Math 118A: Partial Differential Equations.

Introduction.

Details.

Course name	Math 118 A: Partial differential equations
Instructor	Karl Håkan Nordgren
Lecture time and location	Monday, Wednesday and Friday, 2:10pm-3:00pm in CRUESS 107.
Instructor email-address
Instructor office-location and office-hours	2107 (in the Mathematical sciences building), Mondays and Wednesdays, 3:10pm-4:00pm.

Announcements.

Detailed syllabus.

• After the first, introductory, lecture, we will begin our study of partial differential equations (PDE) by looking at first order equations in two dimensions:
  • First up are first order equations with constant coefficients which are solve using characteristics. This solution will be used to obtain hints as to how to approach more complicated first order equations.
  • Second, we cover first order equations where the coefficients depend on the coordinates. These we will again solve using characteristics.
  • Third, we state a theorem which assures the existence of a solution to first order equations where the coefficients depend on the coordinates and the unknown function we are looking for.
• Next we look at general second order equations in two dimensions and find at a way to divide these equations into three categories - parabolic equations, elliptic equations and hyperbolic equations. This motivates the study of the big three second order PDE - the heat equation, the Poisson equation and the wave equation - as examples of the abovementioned types.

• Next we look at general second order equations in two dimensions and find at a way to divide these equations into three categories - parabolic equations, elliptic equations and hyperbolic equations. This motivates the study of the big three second order PDE - the heat equation, the Poisson equation and the wave equation - as examples of the abovementioned types.
  • We begin with elliptic equations. That is, with the Laplacean and the Poisson equation. Here we will define two norms and define what we mean by well-posedness. Then we will prove the mean value theorem and the maximum principle and, using these, a uniqueness result for the Dirchlet and Neumann boundary value problem. From the maximum principle we also obtain continuity with respect to data. A short aside on Fourier series follows. We then obtain existence on a square using Fourier series.
  • Second we cover the heat equation. We again prove uniqueness first, but this time using an energy argument. This also gives us continuity with respect to data. Next we obtain a solution the heat equation on an unbounded domain in R using a fundamental solution. The existenxe of such a solution also gives us a solution on the half-line using a reflection argument. Using Fourier series we also obtain a solution on a bounded interval.
  • Finally, we look at the wave equation. We begin by finding a solution to the wave equation on R using the d'Alembert formula. This also gives us existence on the half-line. Next we find existence in three dimensions using the Kirchoff formula and then existence in two dimensions using the method of decent. We then obtain uniqueness and the principle of causality. Finally we use Fourier series to obtain a solution on a finite interval.

Some important dates.

First lecture: Friday, September 26; drop deadline: Wednesday, October 22; midterm: Friday, October 31; thanksgiving holiday: Thursday-Friday, November 27-28; instruction ends: Friday, December 5; final exam: 1:00pm-3:00pm, Tuesday December 9.

Some information about how this thing is going to work.

• Homework rules.... I will make up homework problems and post them in the 'announcements' section above by 9pm on Thursday, every week. The homework will be due in class on Friday, by 3:00pm. Below are some rules regarding homework, most are taken (with permission) from Yana Mohanty's website.
  • If you cannot make it to class on Friday by 3:00pm, then you have to find a way to give me the homework before Friday at 3:00pm. After that time the homework is considered late. Late homework will not be accepted.

  • In order to make sure that your score gets entered correctly, you must include you first name, your last name and the homework number in the upper right hand corner of the first page of your homework assignment.

  • A correct answer without steps logically leading to it earns 0 points (the same is true for grading on exams)

  • Solutions to problems must be written up in the order in which they are assigned. If your solutions to a problem is out of order and the grader does not see it, you will not get any credit for that problem.

  • Any solution that is too illegible or unclear for the grader to understand will not get full credit.

  • The sheets of the homework write up must be stapled together. If loose sheets get lost, the grader is not responsible for locating them and grading the problems on the lost sheets.

  • Academic integrity will be enforced with homework assignments as with any other course assignments. While students are expected to collaborate on the homework assignments and seek help with them, they must write up their own assignments. Any students found copying solutions from a manual or from each other and presenting them as their own work will face the consequences described on the UC Davis student judicial affairs website.

• Exam rules.... There will be one midterm and one final. The homework rules which can be applied to exams also apply to the midterm and the final. If you miss an exam there will be no make-up exams.

• Textbooks... The standard text for this course is Partial Differential Equations: An Introduction by Walter A. Strauss, which can be found here. This text is very nice, and it covers about 80% of the material for the course. I will also steal from Partial Differential Equations by Lawrence C. Evan; Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou and Dale W. Thoe; and Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima. I will provide lecture notes (and post them in the 'announcements' section above) so it should not be necessary for you to buy all four of these. I have intended the lecture notes to be good enough that you should not need to consult books at all, but should you not find my notes as good as I do, I recommend you buy Walter Strauss' book.

• Grade breakdown... Homework will be worth 20%; the midterm will be worth 30% and the final will be worth 50%. As usual, the course will be curved at the end when all the scores are in.

Things I intend to teach you in addition to the stuff contained in the syllabus.

• Clear exposition... When I complain about the state of a homework which I have just been given by a student, the reply is often that 'the homework is tidy for a math homework' (emphasis mine). Now why is it that students take more care with their essays for English class, than they do with their calculus homeworks? Making your work legible benefits you because you are less likely to make careless errors and it benefits the reader because it will improve the readability of your work. Some things to think about are
  • Use real English words to create real English sentences which explain what you are doing. A string of equations is no fun to read.

  • In the Western world we begin our written work on the left at the top of the paper and continue towards the right until we run out of space. Then we go down one line and start again on the left and continue towards the right, etc. This is also how mathematics should be written. A seemingly random collection of equations in no apparent order is also not fun to read.

• To use the equal sign correctly... The equal sign is a very basic piece of notation and most of us have been using it for many many years. Nevertheless, there will be those among you who do not use it correctly. Here are a few errors to look out for:
  • Only use an equal sign if the two things on either side of it are actually equal. That seems obvious but surprisingly many students violate this rule.

  • Use the equal sign when it is necessary. It might be clear to you that by writing one integral below the next you mean that they are equal, but it might not be as clear to whoever is reading your work.

  • Do not use an arrow when you mean an equal sign. An arrow is, by most interpretations, taken to mean 'implies.' If you mean that one thing implies another, by all means use the arrow, but if you mean that they are equal, use the equal sign. That's what it's there for.