Faculty of Science, Engineering and Computing Assessment Form MA6500 Coursework Assignment Module: MA6500 Setters: Dr Peter Soan Title of Assignment: Coursework Assignment Deadline: 18/03/2021 Module...

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Faculty of Science, Engineering and Computing Assessment Form MA6500 Coursework Assignment Module: MA6500 Setters: Dr Peter Soan Title of Assignment: Coursework Assignment Deadline: 18/03/2021 Module weighting 25% Module Learning Outcomes assessed in this piece of coursework This assessment is designed to assess your ability in the following module learning outcomes: · use Fourier method of separation of variables to solve a partial differential equation, · use finite difference methods for solving PDEs and understand limitations of numerical methods. Assignment Brief and assessment criteria (these will be discussed within a timetabled class) This is a group written assignment – please work in groups on this assignment and submit one copy of the assignment per group. Please attempt all of the following tasks (approximate marks breakdown is given) – which explore the solution to a PDE problem approached in different ways (with possible limitations). Where indicated you should copy and paste commands and output from Matlab (or Excel) into a Word document and annotate this appropriately to explain, clearly, your method, results and comments/conclusions to the reader. Apart from this you may hand write solutions if you wish and scan your work into a single document which should be submitted via the dropbox in Canvas by 12.00 noon 18th March 2021. Marks will be awarded for completeness, correctness and clarity of solutions – you should state clearly all major steps in your calculations, explaining your method to the reader and justifying any conclusions. Similarly, annotations to computer output should be clear and comprehensive. Where you are asked to explore, your exploration should be systematic and logically planned – conclusions should be clear and concise. Effectively you are writing a report on this problem and different approaches to its solution – the division into tasks has been made to assist you, but you should remember that this is an exercise in communicating reasoning and conclusions as well as finding solutions. Marking scheme and feedback: Criterion Max mark (points) Your mark (points) Feedback comments Task 1 (Analytical Solution – Separation of Variables) 30 Task 2 (Finite Difference – Formulation and Hand Calculation) 18 Task 3 (Explicit Forward Difference Solution) 24 Task 4 (Matlab Built-in PDEPE) 28 Things that went well overall Things to work on for next time 1. Adapt (very slightly) the separation of variables method to solve the following partial differential equation: with the following conditions: What happens to the solution as t becomes larger? (Greater credit will be awarded for using the general separation approach to obtain a solution without invoking a substitution/transformation and/or a hunt for a solution of a particular form.) This is an exercise in the communication of mathematical ideas and arguments, as well as in a very minor adaptation of a standard approach. You are expected to discuss fully and clearly the method that you are using, stating any assumptions, explaining why steps taken are valid and why they are necessary. Discuss any options that you explore giving logical justification for your choices/conclusions. Little credit will be awarded for simply copying equations from your notes or a textbook. (30 marks) 2. (a)Consider the partial differential equation in task 1 above. Use the explicit (forward) finite difference method to find the approximate solution at and using and and working to 4 decimal place accuracy. Display your method/calculations clearly. Compare your results with the analytical solution. Discuss carefully, in detail, how you have implemented the derivative boundary conditions in your numerical calculations – citing references to any sources that you have researched and used. (15 marks) (b)Discuss very briefly, without going into detail, why you would not use the explicit scheme above to find the solution at t = 1.5 using and . (3 marks) 3.Implement the explicit (forward) finite difference method using Excel or Matlab to find the approximate solution to the partial differential equation in task 1 above at and using . Experiment with different values of between and and discuss your findings in the context of theoretically suitable values. You may wish to display your solution(s) using surface plots. Compare your results with the analytical solution. You may wish to include plots to help illustrate your comparisons. This task will require a spreadsheet or other software to be uploaded for verification. The spreadsheet or software file will need to be uploaded separately, but with your typed solutions (which should include demonstration of all key results and output). Absent or completely non-functioning software will forfeit the mark for the task. (24 marks) 4.The built-in routine pdepe in Matlab may be used to solve this type of partial differential equation numerically – it employs a different strategy from the standard finite difference approximations, converting the partial differential equation into a system of ‘semi-discrete’ ordinary differential equations. Use pdepe to find and plot the solution to the partial differential equation of task 1 above at t = 0.5, t = 1.5 and t=2.5 and compare with the analytical solution. Experiment with different values of between and and different values of between and – to see the effect of changing these parameters on the solution. Credit will be given for exploration that is systematic and logically planned – conclusions should be clear and concise. Include appropriately selected annotated input and output from Matlab showing your method and results. Comment briefly upon your results. (28 marks)