Summary WTW 238 Summer school Memorandum

UNIVERSITY OF PRETORIA
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
WTW 238 - MATHEMATICS ONLINE Summer school Examination 2021

February 2021
Total: 65
Time: 180 minutes to do the examination

External Examiner : Dr Q v d Hoff
Internal Examiners : Dr R Kufakunesu, Ms R Möller, Ms L Mostert, Dr H Ohlhoff

Surname & Initials: Student Nr:
The University of Pretoria commits itself to produce academic work of integrity. I affirm that I am aware of and have read the Rules and
Policies of the University, more specifically the Disciplinary Procedure and the Tests and Examinations Rules, which prohibit any unethical,
dishonest or improper conduct during tests, assignments, examinations and/or any other forms of assessment. I am aware that no student
or any other person may assist or attempt to assist another student, or obtain help, or attempt to obtain help from another student or any
other person during tests, assessments, assignments, examinations and/or any other forms of assessment.
SIGNATURE:

The paper consists of 3 pages and 17 questions (as well as a formula page).
SHOW ALL YOUR WORK
Read the following instructions
Prohibited

1. No notes, textbook or any other help is allowed. It is a closed book examination.
2. Calculators MAY NOT be used.

Format
1. Only handwritten submissions in one pdf file will be accepted.
2. Write the declaration, your surname, initials, student number and your signature clearly on the first page of your
submission.
3. Numerate your pages and keep them in the correct order and the correct way up.
4. Label each question (sub-question) clearly.
5. Draw a line with a ruler at the end of each question.
6. Show all your work and calculations and indicate how you use definitions and/or apply theorem

Submission

1. You have 180 minutes to do the examination.
2. You have 25 minutes to prepare your submission.
3. You have 10 minutes to submit.
4. Name your pdf when scanning your answers (to avoid submitting the wrong document).
5. Make sure that you are satisfied with your answers before submitting.
6. You have only ONE attempt to do the examination. A second attempt can only be considered if the content is the
same as the first attempt and it is clear that you had to submit twice due to submission difficulties.
7. If you can’t submit because the link is not available any more it means that you start submitting after the submission
time lapsed. It will therefore not be accepted.

Copyright reserved

, HEAT EQUATION
∞  nπx  Z L
X
−n2 π 2 kt/L2 2  nπx 
1) u(x, t) = cn e sin ; cn = f (x) sin dx
n=1
L L 0 L

∞  nπx  Z L
X 2
π 2 kt/L2 1
2) u(x, t) = c0 + cn e−n cos ; c0 = f (x)dx;
n=1
L L 0
Z L
2  nπx 
cn = f (x) cos dx, n ≥ 1
L 0 L


WAVE EQUATION
∞  nπx    Z L
X nπat 2  nπx 
1) y(x, t) = cn sin cos ; cn = f (x) sin dx
n=1
L L L 0 L

∞  nπx    Z L
X nπat 2  nπx 
2) y(x, t) = cn sin sin ; cn = g(x) sin dx
n=1
L L nπa 0 L

1
3) y(x, t) = [F (x + at) + F (x − at)]
2
Z x+at Z x
1 1
4) y(x, t) = G(s) ds = [H(x + at) − H(x − at)] with H(x) = G(s) ds
2a x−at 2a 0




EIGENVALUE PROBLEMS



1) y 00 + λy = 0; y(0) = y(L) = 0:

nπ 2 nπx



λn = L , yn (x) = sin L , n = 1, 2, 3, . . ..



2) y 00 + λy = 0; y(0) = y 0 (L) = 0:
 2  
(2n−1)π (2n−1)πx
λn = 2L , yn (x) = sin 2L , n = 1, 2, 3, . . ..



3) y 00 + λy = 0; y 0 (0) = y(L) = 0:
 2  
(2n−1)π (2n−1)πx
λn = 2L , yn (x) = cos 2L , n = 1, 2, 3, . . ..



4) y 00 + λy = 0; y 0 (0) = y 0 (L) = 0:

λ0 = 0, y0 (x) = 1 and
2
λn = nπ nπx



L , yn (x) = cos L , n = 1, 2, 3, . . ..