APM3701 Assignment 2
(COMPLETE ANSWERS) 2025
(700123) - DUE 8 August 2025
100% GUARANTEEED
,APM3701 Assignment 2 (COMPLETE
ANSWERS) 2025 (700123) - DUE 8
August 2025
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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Here’s how you can approach each part of the question systematically:
(a) Initial-Boundary Value Problem (IBVP) Formulation
The heat equation governing the temperature distribution u ( x , t ) in a
horizontal rod of length L is given by:
∂u ∂2 u
= , 0< x < L ,t >0
∂ t ∂ x2
Explanation of Variables and Parameters:
u(x,t ): Temperature at position x and time t .
x : Position along the rod, 0 ≤ x ≤ L.
t : Time, t ≥ 0.
∂u
∂t
: Rate of change of temperature with respect to time.
∂2 u
: Spatial diffusion of heat (assuming thermal conductivity is 1).
∂ x2
Initial Condition:
u ( x , 0 )=f ( x ) , 0≤ x ≤ L
This describes the initial temperature distribution along the rod.
Boundary Conditions:
u ( 0 , t ) =g1 ( t ) ,u ( L , t ) =g 2 ( t ) ,t >0
These describe the time-dependent temperatures maintained at the ends
of the rod.
(COMPLETE ANSWERS) 2025
(700123) - DUE 8 August 2025
100% GUARANTEEED
,APM3701 Assignment 2 (COMPLETE
ANSWERS) 2025 (700123) - DUE 8
August 2025
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
Downloaded by Corona Virus () lOMoARcPSD|
Here’s how you can approach each part of the question systematically:
(a) Initial-Boundary Value Problem (IBVP) Formulation
The heat equation governing the temperature distribution u ( x , t ) in a
horizontal rod of length L is given by:
∂u ∂2 u
= , 0< x < L ,t >0
∂ t ∂ x2
Explanation of Variables and Parameters:
u(x,t ): Temperature at position x and time t .
x : Position along the rod, 0 ≤ x ≤ L.
t : Time, t ≥ 0.
∂u
∂t
: Rate of change of temperature with respect to time.
∂2 u
: Spatial diffusion of heat (assuming thermal conductivity is 1).
∂ x2
Initial Condition:
u ( x , 0 )=f ( x ) , 0≤ x ≤ L
This describes the initial temperature distribution along the rod.
Boundary Conditions:
u ( 0 , t ) =g1 ( t ) ,u ( L , t ) =g 2 ( t ) ,t >0
These describe the time-dependent temperatures maintained at the ends
of the rod.
