Department of Mathematics and Statistics
Issued: Monday 19th February Due: Noon Friday 2nd March
The assignment counts towards 5% of your grade for this module. There are a total of 10 marks
available for this assignment. The number of available marks for each question is indicated on
the right.
You must complete an assessed work coversheet and submit this attached to your work to the
appropriate submission box at JJT Support Centre by the deadline. Marking will be anonymous
and so please do not write your name on your work. (Your name should however be
written on the assessed work coversheet. This will be later obscured by the Support Centre
before passing onto the marker.) It is important that you write your student number both on
the coversheet and each page of your work. A receipt will be emailed to you. You should
contact the Support Centre immediately if you do not receive the receipt within one
working day of the submission deadline for the work.
If you submit your work after the deadline, but before noon Friday 9th March, 10% of the total
marks available for that piece of work will be deducted from the mark for each working day (or
part thereof) following the deadline up to a total of ve working days. If you submit your work
after noon Friday 9th March it will not be marked and a score of 0 will be recorded. These
deadline and penalties are absolute unless an extension has been agreed by your Senior Tutor
for reasons of extenuating circumstances.
1. Use the heat kernel to solve the initial value problem
ut(x;t) kuxx(x;t) = 0 8x2R;t > 0
subject to
u(x;0) = (x 1)2 8x2R:
(Hint: you may wish to refer back to Example 94 in Handout 8.) [4]
2. Use Duhamel’s principle and the heat kernel to solve the initial value problem
ut(x;t) kuxx(x;t) = xe t 8x2R;t > 0
subject to
u(x;0) = 0 8x2R:
(Hint: you may wish to refer back to Example 98 in Handout 8.) [4]
3. Thus solve the initial value problem
ut(x;t) kuxx(x;t) = xe t 8x2R;t > 0
subject to
u(x;0) = (x 1)2 8x2R:
A full explanation of how you reach your answer must be given. [2]
Total marks available: [10]
Although no marks are allocated for doing so, you are strongly advised to check your
answers to each question by substitution back into the problem.
MA2DE Peter Sweby