Lectures: 2-3:30pm on Tue and Thus, in Henn. 304.

Prerequisites: MATH 215

Textbook (required):"Boundary value problems and Partial Differential Equations" by D. L. Powers, 6th edition. It is available at the bookstore - or you might want to investigate sites such as bookfinder4u to find new/used copies at lower prices. This book is fairly specialized, covering roughly the topics we will study in this class but not many other topics. It has a reasonable selection of problems in each chapter.

Other textbooks that you might find useful are: (1) "Mathematical Methods of Physics and Engineering" by Riley, Hobson and Bence. This book is nice because it covers all math that you are likely to ever encounter as an undergrad, so it should be useful beyond just this course. Also, it has lots of problems and many hints/solutions. I used this as a textbook two years ago. The main problem with it is that because it covers so much material, it devotes rather little space to any one topic. Also, the material of interest to us is midway through and assumed some notation to be familiar from previous chapters; (2) M. Boas, "Mathematical methods in physical sciences" -- this is a classic, maybe somewhat more basic than what we will cover but some of you may find it useful; (3) W. E. Boyce and R. C. DiPrima, "Elementary differential equations and boundary value problems" -- all these, and many many more, are available in our library.

Material: Ordinary and partial differential equations appear everywhere in physics, from Newton's second law, to Maxwell's equations, to Schrodinger's equation and so on and so forth. Having some idea how to go about solving them is, therefore, quite important for any physicist. This course discusses general approaches to dealing with such equations, such as using separation of variables, Green's functions, special functions (Bessel, Legendre) etc. The focus is on problems that can be solved analytically, but if time permits I will also briefly introduce some numerical approaches, so that you also have some idea on how to go about solving problems which do not have analytical solutions. Taking a computational physics course (210 or 410) will further help with that.

The official description of the course material is: The application of ordinary and partial differential equations to physical problems; boundary and initial value problems associated with heat, wave and Laplace equations. Fourier analysis; expansions in Bessel and Legendre functions.

A course covering very similar topics to PHYS312 is MATH316. I suspect that the biggest difference will be on what is emphasized. For example, for a mathematician, a very interesting question is when do certain differential equations with certain types of boundary conditions have a unique solution (sometimes there is no solution consistent with those boundary conditions, and sometimes there can be infinitely many such solutions. If you don't know what boundary conditions are, do not worry, we will learn this soon enough). Such analysis has all sorts of lovely math in it. However, for a physicist, all this discussion is a bit irrelevant, since the equations that describe physical systems surely have a solution, and surely it is unique. If I have a metallic rod at a given temperature and at t=0 I put its two ends in contact with two objects held at other temperatures, then the temperature in the rod will change so that after a long-enough time, each end has the same temperature as the object it is in contact with (these are the boundary conditions). There is a certain differential equation describing how this happens, and we'll study it quite a bit. But there is no doubt that there is a solution to this problem, since we know that we can carry out the experiment and there will be some temperature distribution in the rod at all times. And moreover, the solution is unique -- it would be quite astounding if identical experiments carried out at different times would give different results! Because problems without solutions, or with infinitely many solutions, do not tend to appear in physics, we won't spend any time worrying about them, although in a math course on partial differential equations this might be a topic covered in great detail. There are other such differences and it should be your personal inclinations -- whether you think more like a mathematician or like a physicist -- that should dictate which of these two courses would be most suitable for you.

Homework: Weekly sets. All will be graded, however the lowest two assignment grades will not count towards your final grade. The problem sets will be given to you in class; they will also be posted on-line here. Your solutions should be neatly written, with problems in the order assigned, on pages stapled together (no torn edges or paperclips). The homework must be turned in on the due date in class. The solutions will be posted on-line immediately; as a result, late homeworks will not be accepted.

Discussions regarding the homework problems are encouraged. However, the writing of the homework must be done by each student on his/her own. Under no circumstances should you copy or even look at someone else's solution to a problem while writing your own homework, quiz, or exam for credit. Identical homeworks will be severely penalized.

If you fail to find the correct and complete solution to any assignment problem, make sure right away that you understand the posted solution and are able to solve similar problems by yourself. If you cannot follow the posted solution, come and see me, or the grader, or discuss it with your best friends from among PHYS312 students. Do not procrastinate! Things will not get any easier later on. As an incentive towards re-doing assignments and checking out the posted solutions, one of the final exam problems will be identical to one of the assignment problems.

Pre-reading and in-class assessments: For many lectures, I will ask you to do a limited amount of pre-reading. This will cover, for instance, various definitions and simple methods/concepts. The class time freed by this will be used to check that these definitions/concepts have been understood properly and to solve more problems. The pre-reading material (if any) will be posted at least by Sun night (for a Tue lecture) and Tue night (for a Thus lecture) here. It is your responsability to check whether there is any pre-reading assigned for a class, and to read the material.

To verify that you are all doing this pre-reading, I will give you simple short tests at a few random times of my choosing. The questions will be easy to answer if you did the pre-reading. These will be graded, with credit given both for the accuracy of the answer but also for participation (because I prefer that you come to class even if you did not do the pre-reading). There will also be in-class activities, eg solving problems in a group with 1-2 other students, so that I can see if there are any issues. For such activities you will receive credit for participation, only. The hope is that this will encourage you to try your best without the fear of what happens if you are wrong.

Exams: the midterm will be scheduled during a regular class and will be 1h long. The final will be scheduled by the University during the exam period. A list of useful formulae will be provided by me; no extra materials will be allowed. Calculators are not needed. If there is any potential scheduling conflict, let me know as soon as possible before the exam. Makeups are available ONLY for students with written evidence for emergencies such as sudden illness, accident, death in the family, etc.

Grading:

• 20%: best 10 out of the 12 assignments
• 5%: in-class assessments
• 25%: midterm
• 50%: final exam