# Math 372: Partial Differential Equations

### Homework and Solutions

• Assignment 12.
• Previous assignments.
• Flute and Birds

The file mixed.wav contains a mix of two audio clips: One of birds chirping, and the other of a flute. Your task is to remove the chirping birds, and recover (as well as you can) the original clip of the flute alone. (I would need to see an algorithm and/or code, as well as the final result.)

I will measure the error as follows: Convert your wave file to a vector, and normalise it to have length 1. (If your WAV file is stereo, both channels will be averaged to convert it to mono.) I will take the original wave file for the flute and convert it to a vector (again normalised to have length 1). The error will be half the distance between the above two normalised vectors. (The distance (as measured above) between the mixed clip and the original flute sample is about 0.2578. My code currently reduces this distance to 0.0446, at which point the birds are essentially inaudible, with very little "artifacting". See if you can do better!)

I'd recommend using Matlab, Octave or SciPy/NumPy to do this. If you choose to use Matlab/Octave, a few functions you'll find helpful are `wavread`, `wavwrite`, `fft` and `ifft`. If you use a different language, don't reinvent the wheel: find a library that read wave files and Fourier transforms functions.

• Solutions

Access to solutions is restricted, and you have to log in with your Andrew ID view them. If you get a "Permission Denied" error message message, then send me an email with your Andrew ID to get access for the semester.

### Logistical Info

 Instructor Gautam Iyer.  Office Wean Hall #6121 412 268 8419 eMail For MINOR clarifications and logistical queries, my address is: `gi1242+372@NoSPAM.edu` (replace "`NoSPAM`" with "`cmu`"). For hints or longer questions, come by my office hours and speak to me in person. Anonymous Feedback Feedback at any time (either anonymous or signed) is always appreciated. Use this form to send me anonymous (or signed) feedback. [PS: In a desparate attempt to thwart evil spammers, I've restricted access to this form. Your public IP address should resolve to a hostname ending in `cmu.edu`.] Lectures MWF 11:30--12:20. Office Hours Mondays 3:30--5:00, Fridays 10:30--11:27 Mailing list math-372 This list will be used for all announcements related to this class. All students (and anyone auditing) should subscribe to this list using the above link. I will NOT use BlackBoard. NOTE: Any student who is registered for this course on the first day of class will be automatically subscribed this mailing list. Everyone else should subscribe themselves using the above link. Please subscribe to this list!

### Homework Schedule, Exams Dates and Grading Rubric

Homework due Wednesdays, beginning of class.

Late homework will NEVER be accepted. Really. Read the homework policy for more details.

Midterm 1 Mon, Feb 10. (Closed book, in class.)
Midterm 2 Wed, Apr 2. (Closed book, in class.)
Final Thu, May 8, 8:30am--11:30am in PH A18C (Closed book.)
Grading Homework: 20%, better midterm: 30%, final: 50%.

### Choosing the "better" of your two midterms

When computing your final grade, I will only use the score from the midterm in which you received a higher percentile rank (and consequently, the higher grade). I will then use statistical methods so that your scores from your homework and exams are all comparable, and then average these corrected scores (as described) to compute your final grade. If you want to know the exact details, read the source of the scripts I use.

Bottom line -- you can safely "bomb" one midterm with no consequence to your grade, and let me worry about how the numbers work.

### Course Description

A Partial Differential Equation (PDE for short), is a differential equation involving derivatives with respect to more than one variable. These arise in numerous applications from various disciplines. A prototypical example is the `heat equation', governing the evolution of temperature in a conductor.

Usually finding explicit solutions for even the simplest (LINEAR) PDE's is a formidable task, which doesn't always have a tractable solution. The mathematical study of PDE's usually focuses on deducing properties of solutions, without use of an explicit solution formula. For instance, the fact that heat doesn't collect at hot points is a consequence of the "Maximum principle"; a fundamental theorem about solutions to the heat equation, which also applies to solutions of a more general class of equations.

This course will serve as a conceptual introduction to PDE's differential equations, focussing more on studying properties of solutions and less on finding explicit (and horrendously complicated) solutions. It is aimed at undergraduate Math majors, however is suitable for students from Physics, Engineering and other disciplines who want to develop a more conceptual understanding of the subject.

### Syllabus

• Introduction, initial data, boundary conditions, classification.
• Method of characteristics.
• Wave equation (1D), and finite speed of propagation.
• Heat equation, maximum principle and the heat kernel.
• Separation of variables, Fourier series and convergence.
• Laplace and Poisson equation, (strong) maximum principle.
• Harmonic functions, Mean value property, Greens functions.
• (Time permitting) Introduction to numerical methods, and applications

### Pre-requisites and Difficulty.

Ideally, you should have seen a good course multi-variable calculus and ODEs (i.e. 21-259 and 21-260 or equivalent). At the very least, you should know COLD the chain rule for partial derivatives, the divergence theorem and how to solve basic ODE's (variation of parameters, etc.)

Though it isn't essential, it would be 'helpful' if you're familiar with some elementary analysis. You will find the later topics (heat kernel onwards) a lot easier if you've taken 355/356, and for instance, know the formal definition of a limit, uniform convergence and Riemann integrals. It is, however, possible to 'survive' this course with no prior knowledge of these topics.

### References

• Introduction to PDE by Walter Strauss. (Strongly recommended! Homework problems will be assigned from here.)
• Basic Partial Differential Equations by Bleecker and Csordas.
• An Introduction to Partial Differential Equations by Pinchover and Rubinstein.

### Lectures

• If you must sleep, don't snore :).
• Be courteous when using mobile devices.

Your behaviour should not disturb or distract anyone else. Make sure your cell phone is turned off, or silent. If you must use a laptop, then:

• Turn off the sound
• Do not type on laptop keyboards! Key presses are both noisy and distracting.
• Sit in the last row if you're going to be doing anything other than taking notes.

Repeat offenders of this policy will be penalized.

• Study material from lectures you miss.

Usually each lecture relies on all the previous ones, and missing one could easily lead to a disastrous domino effect. If you have to miss a lecture, then I strongly recommend you study the material you missed before you return to class. Note, while I don't require "mandatory attendance", I require that you know all material covered in class. You are responsible for making up anything that was covered in lectures you missed.

If you miss a lecture, I recommend doing the following:

• Photocopy, and read notes from someone who was in class.
• Reading the relevant sections from the text, Wikipedia, etc.
• Look at the lecture schedule posted online.

After you have done this, you should contact me if you need clarification on any material. You don't have to notify me in advance of occasional absences.

• Also study material from lectures you don't miss.

A good practice in any math course is to look over your notes from class, and/or the relevant sections from the text every class day. Again, this is because each lecture will build on the previous. A gap in your understanding in one lecture will cascade catastrophically through future lectures!

### Homework

• Late homework will never be accepted.
• Homework not turned in at the beginning of class is considered late.

To accommodate special and extreme circumstances, I will not count your lowest two homework scores towards your grade. So, for instance, if you have a family emergency you need not turn in the homework for that particular week, and it will not affect your grade. If you turn in every homework on time, then I will automatically drop the bottom two scores from your grade.

Keep in mind, this policy is to accommodate special, and extreme circumstances. If you use up your two freebies early in the semester, then please ensure you and your extended family remain in good health for the remainder of the semester.

• Do, but don't turn in optional problems.

Your assignments will frequently contain optional problems (marked with a star). These problems are helpful to think about, but you should not turn them in with your regular homework. Problems are made optional for a variety of reasons: Some problems are optional because they are (easy?) standard facts which I did not have time to do in class. Others are optional because they are interesting `challenge' problems, which may or may not have a tractable solution in the scope of this course.

Optional problems won't be graded, and won't count towards your grade. There is no "extra credit" for doing optional problems. You're welcome to discuss any optional problem with me or your classmates, but don't turn it in with your regular homework.

• Working in groups is encouraged, but solutions must be written up on your own.

You're encouraged to work in groups, however you must write up the solution on your own. Photo-copying or blindly plagiarising solutions from members of your study group (or anyone else for that matter) will be treated as cheating, and dealt with severely.

Homework is probably the most important part of this course. Trying such problems on your own is the only way to get a good conceptual understanding of this material. So be sure you give it a good shot!

• Nearly perfect student solutions may be scanned and hosted.

I will usually write up solutions to the harder problems on each homework. For the remaining problems, perfect, or nearly perfect student solutions may be scanned in and hosted here, with identifying information removed. If you would not like your homework assignments scanned in and hosted, then please let me know.

### Exams

• All exams are closed book, in class exams.
• You are not allowed any computational aids, cheat sheets, etc.
• Exams from previous years are (or will be) linked to in the Handouts section.
• Some advice on preparing for math exams is here.