,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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, Let's go through this problem step by step.
Part (a) - Initial-Boundary Problem
We need to set up an initial-boundary value problem (IBVP) for
the temperature distribution u(x,t)u(x,t)u(x,t) in the rod at any
point xxx and time ttt.
Explanation of Variables and Parameters:
u(x,t)u(x,t)u(x,t) is the temperature at point xxx on the rod
at time ttt.
LLL is the length of the rod.
f(x)f(x)f(x) is the initial temperature distribution along the
rod at t=0t = 0t=0. So, u(x,0)=f(x)u(x, 0) = f(x)u(x,0)=f(x).
g1(t)g_1(t)g1(t) is the temperature of the medium at the left
end of the rod (x=0x = 0x=0) at time ttt, so u(0,t)=g1(t)u(0,
t) = g_1(t)u(0,t)=g1(t).
g2(t)g_2(t)g2(t) is the temperature of the medium at the
right end of the rod (x=Lx = Lx=L) at time ttt, so
u(L,t)=g2(t)u(L, t) = g_2(t)u(L,t)=g2(t).
The heat conductivity is 1, meaning the rod’s material has
constant conductivity.
The heat equation for temperature distribution in the rod,
assuming the heat conductivity is constant, is given by the heat
equation:
∂u∂t=∂2u∂x2,0<x<L, t>0.\frac{\partial u}{\partial t} =
\frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, \, t > 0.∂t∂u
=∂x2∂2u,0<x<L,t>0.
Now, let’s add the initial and boundary conditions:
(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
Downloaded by Corona Virus () lOMoARcPSD|
, Let's go through this problem step by step.
Part (a) - Initial-Boundary Problem
We need to set up an initial-boundary value problem (IBVP) for
the temperature distribution u(x,t)u(x,t)u(x,t) in the rod at any
point xxx and time ttt.
Explanation of Variables and Parameters:
u(x,t)u(x,t)u(x,t) is the temperature at point xxx on the rod
at time ttt.
LLL is the length of the rod.
f(x)f(x)f(x) is the initial temperature distribution along the
rod at t=0t = 0t=0. So, u(x,0)=f(x)u(x, 0) = f(x)u(x,0)=f(x).
g1(t)g_1(t)g1(t) is the temperature of the medium at the left
end of the rod (x=0x = 0x=0) at time ttt, so u(0,t)=g1(t)u(0,
t) = g_1(t)u(0,t)=g1(t).
g2(t)g_2(t)g2(t) is the temperature of the medium at the
right end of the rod (x=Lx = Lx=L) at time ttt, so
u(L,t)=g2(t)u(L, t) = g_2(t)u(L,t)=g2(t).
The heat conductivity is 1, meaning the rod’s material has
constant conductivity.
The heat equation for temperature distribution in the rod,
assuming the heat conductivity is constant, is given by the heat
equation:
∂u∂t=∂2u∂x2,0<x<L, t>0.\frac{\partial u}{\partial t} =
\frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, \, t > 0.∂t∂u
=∂x2∂2u,0<x<L,t>0.
Now, let’s add the initial and boundary conditions:
