MA2605代做、代写MATLAB编程语言

日期：2023-12-28 10:44

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, ???? and ???? 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:

????????

???????? = cos  

????????

4   , ????(0) = 0, 0 ≤ ???? ≤ ????.

a. By showing that cos  

????????

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(????) ???????? = ln  tan  

????

2 +

????

4

  + ???? ,

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 ???? = ???? for an appropriate range of

time steps. Calculate: ???? = log2  

???????????????????? ???????????????????????????????????????? ???????????????????? ???????? ???????????????????? ???? 2????

???????????????????? ???????????????????????????????????????? ???????????????????? ???????? ???????????????????? ???? ????   for appropriate

values of τ, and explain how this could be used to test the conjecture: ???????????????????? = ????????????,

where ???? 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

? ? 0 0 0

? 0 ? 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:

? ????2????

????????2 = ????????2 ? ????, ???? ∈ (?1,1),

????(?1) = ????(1) = 0.

a. Determine the exact solution, by direct integration or otherwise. [10 marks]

b. Suppose ???? is a positive even integer, ? = 2

????, and define ???????? = ?1 + ?????,???? = 0, … , ????.

Consider the following finite difference scheme for the numerical solution of the

boundary value problem:

?  ????????+1 ? 2???????? + ?????????1

?2   = ????????????

2 ? ????, ???? = 1, … , ???? ? 1,

????0 = 0, ???????? = 0,

where ???????? ≈ ???? ???????? , ???? = 0, … , ????. Rewrite this difference scheme as a system of linear

equations in matrix form with a vector of unknowns ???? = (????1, … , ?????????1)????, and

comment on the structure of the matrix. [10 marks]

c. Write a code to compute ???? for any given input ????, and plot ???? and the error on

different graphs, each for an appropriate range of values of ????. 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.