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APM3701
Assignment 1
(COMPLETE
ANSWERS) 2025
() - DUE 29 May
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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 1 (COMPLETE
ANSWERS) 2025 (608471) - DUE 29 May Save 3
minutes reading time
Course
Partial Differential Equations (APM3701)
Institution
University Of South Africa (Unisa)
Book
Differential Equations
APM3701 Assignment 1 (COMPLETE ANSWERS) 2025 (608471) - DUE 29 May
2025; 100% TRUSTED Complete, trusted solutions and explanations.. Ensure
your success with us...
QUESTION 1 Solve the following (initial)-boundary value problem, a. uxy (x,
y) = xy3, x, y 0. u (x, 0) = f (x) , and uy (0, y) = g (y) . Determine u (x, y) , if
f (x) = cosx and g (y) = y+sin y. (Check your answer by substituting, and
explain all the steps clearly) (15 Marks) b. xux + yuy = yu u (2x2, x) = x2 −
1. (1) (Check your answer by substituting before applying the boundary
condition, and explain all the steps clearly) (15 Marks) [30 Marks] QUESTION
2 Consider the heat flow in a rod of length a unit and heat conductivity c. a.
If initially the rod was submerged in a fluid which is kept at a temperature
K◦C, write down the initial boundary value problem that is satisfied by the
rod, if the heat flux is of k unit at the left end and A units at the right end.
(Explain the meaning of every constant and variable). (5 Marks) b. Determine
the temperature of the rod at any point x of the rod at time t > 0. (Explain all
the steps). (20 Marks) [25 Marks] QUESTION 3 Consider the temperature
distribution in a rectangular plate as described in the figure below.
Determine the steady-state temperature distribution u (x, y) in the plate.
Calculate the steady-sate temperature at the centre of the plate. [25 Marks]
W L Ambiant temperature Ice at 00 Ice at 00 Ice at 00 5 Downloaded by
Corona Virus () lOMoARcPSD| QUESTION 4 A homogeneous and infinitely long
string is stretched horizontally and vibrates vertically. If its vertical vibration
at position x and at time t of is determined by the function u = u (x, y).
Suppose that when the string is straight, it has a linear density of 9 and the
tension at any given point of the string is 3. a. Derive the partial differential
equation satisfied by u . [Do not repeat the derivation done in the Study
Guide, you may use the appropriate formula and explain the constants and
APM3701
Assignment 1
(COMPLETE
ANSWERS) 2025
() - DUE 29 May
[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 1 (COMPLETE
ANSWERS) 2025 (608471) - DUE 29 May Save 3
minutes reading time
Course
Partial Differential Equations (APM3701)
Institution
University Of South Africa (Unisa)
Book
Differential Equations
APM3701 Assignment 1 (COMPLETE ANSWERS) 2025 (608471) - DUE 29 May
2025; 100% TRUSTED Complete, trusted solutions and explanations.. Ensure
your success with us...
QUESTION 1 Solve the following (initial)-boundary value problem, a. uxy (x,
y) = xy3, x, y 0. u (x, 0) = f (x) , and uy (0, y) = g (y) . Determine u (x, y) , if
f (x) = cosx and g (y) = y+sin y. (Check your answer by substituting, and
explain all the steps clearly) (15 Marks) b. xux + yuy = yu u (2x2, x) = x2 −
1. (1) (Check your answer by substituting before applying the boundary
condition, and explain all the steps clearly) (15 Marks) [30 Marks] QUESTION
2 Consider the heat flow in a rod of length a unit and heat conductivity c. a.
If initially the rod was submerged in a fluid which is kept at a temperature
K◦C, write down the initial boundary value problem that is satisfied by the
rod, if the heat flux is of k unit at the left end and A units at the right end.
(Explain the meaning of every constant and variable). (5 Marks) b. Determine
the temperature of the rod at any point x of the rod at time t > 0. (Explain all
the steps). (20 Marks) [25 Marks] QUESTION 3 Consider the temperature
distribution in a rectangular plate as described in the figure below.
Determine the steady-state temperature distribution u (x, y) in the plate.
Calculate the steady-sate temperature at the centre of the plate. [25 Marks]
W L Ambiant temperature Ice at 00 Ice at 00 Ice at 00 5 Downloaded by
Corona Virus () lOMoARcPSD| QUESTION 4 A homogeneous and infinitely long
string is stretched horizontally and vibrates vertically. If its vertical vibration
at position x and at time t of is determined by the function u = u (x, y).
Suppose that when the string is straight, it has a linear density of 9 and the
tension at any given point of the string is 3. a. Derive the partial differential
equation satisfied by u . [Do not repeat the derivation done in the Study
Guide, you may use the appropriate formula and explain the constants and
