Syllabus for MA 776 Partial Differential Equations

Syllabus for MA 776 Partial Differential Equations, Spring 2022

Course Information

Lecturer: Ryan Goh

Email: rgoh@bu.edu
Office: MCS 243
Office Hours: W: 9-10, 1-2pm, Th: 9-10 (virtual, email for zoom link) and by appointment.
Web page: http://math.bu.edu/people/rgoh/

Lectures:

Time: Monday, Wednesday, Friday, 11:15-12:05am;
Location: CAS B25B

Textbook: Partial Differential Equations by Lawrence C. Evans, 2nd Edition

Additional References
Here are some other references which may provide an alternative viewpoint and presentation of the course material and which you might find useful. More will be added to the course page as needed.

Logan, D. Applied Partial Differential Equations

Renardy, M., Rogers, R., An Introduction to Partial Differential Equations

Taylor, M., Partial Differential Equations

Olver, P. Introduction to Partial Differential Equations

Course Webpage: http://math.bu.edu/people/rgoh/teaching/ma776-ss22/course-page.html

Course Description:

Broadly this course will focus on the fundamental tools and approaches used to study Partial Differential Equations. Such equations are fundamental in the modeling and understanding of phenomena observed across the natural world. Furthermore, the study of such equations has motivated the development of many novel mathematical tools.

We will focus on two general areas: I: Representation formulas for specific (but important) linear and nonlinear PDEs, and II: Theoretical tools for studying linear equations. This roughly consists of the first half of the course textbook.

Prequisites This course assumes you have a solid background in undergraduate analysis (say at the level of BU CAS MA 511-512) and some background in graduate analysis (BU CAS 711) including measure theory. While we will not give proofs of all background statements, relevant concepts will be refreshed and reviewed as needed. Before the class begins, I suggest you read through the appendices of the textbook to jog your memory of various concepts. Furthermore, some knowledge of ordinary differential equations will come in handy once in a while, but we will go over any topics required. A coarse schedule of the course is as follows:

Part I: Representation Formulas

Introduction (Ch. 1)
Four Linear PDE (CH 2.)
- Transport equation
- Laplace's equation
- Heat equation
- Wave equation
Nonlinear First-Order PDE (CH 3.)
- Method of Characteristics
- Introduction to conservation laws
Other ways to represent solutions (CH. 4)
- Separation of variables
- Similarity solutions
- Transform methods (Fourier and Laplace transform)
- Power series and Cauchy-Kovalevskaya theorem (if time)

Part II: Linear PDE

Sobolev Spaces (Ch. 5.1 - 5.7, 5.8.1, 5.8.5, 5.9)
- Weak derivatives and Sobolev spaces
- Approximation
- Extensions and traces
- Important inequalities
- Compactness
- Relation to Fourier transform
2nd-order elliptic Equations (Ch. 6) - (all topics)
Linear evolution equations (Ch. 7) - as much as time allows

Homework

There will be assigned homework every week or two throughout this semester. Assignments will be posted on the course webpage . Please submit all homeworks to me via email in PDF form (either scanning handwritten solutions or outputting a PDF of your LaTeX file) on the date they are due. Solutions must be clearly written, using complete sentences when necessary, and composed in a concise manner. (Once you complete a solution, think about how you can explain in the clearest and most concise way possible). Solutions to problems will be emailed out after problems are graded. You are welcome to collaborate with classmates but must writeup solutions on your own and in your own words. The idea here is that you should be able to explain any solution you writeup without the aid of anything else. If I suspect that this is not the case I reserve the right to have you come into my office and present a solution on the blackboard. Furthermore, solutions copied directly from another student or from the internet will receive no points. Here is the BU CAS Code of Conduct.

Midterm and Final

There will be one midterm and a final exam. Both will be take-home exams and you will be given 48 hours to complete and submit the problems. The dates for these exams will be announced soon. You are not allowed to collaborate with each other on these problems, but can use notes from class and the textbook.
Update: The first exam will be take-home, and distributed on Wednesday, March 16th and is due on Friday, March 18th. You may use any textbook, but may not discuss the exam with any other students (or others online). In other words the exam must be your own work.

Grades

Your course grade will assigned as follows

Homework scores (50%)

Midterm (20%)

Final (30%)

Learn From Anywhere and Classroom policies

Following BU's spring 2022 covid policy, please let me know as soon as possible if you have to miss class due to a covid related issue (either quarantine or taking care of a relative). I will then record the lectures you have to miss and provide them to you.

If I have to miss class for a covid-related issue I will contact the class by email as soon as possible to inform you of the plan for lectures (mostly like remotely given on zoom). Please check your BU email at least once a day.

We will observe all University mandated COVID-19 policies on social distancing and mask wearing.

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