Math 302

Partial Differential Equations

Fall 2007

Homework assignments

Daily notes

Monday, December 10

Friday, December 7

Thursday, December 6

Tuesday, December 4

Monday, December 3

Friday, November 30

Tuesday, November 27

Monday, November 26

Tuesday, November 20

We've covered most of the ideas in Section 7.2 with the exception of convolution. We'll discuss convolution next week. I've assigned some problems from Section 7.2 and will later assign a few more on convolution.

We've also talked about the main idea in Section 7.3 but haven't finished because we need the convolution result from Section 7.2. I started in on an example of applying the Fourier transform method to PDEs in order to motivate need for the convolution result. Next week, we'll take care of the convolution result and then finish off our first example, which is the same as Example 2 in Section 7.3.

Monday, November 19

Friday, November 16

Thursday, November 15

Tuesday, November 13

Monday, November 12

Friday, November 9

Thursday, November 8

Tuesday, November 6

Monday, November 5

Friday, November 2

Thursday, November 1

Tuesday, October 30

Sometime in the next week, e-mail me a project topic proposal or a request for help finding a topic. If you're looking for help, let me know if you are more interested in theory or application. Here's some topic ideas:

• Proof of the Fourier series convergence theorem
• Detailed derivation of the wave equation
• Detailed derivation of the heat equation
• McKenna model of the first Tacoma Narrows Bridge
• Black-Scholes model of financial derivatives
• Method of characteristics
• Modeling traffic flow
• The role of Fourier series and partial differential equations in the development of the function concept
• Signal analysis
• Laplace transforms
• "Can one hear the shape of a drum?"
• PDE models of biological systems

Monday, October 29

I distributed Exam #2 in class today. It is due in one week.

Friday, October 26

Thursday, October 25

Friday, October 19

Section 3.7 deals with the wave equation for a rectangular spatial domain and the heat equation for a rectangular spatial domain. In class, we worked through the details of the "standard" IBVP for the wave equation. Problem 17 asks you to do something similar for the "standard" IBVP for the heat equation.

We'll meet in the Jones classroom next Thursday. Have a good break.

Thursday, October 18

Tuesday, October 16

On Thursday, we will meet in the TH 189 computer lab.

Monday, October 15

We will have a regular class session tomorrow (Tuesday) and then meet in a computer lab on Thursday. Among other things, we'll play with 3D plots in Mathematica.

Friday, October 12

Thursday, October 11

Tuesday, October 9

Monday, October 8

Friday, October 5

Thursday, October 4

Exam #1 is available now.There are four problems (on two pages). Read the instructions carefully. Ask me questions as you think of them. The exam is due on Tuesday, October 10.

Tuesday, October 2

Exam #1 will be take-home. My tentative plan is to distribute it on Thursday and have it due next Tuesday.

Monday, October 1

Friday, September 28

Section 3.1 generalizes the ideas of linear and homogeneous to apply to partial differential equations. You should read this section and be able to use the language correctly.

The wave equation is just Newton's second law applied to a specific physical situation. In class, I tried to give you a few connections between the physical situation and the wave equations. Section 3.2 gives a more detailed derivation. I encourage you to read this, particularly if you are in physics, but we will not cover these ideas in detail. Come talk with me outside of class if you have questions.

Thursday, September 27

We will skip the material in Sections 2.8-2.10. These sections give a proof of the main convergence theorem for Fourier series and more details on uniform convergence. The material in these sections would be appropriate for a course project topic.

Tuesday, September 25

Monday, September 24

Friday, September 21

If you compute the coefficients c_n for the complex form of the Fourier series of a (real-valued) function, you can extract the coefficients a_n and b_n for the real form. One way to do this is using the formulas given in Display (7) on page 62 of the text. Another way is to note that c_n=a_n/2-ib_n/2 for n>0. From this, we can read off that a_n is two times the real part of c_n and b_n is negative two times the imaginary part of c_n. (Keep in mind that the imaginary part of a complex number is the real coefficient on i. For example, the imaginary part of 5-13i is -13, not -13i.) If you finish the example you started at the end of class, you should be able to recover the coefficients we calculated for the Fourier series of this function as our first example a week or two ago.

Thursday, September 20

Tuesday, September 18

In class, I mentioned an intriguing fact about what can happen with rearranging the terms in certain series. Here's some statements with keywords you can search on to explore these ideas:

• If a series converges absolutely, then a rearrangement of the terms does not change the sum.
• If a series converges conditionally, then a rearrangement of the terms might change the sum.
• Riemann series theorem: If a series converges conditionally, then for any number S, there is a rearrangement of the terms such that the resulting sum is S.

Monday, September 17

Friday, September 14

Mathematica is a powerful computational tool. It is designed to do a lot of things and be very flexible. As a consequence, Mathematica takes a substantial investment to learn. The program does include extensive documentation and help under the Help menu. One thing you might want to try is the First Five Minutes with Mathematica tutorial. As with any command-line environment (as opposed to menus), you will need to be precise with syntax. This will become easier as you become familiar with the common styles and patterns in Mathematica.

There are other software packages available with roughly the same functionality as Mathematica. One example is Maple. Although most of my experience is with Mathematica, my general sense is that no one of these is clearly better for all things. I will not require you to use any specific package, but I will expect you to have proficiency with some computational technology that can things comparable to the features I will use in Mathematica.

I've assigned Problems 25 and 26 from Section 2.3 to be submitted by Thursday.

We'll meet in the TH 191 computer lab for Friday's class. You'll have a chance to get familiar with some basic features of Mathematica.

I will have you submit a few homework problems next week. More details soon.

Tuesday, September 11

In class, I made reference to a convergence theorem for Fourier series, but did not state it precisely. A precise statement is given in Theorem 1 on page 30 of the text. You should read and understand this result. A proof of this result is given in Section 2.8. We'll decide later whether or not to work through the proof. I'll ask you about your interest in doing this.

In class, I handed out a paper copy of a Mathematica file. Above is a link to the original Mathematica file. You should be able to download the file and then open it using Mathematica. Note that Mathematica should be available in several computer labs across campus. I don't know the exact locations but can find out if you are interested. Let me know.

Monday, September 10

Friday, September 7

Thursday, September 6

I've assigned a few problems from Section 1.2. These involve playing with simple solutions to the wave equation. I'll address questions on these in class tomorrow.

As I mentioned in class, we will approach our study of Fourier series from a linear algebra perspective. In particular, we will view a Fourier series as a linear combination of basis vectors. All of the action will take place in a specific vector space. The (abstract) vectors in this vector space are functions. The specific vector space is a bit hard to define precisely right away. For now, you can think of it as something like the vector space C[a,b] (i.e., the set of all continuous functions defined on the interval [a,b] with the natural way of adding functions and multiplying by a scalar) that I used as an example in class today.

You can read the first part of Section 6.1 or review your linear algebra text.

Tuesday, September 4

Fun Stuff

The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.

If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.

Check out the Astronomy Picture of the Day.