Math 302

Partial Differential Equations

Fall 2008

Homework assignments

Daily notes

Thursday December 18

This morning, I'll be around my office until about noon. This afternoon, I'll be here from about 1:30 until at least 3:00.

Tuesday, December 9

Exam #4 and your course project report are due by 3 pm on Friday December 19. You can bring these to my office anytime before then. If I'm not in, just slide your work under the door.

I'll be around during reading period and final exam week. E-mail or come find me to ask questions on your project or the exam.

Monday, December 8

Friday, December 5

Thursday, December 4

Tuesday, December 2

Monday, December 1

Tuesday, November 25

Here's the animation we looked at showing the D'Alembert view of a solution for the wave equation on a bounded interval.

Monday, November 24

Friday, November 21

Thursday, November 20

Note that I've assigned problems from Section 4.1

Tuesday, November 18

Monday, November 17

I've assigned a few problems from Section 4.1. We'll wait until Friday to look at these in class since you're likely to be busy with the exam earlier in the week.

If you haven't picked a course projec topic already, you should do so soon and then check with me for approval. Come talk with me if you're not sure about what to do.

Friday, November 14

We've now seen one full example of the last (or maybe next to the last) big idea for the course. The idea is that given a PDE problem involving two boundary conditions for one of the independent variables, we can

• Look for solutions in a product form.
• Separate variables in the PDE to produce ODEs for each factor in the product form. These ODEs share a common separation factor (often denoted &lambda).
• Separate variables in the BCs to get boundary conditions for the relevant factor in the product form. Ideally, this gives us a regular Sturm-Liouville problem.
• Find the eigenvalues and eigenfunctions for the SLP.
• Feed the eigenvalues into the other ODE and solve it.
• Form a "general solution" as an infinite linear combination of product solutions.
• Apply the remaining IC to determine values of the coefficients in the "general solution".

Thursday, November 13

I plan to distribute Exam 3 in class tomorrow. It will be due next Thursday.

Tuesday, November 11

Monday, November 10

The example we worked out in class is the prototypical example of how boundary value problems for ODEs come out of the separation of variables approach to solving a PDE problem. Tomorrow, we'll look at some results from Sturm-Liouville theory. S-L theory gives results about the nature of eigenvalues and eigenfunctions for a general class of boundary value problems.

Friday, November 7

I've assigned problems from Section 3.3

Thursday, November 6

If you are looking for an alternative to Mathematica, you might consider Sage. Sage is "a free open-source mathematics software system".

Tuesday, November 4

In this part of the course, we are developing an idea we will be using in the process of solving PDE problems on bounded domains (as opposed to the unbounded domains we worked with in Chapter 2). The idea is to expand a given function f in terms of an orthogonal basis of functions. When this arises in the process of solving an initial-boundary value problem for a PDE, the PDE and boundary conditions will dictate which orthogonal basis we use. In our first examples, our bases will contain sine functions or cosine functions or both. In other examples, our bases will contain functions that you haven't met yet such as Bessel functions.

Monday, November 3

I've changed the due date for the problem to be submitted from Section 3.2

Friday, October 31

In class, we looked at the mean-square error. (Here, "mean" is being used in the sense of "average".) The mean-square error is one way of measuring how close two functions are to each other. In our case, we are looked at how close a function f is to the partial sums for an orthogonal expansion of f.

When we looked at things graphically using Mathematica, we found evidence that choosing an expansion coefficient to minimize the mean-square error gives us the same result as we get by computing the expansion coefficient using the inner product. This is not an accident. See the discussion on pages 103-104 of the text for a proof of this fact.

Speaking of the text, this is a good time to read Section 3.2 very carefully. Make note of any questions you have on this reading and ask them in class on Monday (or by email before Monday).

Thursday, October 30

I've assigned additional problems from Section 3.2. Some of these involve looking at pointwise error and mean-square error. We'll talk about these ideas in class tomorrow as part of our discussion of convergence issues.

Tuesday, October 28

Monday, October 27

Today, you experimented with estimating coefficients for expanding a particular function in terms of the basis functions sin(kx). Tomorrow, we'll see how to compute these coefficients.

Friday, October 24

I'll hold off assigning new problems until after the exam is due on Tuesday.

Thursday, October 23

I distributed Exam #2 in class. It will be due Tuesday, October 28.

Friday, October 17

When we went to the computer lab at the end of class, nobody got to the point of plotting the solution we had found in class. Work on this as part of your homework assignment. You might find it useful to look back at what we did the last time we went to work with Mathematica.

We'll delay looking at Fouier transforms until the end of the semester. We won't cover the idea in Section 2.8. So, we are done with Chapter 2 for now. Done, that is, except for homework and a take-home exam. I'll have the take-home exam ready when you return from break. (If I get it done earlier, I'll post it here and send out an email to alert you.) It will be due on Tuesday, October 28.

From one point of view, the material we'll start in on after break is a big application of linear algebra. Here's a few of the ideas from linear algebra we'll be using: vector space, inner product, orthogonal, linear combination, linear independence, basis, dimension. In our applications of these ideas, we'll be working with vector spaces of functions. The vectors spaces will not have finite dimension. This makes it much harder to determine whether or not a given set of vectors is a basis. We'll get to deal with linear combinations that are infinite series rather than finite sums.

Thursday, October 16

Tuesday, October 14

When we discussed Problem 3 from Section 2.4 in class, I forgot to note that ultimately we only need the region of the xt-plane with x>0 and t>0. For this quadrant, there are only two different regions to consider. The third region we looked at in class has x<0 which is not relevant for the original problem.

Monday, October 13

You should finish off the problem we started in class by determining the inverse Laplace transform that we need. For this, you can use InverseLaplaceTransform[ftcn,variable1,variable2] in Mathematica. If you sit down with Mathematica, you might as well also experiment with LaplaceTransform[ftcn,variable1,variable2]

The problems I've assigned are from a handout with a few problems on Laplace transforms.

Friday, October 10

We'll skip over the material in Section 2.5. In class, we started looking at Laplace transforms. Some of you have previously seen Laplace transforms in your ODE course. Laplace transforms can be used for ODEs and for PDEs. The Laplace transform is an example of an integral transform. We'll next study another integral transform called the Fourier transform.

Thursday, October 9

The technique we used in class today to get a solution for an IBVP for the heat equation on the half-line is called the method of reflection. I'm leaving it to you to read about using this idea for solving an IBVP for the wave equation on the half-line. I've assigned a few problems from Section 2.4

Tuesday, October 7

Here are some bits and pieces I pulled from responses to Question 5 on Exam #1. Some of these are quotes and others are paraphrases. There is some redundancy here in terms of distinct ideas:

• ODE involves function of one variable and its derivatives; PDE involves function of two or more variables and its derivatives
• some terminology same for ODEs and PDEs: linear/nonlinear, order, homogeneous/nonhomogeneous
• for both, first find general solution, then specific solution when some kind of auxillary conditions are applied
• general solution of ODE includes constants; general solution of PDE includes functions of some variables
• auxillary conditions for ODEs are constants; auxillary conditions for PDEs are functions
• unique solution for PDE requires both initial and boundary conditions whereas ODE needs initial conditions
• main methods we have for solving PDEs involve making substitutions that allow us to form an ODE out of the PDE
• so far, main tactic has been breaking down a PDE into ODEs
• methods for translating PDEs to ODEs that are solvable
• many PDEs can be re-expressed in terms of one independent variable by way of characteristic coordinates, thus rendering the techniques and methods of ODEs applicable
• solutions to PDEs are often found by changing independent variables to change the PDE into a more easily solved ODE
• to solve a PDE we often use methods to reduce it down to an ODE and the way we solve it depends on the type of problem
• for both ODEs and PDEs, can classify equation to gain insight on how to solve and nature of solution
• 2nd order PDEs can be put in canonical form (heat, wave, Laplace)
• for both, useful to plot solutions; for ODEs plot is y vs x; for PDEs, plot is u vs (x,t) or (x,y) so different views can be useful (surface, snapshots, animation)
• both ODEs and PDEs can be used in modeling real-world applications from physics, biology, finance, traffic flow; PDEs allow for more complex modeling

Monday, October 6

We talked about the idea of well-posed problems in class today. The motivation here is to give a set of mathematical conditions that are necessary in order that a differential equation model will "physically reasonable." One way to think about this is that the well-posed conditions are necessary in order for a differential equations model to have predictive power.

I've designated problems to submit from Sections 2.1 and 2.2.

Friday, October 3

There is a Math/CS Department Seminar on Monday at 4 pm. Manley Perkel will speak on "Embedding Complete Binary Trees into Grids". I'll bring the abstract to class on Monday.

Thursday, October 2

We'll meet in the computer lab for class tomorrow.

Tuesday, September 30

Here's a recap of how we arrived at a solution to the general Cauchy problem for the heat equation:

• Used dimensional analysis to guess a form for a solution.
• Used this guess to find a solution to a specific Cauchy problem with initial condition given as a step function. We denoted this solution w(x,t).
• Took the partial derivative of w with respect to x to get a solution to a different Cauchy problem. We denoted this solution G(x,t) and deduced that the initial condition for this solution is the delta "function" δ(x).
• Built a solution for a general initial condition as a "linear combination" of shifted functions G(x-y,t). Here, "linear combination" is generalized from a sum to an integral.

I've assigned problems from Section 2.1. Here's some comments:

• For Problem 1, you can go straight to the solution as given in Equation (2.8). The idea here is to attempt to simplify the resulting integral.
• For Problem 5, we'll go to the computer lab later this week and learn some pieces of Mathematica that will be relevant for this.

Monday, September 29

I'll assign problems from Section 2.1 after class tomorrow. In the meantime, here's three problems you should do:

1. Verify directly that the function we've called w(x,t) is a solution to the heat equation for all x and all positive t.
2. Verify directly that the function we've called G(x,t) is a solution to the heat equation for all x and all positive t.
3. In terms of the function G, write down a solution of the heat equation that corresponds to starting with 1 unit of heat energy at the point x=3. In terms of the function G, write down a solution of the heat equation that corresponds to starting with 4 units of heat energy at the point x=-5.

Friday, September 26

I'll assign problems from Section 2.1 after class on Monday so we can discuss more details. I'm sure your take-home exam will be enough to keep you out of trouble in the meantime.

Thursday, September 25

Tuesday, September 23

I've finished writing the take-home exam. Please read the instructions carefully. Feel free to ask for clarification on the instructions or problems.

I'll have homework graded and available for you to pick up tomorrow at my office.

Monday, September 22

In class, I mentioned this handout that gives more details on classifying quadratic equations in two variables. I wrote this handout for a Math 301 class so the notation is slightly different and the last example is not directly relevant to what we looked at. In the handout, the general quadratic form is written as Ax²+2Bxy+Cy² rather than as Ax²+Bxy+Cy². Using 2B in place of B is a matter of convenience. The handout gives more detailed references results from linear algebra (with specific references to results in Rob Beezer's A First Course in Linear Algebra.)

I'll have a take-home exam to distribute in class tomorrow. It may be available here later today if I finish it.

Friday, September 19

I've assigned one problem from Section 1.8. It involves using the mean value property for solutions of Laplace's equation to get numerically approximate a solution.

Our first exam will be a take-home exam on the material from Chapter 1. I'll distribute it early next week and have it due early the following week.

Thursday, September 18

In class, we went through a quick and dirty motivation for the basic wave equation as a model for a vibrating string. Section 1.5 has a much more detailed derivation. Section 1.5 also derives the basic wave equation as a model for acoustic (i.e., sound) signals. You are welcome to read these derivations but we will not cover them in detail in this course. I encourage physics students to look at the details. If you do read them and have questions, feel free to come ask me.

Tuesday, September 16

Monday, September 15

I'll post an assignment from Section 1.3 after class tomorrow.

Friday, September 12

We got some hands-on experience with Mathematica today by inputting each of the commands from this handout. Mathematica has many capabilities. We'll mainly use its graphics capabilities to help visualize solutions.

Thursday, September 11

Note added late: I just realized that I drew a picture incorrectly in class today. When I was drawing the ξ=C curves in our example, I was thinking of x as the vertical axis and t as the horizontal axis when, in fact, there were the other way around. I'll fix this up in class tomorrow.

I've assigned a few more problems from Section 1.2, including one to be submitted next week.

Note that in Section 1.2, the author of our text recycles some notation so certain symbols have different meanings at different places. In particular,

• F first appears in Equation (1.10) on page 11 as arbitrary function in the general solution of the simple advection equation (1.9). Later F is used with a different meaning on page 13 to denote the source term expressed in the new coordinates ξ and τ. In class, I've been using α rather than F for the first of these uses.
• φ first appears on page 10 to denote flux. Later φ is used on page 14 to denote something different. In class, I used Φ rather than φ for this second use.

Tuesday, September 9

In class, we finished deriving what our text calls the fundamental conservation law. To go further, we need a relationship between flux φ and density u. For a simple model of advection, we use the simple relationship φ=cu. Substituting this into the fundamental conservation law gives us a partial differential equation for the unknown density u. We can use the method of characteristics to solve equations of this form. We did the simplest case of this in class. On Thursday, we'll look at more general, and more interesting, cases.

I've assigned a few problems from Section 1.2. I'll assign more later in the week, including some to submit.

Monday, September 8

In class today, we started in on a very typical type of reasoning that leads to a partial differential equation model. The key elements involve thinking about density u and flux φ. We also throw in a quantity f that is related to creating new stuff (or injecting new stuff in the system.) Flux φ and the creation/injection quantity f are somewhat subtle since each is a density. Further, each is a density in which the relevant quantity is a rate. Here's specific descriptions:

• Flux φ is an area density for the rate at which stuff moves across a surface; that is, flux is the rate at which stuff moves across a surface per unit area of that surface.
• The quantity f is a volume density for the rate at which stuff is created; that is, f is the rate at which stuff is created per unit volume.

• density u has units of kg/m³
• flux φ has units of (kg/s)/m³
• f has units of (kg/s)/m³

I'll assign some problems from Section 1.2 after tomorrow's class.

Friday, September 5

I've assigned a few problems from Section 1.1 to submit on Monday. For Problem 6, pay attention to whether or not "constants" of integration are really constant.

Thursday, September 4

I have not assigned any new problems. For now, keep working on the problems from Section 1.1. We'll address more questions from these in class tomorrow.

Tuesday, September 2

I've assigned problems from Section 1.1. Later in the week, I'll pick out a few for you to submit.

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.