time ttt, i.e., at x=0x = 0x=0.
g2(t)g_2(t)g2(t) is the temperature at the right end of the
rod at time ttt, i.e., at x=Lx = Lx=L.
LLL is the length of the rod.
Heat Equation:
The temperature distribution u(x,t)u(x,t)u(x,t) in the rod satisfies
the heat equation, which describes the flow of heat:
∂u(x,t)∂t=∂2u(x,t)∂x2\frac{\partial u(x,t)}{\partial t} =
\frac{\partial^2 u(x,t)}{\partial x^2}∂t∂u(x,t)=∂x2∂2u(x,t)
This equation models the temperature evolution in a material
with constant thermal conductivity.
Initial Condition:
At t=0t = 0t=0, the temperature distribution is given by
f(x)f(x)f(x):
