MA2605 – Professional Development and Project Work
Assignment 3
Distribution Date: Friday December 1st , 2023
Submission Deadline: 23:59 Friday December 29th, 2023
(through Wiseflow)
Feedback by: After exam panels and boards
Contribution to overall module assessment: 50%
Indicative student time working on
assessment:
20 hours
Main objective of the assessment: The objective of this task is to solve a range of problems
involving the numerical solution of differential equations. Solutions must be written up using LaTeX,
and numerical methods must be coded using MATLAB.
Description of the Assessment: Each student must submit a report (a single .pdf file), written using
LaTeX (article style). There is no hard page limit, but it should be possible to answer all questions
successfully without writing more than 10 pages. All MATLAB codes used to generate results in the
report should also be submitted in a .zip file, and it should be clearly stated in your answer to each
question which code(s) correspond(s) to that question. The report should be clearly titled, and should
address the solution of the following problems (in each question, 𝛼𝛼 and 𝛽𝛽 are, respectively, the last
and second to last non-zero digits of your student number - note also that most parts can be solved
independently, i.e. if you get stuck on one part then that should not prevent you from attempting the
other parts):
1. Consider the initial value problem:
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑 = cos
𝛼𝛼𝛼𝛼
4 , 𝑦𝑦(0) = 0, 0 ≤ 𝑡𝑡 ≤ 𝛽𝛽.
a. By showing that cos
𝛼𝛼𝑦𝑦
4 satisfies a particular condition (which you should state),
show that the problem has a unique solution. [10 marks]
b. Find the exact solution, showing your working. (Hint: you may find the following
formula helpful:
sec(𝑦𝑦) 𝑑𝑑𝑑𝑑 = ln tan
𝑦𝑦
2 +
𝜋𝜋
4
+ 𝐶𝐶 ,
where C is a constant.) [10 marks]
c. Use the Forward Euler Method to approximate the solution to the initial value
problem, and draw up a table comparing the error at 𝑡𝑡 = 𝛽𝛽 for an appropriate range of
time steps. Calculate: 𝑝𝑝 = log2
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 2𝜏𝜏
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝜏𝜏 for appropriate
values of τ, and explain how this could be used to test the conjecture: 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐶𝐶𝜏𝜏𝑝𝑝,
where 𝐶𝐶 is constant. [10 marks]
d. Repeat part c for the Trapezoidal method (an implicit method), using Fixed Point
Iteration to compute the results at each step. [10 marks]
e. Repeat part c using the modified Euler (predictor-corrector) method, for which you
should use the forward Euler method as a predictor, inserting that solution into the
right-hand side of the trapezoidal method equation. [10 marks]
f. Explicitly write out the steps of the four stage Runge Kutta method given by the
following Butcher Tableau, and then repeat part c using this method: [10 marks]
0 0 0 0 0
½ ½ 0 0 0
½ 0 ½ 0 0
1 0 0 1 0
1/6 1/3 1/3 1/6
g. Comment on the advantages and disadvantages of using each of the methods from
parts (c)-(f) above, for solving initial value problems such as the one in this question
[10 marks]
2. Consider the boundary value problem:
− 𝑑𝑑2𝑦𝑦
𝑑𝑑𝑥𝑥2 = 𝛼𝛼𝑥𝑥2 − 𝛽𝛽, 𝑥𝑥 ∈ (−1,1),
𝑦𝑦(−1) = 𝑦𝑦(1) = 0.
a. Determine the exact solution, by direct integration or otherwise. [10 marks]
b. Suppose 𝑁𝑁 is a positive even integer, ℎ = 2
𝑁𝑁, and define 𝑥𝑥𝑗𝑗 = −1 + 𝑗𝑗ℎ,𝑗𝑗 = 0, … , 𝑁𝑁.
Consider the following finite difference scheme for the numerical solution of the
boundary value problem:
− 𝑌𝑌𝑗𝑗+1 − 2𝑌𝑌𝑗𝑗 + 𝑌𝑌𝑗𝑗−1
ℎ2 = 𝛼𝛼𝑥𝑥𝑗𝑗
2 − 𝛽𝛽, 𝑗𝑗 = 1, … , 𝑁𝑁 − 1,
𝑌𝑌0 = 0, 𝑌𝑌𝑁𝑁 = 0,
where 𝑌𝑌𝑗𝑗 ≈ 𝑦𝑦 𝑥𝑥𝑗𝑗 , 𝑗𝑗 = 0, … , 𝑁𝑁. Rewrite this difference scheme as a system of linear
equations in matrix form with a vector of unknowns 𝑌𝑌 = (𝑌𝑌1, … , 𝑌𝑌𝑁𝑁−1)𝑇𝑇, and
comment on the structure of the matrix. [10 marks]
c. Write a code to compute 𝑌𝑌 for any given input 𝑁𝑁, and plot 𝑌𝑌 and the error on
different graphs, each for an appropriate range of values of 𝑁𝑁. Comment on your
results. [10 marks]
Learning outcomes to be assessed: The module learning outcomes relevant to this assessment are:
• Plan and implement numerical methods for differential equations using an appropriate
programming language. Illustrate the results using the language's graphics facilities. Analyse
and interpret the results of the numerical implementation in terms of the original problem;
• Choose with confidence and manipulate accurately the appropriate techniques to solve
problems with linear differential equations, including providing criteria for the accuracy of
numerical methods;
• Demonstrate the knowledge and understanding of the multiple skills necessary to operate in a
professional environment
Marking: the total mark available for this assignment is worth up to 50% of the available overall
mark for the module. Marks (out of 100) will be awarded for answers to the questions listed above
according to the stated mark distribution.
Submission instructions: Submission should be through WISEflow. Each student should submit
two files:
1. A single .pdf file, containing the full report. The name of this file should include the module
code and your student ID number, e.g. MA2605_1234567.pdf.
2. A zip file containing all MATLAB (.m) files used to generate the results in the .pdf. The
name of this file should also include the module code and your student ID number, e.g.
MA2605_1234567.zip.
If you are unsure how to download your .pdf file from Overleaf into a folder on your computer, then
please follow the instructions given in the following link:
https://www.overleaf.com/learn/how-to/Downloading_a_Project
Note that the first part of the instructions creates a .zip file containing all of the source files but not the
.pdf file. You will need to download the .pdf file separately by following the instructions on how to
download the finished .pdf. Please remember to back up your files periodically; it is your
responsibility to make sure that your files are securely backed up, and the safest way to do this is by
using the filestore at Brunel – details of how to do this can be found at:
https://intra.brunel.ac.uk/s/cc/kb/Pages/Saving-work-on-your-filestore-at-Brunel.aspx
You can login into Wiseflow directly at https://europe.wiseflow.net/login/uk/brunel.
Plagiarism and references: The university’s standard rules on plagiarism and collusion apply (see
https://www.brunel.ac.uk/life/library/SubjectSupport/Plagiarism for more information). This is an
individual assignment, and work submitted must be your own. Information from any research
undertaken (e.g., in text books or online) should be given credit where appropriate. The lecture on
academic misconduct and plagiarism, given on Thursday October 19th (week 6, lecture 13) is
available for viewing via the course Brightspace page, and you are strongly encouraged to watch this
if you have not done so already.
Please familiarise yourself with the university’s guidelines to students on the use of AI,
see https://students.brunel.ac.uk/study/using-artificial-intelligence-in-your-studies.
Late submission: The clear expectation is that you will submit your coursework by the submission
deadline. In line with the University’s policy on the late submission of coursework, coursework
submitted up to 48 hours late will be capped at a threshold pass (D-). Work submitted over 48 hours
after the stated deadline will automatically be given a fail grade (F). Please refer to
https://students.brunel.ac.uk/study/cedps/welcome-to-mathematics for information on submitting late
work, penalties applied, and procedures in the case of extenuating circumstances.