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MA2605 – Professional Development and Project Work Assignment 3

Main objective of the assessment: The objective of this task is to write a report (maximum 10 pages long) on the solution of a set of problems involving differential equations, their numerical solution, and related numerical approximation techniques. Solutions must be written up using LaTeX, and numerical methods must be coded using MATLAB. In this task, wherever the parameters α and β appear, you should replace them with the last (α) and second to last (β) non-zero digits of your student number (e.g., if your student number is 1234567 then you should take α= 7 and β = 6 wherever α and β appear below, or if your student number is 1234050 then take α = 5 and β = 4).

Description of the Assessment: Each student must submit (as a .pdf file) a report, written using LaTeX (article style). The maximum length of the report is 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 labelled with your student number, and should address the solution of the following problems. Note 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. Marking will be out of 100, with parts (a), (b), (c) and (d) each worth 20 marks, and with up to 20 available for the professionalism with which the report is presented, including a consideration of whether instructions have been followed, quality of LaTeX, and general presentation, including spelling, grammar and punctuation.

It is good practice to include the Matlab code in the report itself with explanations.

The report should have the following structure

1. Title: Choose a title appropriate to the content of the report.

2. Introduction: In this section you should introduce the report, and explain what you are going to do in it.

3. Main body: In this section (to which you should give an appropriate title), you should answer the following questions, which are concerned with how the population size of two different species of fish varies with time, depending on a number of factors, as follows:

a) The (nonnegative) size of a population of small fish in a lake, u(t), can be modelled by the

initial value problem:

where α on the right-hand side represents population growth proportional to the population size (due to breeding), and −βu2 on the right-hand side represents population decline proportional to the square of the population size (due to limited access to resource). Find theexact solution to this initial value problem,  showing your working, plot your solution for 0 ≤t≤ 5/α using Matlab, include the plot in your report, and explain how you would expect the solution to behave as t → ∞.

b) Use the Forward Euler Method, Backward Euler method, modified Euler (predictor-corrector) and the four stage Runge Kutta method given by the following Butcher Tableau, to approximate the solution to the initial value problem for 0 ≤ t ≤ 5/α , and draw up a table comparing the error at t = 5/α for all four methods and for at least N=10,100 and 1000 time steps. Do more time steps if you deem it suitable.

Hint: if you are unable to find the exact solution to the problem, try solving numerically anyway, and check that your numerical solution is behaving as expected by checking that its gradient matches what you might expect from the formula for du/dt, given above.

Hint: The implicit problem for Backward Euler can be solved explicitly, but you need to state the formula for Backward Euler explicitly and show how you have solved the implicit problem.

c) After a period of time, a new species of larger carnivorous fish, with (nonnegative) population size v(t), is introduced to the lake. These fish like to eat the smaller fish, and the size of the two populations then satisfies the initial value problem:

Explain in your own words what you think the new terms in the differential equations might represent. Solve this pair of coupled differential equations numerically with a single appropriate method of your choice, describing your approach clearly, plot the two populations over a suitable time interval (depending on α and β, you may need a much larger time interval than in parts (a) and (b)) using Matlab with a sufficiently large number of time steps, include the plot in your report, and explain your findings.

d) After a further period of time, some people decide to start fishing on the lake. They catch the smaller fish at a rate f, but are not interested in catching the larger carnivorous fish. The two populations of fish then satisfying the following differential equations:

By solving these differential equations numerically, using initial conditions and an appropriate numerical method of your choice, consider how the population size of both the smaller and larger fish is affected by the value of f, and determine an approximation to the maximum value of f so that neither species of fish is in danger of becoming extinct. Explain 3 your working carefully. Hint: to aid your explanations, you may find it helpful to include some plots of the two populations over suitable time intervals, for different values of f .

4. Conclusion: In this section, you should summarise your report, and draw some conclusions about how numerical approximation methods may be useful for solving practical problems.

5. References: Include a reference to at least one suitable resource (e.g., a book, or lecture notes etc), which should be cited at the appropriate location in the main text. This reference should be added to the bibliography using BibTeX.

Learning outcomes to be assessed: The module learning outcomes relevant to this assessment are:

• Plan and implement numerical methods 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 appropriate techniques for solving problems with 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 (marked out of 100) is worth up to 50% of the available overall mark for the module. To repeat what was stated above, up to 80 marks (out of 100) will be awarded for answers to the questions listed above, with up to 20 marks available for each of parts (a), (b), (c) and (d), and with up to 20 additional marks available for the professionalism with which the report is presented, including a consideration of whether instructions have been followed, quality of LaTeX, and general presentation, including spelling, grammar and punctuation.

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 follow the links to WISEflow through the module’s section on Blackboard Learn or login directly at https://europe.wiseflow.net/login/uk/brunel.Academic misconduct: This is an individual assignment, and work submitted must be your own.

Information from further research undertaken (e.g., online) should be given credit where appropriate.

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. The lecture slides on academic misconduct and plagiarism, given in Lecture 14 in week 7, is available via the course

Brightspace page, and you are strongly encouraged to read this if you have not done so already.

Misconduct in assessment is taken very seriously by the University. You are expected to abide by Senate Regulation 6 - Student Conduct (Academic and Non-Academic), which can be found here:

https://www.brunel.ac.uk/about/administration/governance-and-university-committees/senateregulations. Further advice on understanding what plagiarism and collusion are and how they can be avoided can be found here: https://www.brunel.ac.uk/life/library/SubjectSupport/Plagiarism

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 incur a 10% deduction from the original mark, if the mark is higher than 44 out of 100; marks between 44 and 40 will be capped at 40 (D-), marks below 40 will be unchanged. 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.