,APM3701 Assignment 2 (COMPLETE ANSWERS)
2025 (700123) - DUE 8 August 2025; 100%
TRUSTED Complete, trusted solutions and
explanations.
MULTIPLE CHOICE,ASSURED EXCELLENCE
apm3700 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|
QUESTION 2 Find the displacement u (x, t) of a semi–infinite
vibrating string, if the finite end is fixed, the initial velocity is
zero and the initial displacement is xex at every point x of the
string. (Explain all the details) [25 Marks]
Question 1: Heat Flow in a Rod
We model the heat conduction using the heat equation:
(a) Initial-Boundary Value Problem (IBVP)
The temperature u(x,t)u(x,t)u(x,t) in a rod of length LLL satisfies
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, \quad t > 0.∂t∂u
=∂x2∂2u,0<x<L,t>0.
Boundary conditions:
The temperature at the left end x=0x=0x=0 is given by
u(0,t)=g1(t)u(0,t) = g_1(t)u(0,t)=g1(t).
The temperature at the right end x=Lx=Lx=L is given by
u(L,t)=g2(t)u(L,t) = g_2(t)u(L,t)=g2(t).
Initial condition:
u(x,0)=f(x),0≤x≤L.u(x,0) = f(x), \quad 0 \leq x \leq
L.u(x,0)=f(x),0≤x≤L.
2025 (700123) - DUE 8 August 2025; 100%
TRUSTED Complete, trusted solutions and
explanations.
MULTIPLE CHOICE,ASSURED EXCELLENCE
apm3700 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|
QUESTION 2 Find the displacement u (x, t) of a semi–infinite
vibrating string, if the finite end is fixed, the initial velocity is
zero and the initial displacement is xex at every point x of the
string. (Explain all the details) [25 Marks]
Question 1: Heat Flow in a Rod
We model the heat conduction using the heat equation:
(a) Initial-Boundary Value Problem (IBVP)
The temperature u(x,t)u(x,t)u(x,t) in a rod of length LLL satisfies
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, \quad t > 0.∂t∂u
=∂x2∂2u,0<x<L,t>0.
Boundary conditions:
The temperature at the left end x=0x=0x=0 is given by
u(0,t)=g1(t)u(0,t) = g_1(t)u(0,t)=g1(t).
The temperature at the right end x=Lx=Lx=L is given by
u(L,t)=g2(t)u(L,t) = g_2(t)u(L,t)=g2(t).
Initial condition:
u(x,0)=f(x),0≤x≤L.u(x,0) = f(x), \quad 0 \leq x \leq
L.u(x,0)=f(x),0≤x≤L.
