Extra class to discuss problem sheet 2

There will be an extra class to spend some time going through the second problem sheet. It will start promptly at 3pm in DHSR2 on Monday 4/14.

I hope to see you there. (You will gain more out of this if you work on the problems before then.)

Nobody submitted a write-up for problem sheet 2?

I checked the box for this lecture course, and it looks like nobody submitted a write-up for the second set of practice problems. Please correct me if I'm wrong, and I'll go and mark the problems. Sigh...

Anyway, we'll still go over these problems in week 0. (Let me know about good times...) But please be aware that you'll get a lot more out of it if you try the problems between now and then.

Review class for homework sheet 2

As we discussed in lecture, I am going to hold a review class for homework sheet 2 during week 0. The format will be similar to the review class for the other half of the class, but I anticipate that the big picture summary of material from the lectures will be shorter than for distributions because I think that you found those lecturers easier to follow than the ones on distributions. (You should let me know if I am mistaken. We have a limited time, and I want to try to maximize how much it helps you.)

You are supposed to take the Exam on Thursday April 17th, so we are going to have this class on Monday April 14th (when you are required to be back). If you have questions before then, you can ask them here or by e-mail. I will be around all break. I'm sorry that I am unable to do the review this week---I need to finish the marking first, I am out of town on Thursday, and I am required to sit in on the MSc project presentations on Friday because I will be marking them in September.

Will 3pm on April 17th work for everybody or do I need to adjust this time to make sure that everybody who wants to come can come. Please let me know.

Solution Set 2 posted

I have posted solution set 2. I tried to add explanations for various notational items and remove as many typos as I could, but if there are any issues remaining, please bring them to my attention ASAP (and, in particular, {\it before} the week 0 review class for this sheet) so that I can update the document appropriately. You can either post these here or send them to me by e-mail.

I will pick up the submitted problem sheets tomorrow and will mark them in the next couple of days. I'll let you know when they are available.

Bicycles in London: eBourbaki mathematical modeling contest

Eliana Hechter of eBourbaki just sent me an e-mail about the following:

e-Bourbaki, "a mathematical problem-solving firm whose mission is to solve the world's mathematical problems using contests to inspire innovation and creativity", sponsors a mathematical modeling contest every spring and the recently-announced 2008 contest pertains to bicycles in London.

I encourage any and all mathematics students to participate in this!

Here is what Eliana wrote in her e-mail:

I am writing to inform you about an upcoming contest for students of
mathematics, computer science, and engineering in the United
Kingdom. The contest, which asks students to mathematically model a
low-cost bicycle rental service for the City of London, will
commence at 5pm on Monday, May 5, 2008 and end at 5pm on Monday, May
12, 2008. We hope that you will encourage the students with whom you
have contact to participate.

The first prize is ?1000 and winning projects may have the potential
to be implemented as London strives to become more sustainable and
efficient in its transportation infrastructure.

The contest is hosted by eBourbaki, a mathematical problem-solving
website I founded dedicated to increasing interest and involvement
in mathematics in the global community, and is sponsored by Winton
Capital. Further information can be found at our website,
www.ebourbaki.com. Participation in the contest requires contestants
to register with eBourbaki website and we encourage participants to
do so as soon as possible, to stay informed about contest details.

In the past others have used eBourbaki questions as part of their
teaching program, or as a 'final project' for applied modeling
classes. We are happy to work with professors if they choose to
consider this option.

Supplementary class on Monday Feb 25 (week 7), 4pm

There is a supplementary class scheduled for Monday of week 7 (Feb 25th) starting at 4pm in DHSR2 (seminar room 2 in Dartington House; it's labeled as room 18 --- it's right across from my office). I'll spend a little bit of time going through the big picture on distributions and then we'll go through some of the problems on sheet 1. (Note that we probably will not have time to go through all of them.)

Update to Solution Sheet 1

I made a small update to problem (2a) in the solution sheet 1. Namely, it now includes a precise statement of weak convergence, as I think that will help you understand the solution better. (Weak convergence means that it converges in inner product for all test functions. In the context of this course, this can be expressed in integral form. The solution set shows both of these.)

Solution Sheet 1 posted

The solution sheet for the first problem sheet has been posted. Only 4 people submitted their work on this sheet. That's not a particularly optimal way to get potentially very useful feedback.

I am still awaiting more feedback for when to have the extra class to go over the big picture a bit and also to spend some time on the first homework sheet (though we won't have time to go over everything). Four people have expressed opinions about that. I will wait a little bit more time for others to indicate when they're available, and then I will figure out a day and time from this input.

Supplementary class for distributions

I have received an e-mail requesting an extra lecture to review some of the big picture from the unit on distributions.

That sounds like a good idea to me, though the format will be slightly different. I can spend a little bit of time reviewing distributions (beyond the 15 minutes from today's lecture) and then I would also like to spend some time going through the first problem sheet. We can't have this until after the first sheet gets marked, but I'm certainly happy to do this.

I need you to tell me when you are available and then I'll need to get a room. Please provide comments about time constraints in the comments in this spot, and I'll use that to help figure out a time that hopefully everybody can make.

Problem Sheet 2 available

Problem sheet 2 (due Friday of week 8 at 4pm) is now available.

Note that there is a box in the basement of the Mathematical Institute where you should turn in problem sheet 1 (due Friday of week 4 at 4pm).

Solutions will eventually be posted for problem sheet 4, and I'd also like to have a class to discuss these problems. I have the MSc lecture schedule, so I will try to schedule something around that. Hopefully we can find a time that everybody can make. Tentatively, I'd like to do this during week 6. (Note that the class for the second problem sheet will need to be early in Trinity term.)

Note that the 6th problem in problem sheet 2 is quite long. It is meant to guide you to the original analytical derivation of solitons and various things (including insights into shock phenomena) that occurred along the way. It will also be extremely illuminating conceptually, so I hope you'll take the time to go through it to a decent extent. Anyway, while doing it fully will take some time, it's something that will be educational and I am certainly open towards turning that into one of the proverbial class project things you're supposed to be doing.

Problem Sheet 1 available

Problem Sheet 1 is now available. It is due by 4pm on Friday of 4th week. Please submit it to a box that will be set up by the TA.

Welcome to B5b - supplementary lectures (topics in applied differential equations)

This is the website for Maths B5b - supplementary lectures (topics in applied differential equations) for the Hilary 2008 term.

The lecture meets on Thursdays at 11:00 am in SR2 in the Mathematical Institute basement. The plan is to start lectures at 11:05 and finish them at 11:55.

On this blog, I'll be posting links, answering questions that are of general relevance to the course, etc.

The TA for this lecture course is Dave Hewett, who some of you might remember as the TA for C6.3a in the Michaelmas term. He will be marking your assignments. (My plan is to have two assignments: the first will be due on Friday of week 4 and the second will be due on Friday of week 8---though the due date of the second set of problems is still being determined. I will be posting the problems shortly.)

The basic topics for the course will consist of an introduction to the theory of distributions and then an introduction to nonlinear partial differential equations. The course will be divided as follows:

1) Distributions (roughly 3 lectures): This will take what you learned in the past on delta functions, Green's functions, etc. and formalize them mathematically.

2) Further development of hyperbolic equations (roughly 3 lectures): This will be roughly along the lines of what is listed in the official syllabus, which mentiones "the Cauchy-Kovalevskaya theorem, Riemann invariants, shocks and weak solutions, causality." I also hope to discuss (and/or include in the problem sets) rarefaction waves and how this stuff is relevant to a modern, very important numerical method called the level set method. (I can't promise exactly what topics we'll cover, but these are the lines along which I am thinking.)

3) Introduction to dispersive equations (roughly 2 lectures): solitary waves, KdV equation (and how to derive it from the FPU model), NLS equation


I haven't assembled a huge list of references, but here are some that may prove helpful (titles are approximate):

1) Butkov, Mathematical Physics
2) Fritz John, Partial Differential Equations
3) Paul Garabedian, Partial Differential Equations
4) Alwyn Scott, Encyclopedia of Nonlinear Science (selected entries)
5) Gerald Whitham, Linear and Nonlinear Waves
6) Various analysis, pde, introductory nonlinear wave textbooks


I'll try to add more recommendations to the list as we go along. If there are any books (or online resources) you see that you like, please let me know and I'll mention them here.

Note that I do not have any specific online lectures notes for this course.