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- 首页 >> Algorithm 算法作业 MATH704 Linear Partial Differential Equations. Assignment 2

Due on Monday 30 March 2020, 4 pm

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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

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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