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- 首页 >> Algorithm 算法 MATH704 Linear Partial Differential Equations. Assignment 2
Due on Monday 30 March 2020, 4 pm
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Students may discuss this assignment with lecturer and other students. However, the entire
student’s work must be their own. No student may receive any part of the assignment from
another person, or let someone copy from their assignment. Failure to abide by this will result in
the assignment not being accepted.
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MATH704. Linear Partial Differential Equations. 2020.
Assignment 2
Question 1. The distribution of heat in a metal rod is modelled by the following nonhomogeneous
heat equation:
ut = 2uxx + 3 + 4 cos x
Solve the equation if the initial temperature u(x, 0) = 5 + 6 cos x + 7 cos 2x, and the ends of
the rod are insulated:
ux(0, t) = ux(π, t) = 0, t > 0.
Question 2. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 1 and fixed ends.
utt = uxx + xu(x, 0) = 3π3sin(πx)ut(x, 0) = x
u(0, t) = u(1, t) = 0, t > 0.
Question 3. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 2π and fixed ends.
utt = 4uxx + 2π − xu(x, 0) = 7 sinut(x, 0) = 0
u(0, t) = u(2π, t) = 0, t > 0.
Question 4. The following non-homogeneous Laplace equation (Poisson equation) models
the distribution of electrical potential when an outside charge is present:
uxx + uyy = π − x
Solve the equation subject to the following boundary conditions:
u(x, 0) = π − x, u(x, π) = 0,
u(0, y) = u(π, y) = 0.
Question 5. The temperature u(x, t) of a narrow metal rod of length L = π with a heat
source is modelled by the following non-homogeneous heat equation:
ut = uxx + x.
Solve the equation if the initial temperature u(x, 0) = 2π + 4x, and the ends are kept at
constant temperatures as follows:
u(0, t) = 2π, u(π, t) = 5π, t > 0.
Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for
w = u − c1x − c2.2
Due on Monday 30 March 2020, 4 pm
Attach this cover page to your assignment
Do not attach any other covers
If you did not attempt a question, write 0 in the
corresponding cell above.
Start each question on a new page. Write the number of each question in the middle of a row.
Do not copy the questions.
Show detailed working for each question, with pages stapled together and solutions in order
from the first question to the last question.
Solutions can be hand written but legible.
Highlight or frame final answers.
Not following these instructions means poor presentation and may reduce the grade by
up to 20 marks.
Completed assignments should be placed in the Assignment box on level 3 of WZ building,
next to the school reception, or submitted to the lecturer in class. Electronic submission is not
available.
Students may discuss this assignment with lecturer and other students. However, the entire
student’s work must be their own. No student may receive any part of the assignment from
another person, or let someone copy from their assignment. Failure to abide by this will result in
the assignment not being accepted.
It is my responsibility to keep a copy of my assignment
This assignment is entirely my own work
MATH704. Linear Partial Differential Equations. 2020.
Assignment 2
Question 1. The distribution of heat in a metal rod is modelled by the following nonhomogeneous
heat equation:
ut = 2uxx + 3 + 4 cos x
Solve the equation if the initial temperature u(x, 0) = 5 + 6 cos x + 7 cos 2x, and the ends of
the rod are insulated:
ux(0, t) = ux(π, t) = 0, t > 0.
Question 2. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 1 and fixed ends.
utt = uxx + xu(x, 0) = 3π3sin(πx)ut(x, 0) = x
u(0, t) = u(1, t) = 0, t > 0.
Question 3. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 2π and fixed ends.
utt = 4uxx + 2π − xu(x, 0) = 7 sinut(x, 0) = 0
u(0, t) = u(2π, t) = 0, t > 0.
Question 4. The following non-homogeneous Laplace equation (Poisson equation) models
the distribution of electrical potential when an outside charge is present:
uxx + uyy = π − x
Solve the equation subject to the following boundary conditions:
u(x, 0) = π − x, u(x, π) = 0,
u(0, y) = u(π, y) = 0.
Question 5. The temperature u(x, t) of a narrow metal rod of length L = π with a heat
source is modelled by the following non-homogeneous heat equation:
ut = uxx + x.
Solve the equation if the initial temperature u(x, 0) = 2π + 4x, and the ends are kept at
constant temperatures as follows:
u(0, t) = 2π, u(π, t) = 5π, t > 0.
Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for
w = u − c1x − c2.2