Syllabus for MA 776 Partial Differential Equations

Syllabus for MA 776 Partial Differential Equations, Fall 2025

Course Information

Lecturer: Ryan Goh

Email: rgoh@bu.edu
Office: CDS 423
Office Hours: Tuesday 10:30 - 12:30pm
Wednesday 1:30-2:30pm
or by appointment (please provide availability when requesting to meet).
Web page: http://math.bu.edu/people/rgoh/

Lectures:

Time: Monday, Wednesday, Friday, 11:15-12:05am;
Location: FLR 122

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-fs25/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 and gradescope. Please neatly write out or type your solutions, turn them into one PDF (scanning if you write them on paper), and submit them through the course Gradescope page. A link to this page can be found on the course blackboard page. 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 at least 72 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.

Grades

Your course grade will assigned as follows

Homework scores (60%)

Midterm (20%)

Final (20%)

Classroom policies

Questions are always welcome during lecture. Please raise your hand or interupt me if I'm facing the board.

Excused absenses and make-up exams

Please let me know about all religious observances at the beginning of the semester.

Students with disabilities

Please contact me as soon as possible. I am happy to work with you and the BU office of Dissability and Access Services.

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