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APM3701
Assignment 2
(COMPLETE
ANSWERS) 2025
(700123) - DUE
8 August 2025
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[Type the abstract of the document here. The abstract is typically a short summary of the contents of
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APM3701 Assignment 2 (COMPLETE
ANSWERS) 2025 (700123) - DUE 8 August
2025Save 3 minutes reading time
Course
Partial Differential Equations (APM3701)
Institution
University Of South Africa (Unisa)
Book
Differential Equations & Linear Algebra
APM3701 Assignment 2 (COMPLETE ANSWERS) 2025 (700123) - DUE 8
August 2025; 100% TRUSTED Complete, trusted solutions and explanations.
Ensure your success with us...
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 3 When there is heat transfer
from the lateral side of an infinite cylinder of radius a into a surrounding
medium, the temperature inside the rod depends upon the time t and the
distance r from its longitudinal axis (i.e. the axis through the centre and
APM3701
Assignment 2
(COMPLETE
ANSWERS) 2025
(700123) - DUE
8 August 2025
[Type the document subtitle]
[Pick the date]
[Type the abstract of the document here. The abstract is typically a short summary of the contents of
the document. Type the abstract of the document here. The abstract is typically a short summary of
, the contents of the document.]
APM3701 Assignment 2 (COMPLETE
ANSWERS) 2025 (700123) - DUE 8 August
2025Save 3 minutes reading time
Course
Partial Differential Equations (APM3701)
Institution
University Of South Africa (Unisa)
Book
Differential Equations & Linear Algebra
APM3701 Assignment 2 (COMPLETE ANSWERS) 2025 (700123) - DUE 8
August 2025; 100% TRUSTED Complete, trusted solutions and explanations.
Ensure your success with us...
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 3 When there is heat transfer
from the lateral side of an infinite cylinder of radius a into a surrounding
medium, the temperature inside the rod depends upon the time t and the
distance r from its longitudinal axis (i.e. the axis through the centre and
