Case study
- Apply the separation of variables method to solve the following partial differential equation:
with the following conditions:
You may assume that the solution consists of a steady state solution and a transient solution , where and its partial derivatives tend to zero as .
This is an exercise in the communication of mathematical ideas and arguments. 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 clear, concise, logical justification for your choices/conclusions. Little credit will be awarded for simply copying equations from your notes or a textbook.
(35 marks)
- 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, explicitly demonstrating how you derive the equations that you use. Compare your results with the analytical solution to assess the accuracy of the approximate solution (do not expect extremely high accuracy).
(30 marks)
- 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 or close to, (depending upon your choice of ) , , , and using . Experiment with different values of to obtain one solution which is quite accurate for all values of up to and one which demonstrates instability (a wildly fluctuating approximate solution). Compare your results with the analytical solution. You may wish to include plots (e.g. profile plots of the solution for all at given values of ) to help illustrate your comparisons, possibly in addition to tables of data. Discuss how the accuracy of your numerical solutions varies with time and in particular discuss how instability manifests itself in your solutions (how and where you can see the effects). You are not expected to produce detailed theoretical analysis (and note that the equation being solved is not the standard dimensionless form of the heat equation so the general theoretical analysis of instability in the notes is not immediately directly applicable). (Note that for some choices of you may not obtain output exactly at or etc. – in such a case it will be acceptable to report your solution for a value of very close to or etc.)
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 – credit will awarded for the clarity of your discussion and insight shown). Absent or completely non-functioning software will forfeit the mark for the task.
(35 marks)
