Ryan Goh
Office Hours: Monday 9-10:30am, Thursday 12:30-2pm, and by appointment (please include availability when requesting an appointment).
Email: rgoh@bu.edu
Office: CDS 423
Web page: math.bu.edu/people/rgoh
Cameron Edgar
Email: cse1@bu.edu
Office: CDS 346
Office Hours: Monday 2:30-3:30pm
Time: Tuesday and Thursday, 11:00 a.m.–12:15 p.m.
Location: PSY B37
These meeting details are based on the official Fall 2026 CAS MA 561 listing.
The required text is listed through the BU Bookstore. Both texts are available electronically through BU Libraries.
Required: Peter J. Olver, Introduction to Partial Differential Equations, 3rd ed. The text is available through BU Libraries. Corrections are available on the author’s page, and the current Springer edition is available here while connected to the BU network.
Supplementary: David J. Logan, Applied Partial Differential Equations, 3rd ed., available through BU Libraries.
math.bu.edu/people/rgoh/teaching/ma561-fs26/course-page.html
This course studies partial differential equations (PDEs), a central tool of applied mathematics. PDEs model phenomena throughout science, from nuclear physics and chemical reactions to fluid interactions and galaxy formation. We will develop analytical and numerical methods for fundamental PDEs and characterize their solutions and behavior.
The official prerequisites are MA 225 or MA 230, together with MA 226 or MA 231, or consent of the instructor. A solid understanding of ordinary differential equations (at the level of MA 226) and calculus (at the level of our Calc I, II, and III courses) is required. Some experience with basic linear algebra will be helpful. There will also be a numerical component (see below) which will require writing basic codes to numerically solve various equations.
Skills and Goals: Broadly, the course will help you develop the following skills:
Our course schedule is subject to student interests and class pace, but will roughly follow the outline below.
Students will be expected to read the corresponding sections of the textbook before and after they are covered in class and work through the examples as they go. The text is quite readable and will help you absorb the material. Precise reading assignments will be announced in class and on Blackboard the week before.
We will use computer simulations to explore and analyze differential equations throughout the course. In my examples, I will mainly use MATLAB, which BU students can download for free here. If you have not used MATLAB, see this quick tutorial and this tutorial on solving differential equations with MATLAB. I will also use Mathematica from time to time, mostly for plotting; see the download instructions and a tutorial to get started.
You are welcome to use other software, such as Python or Julia. Although I may be familiar with the language you choose, I cannot guarantee that I will be able to debug or troubleshoot code written in every language. You may use AI coding tools to help write code, but you must understand the code you use and cite any sources used.
Homework will be posted on the course webpage and Gradescope. Please neatly write or type your solutions, combine them into one PDF, and submit the PDF through the course Gradescope page. If you write on paper, scan your work before submitting it. A link to Gradescope can be found on the course Blackboard page. Assignments are due at 11:00 a.m. on Tuesdays. No late work will be accepted. Please start problems and get help early. Homework will include textbook problems, numerical exercises and explorations, and AI audits in which you identify and correct flaws in an AI-generated solution. Your two lowest homework scores will be dropped.
Doing exercises and problems is the best way to learn mathematics. We will have homework assignments every week. You are welcome and encouraged to work together, but you must write up solutions by yourself and in your own words. You must understand how to solve the problems yourself. You may use AI tools to aid and check your work, as well as to write code, but you must understand the work you submit and be able to explain it in your own words. This is one reason for the PDE Problem Seminars. Please cite all sources used, including AI tools.
In your solution sets, explain your work clearly, concisely, and in complete sentences. Homework should be legible and organized. If you are concerned about the legibility of your handwriting, please contact me about using the mathematical typesetting software LaTeX. The online tool Overleaf is a good way to get started with LaTeX with lots of documentation..
Homework will be lightly graded using the following rubric for each problem, subject to grader availability:
Answer keys will be posted on Blackboard one to two weeks after the due date.
Most weeks, we will spend approximately 15 minutes each class discussing two recently completed homework problems. These short student-led discussions are designed to help us understand the mathematical decisions that make a solution work, rather than simply to reproduce a polished answer.
For each selected problem, one student will serve as the primary presenter and one student will be a "questioner". The presenter should state the problem in their own words, explain the choice of method and the key steps of the derivation, and verify or interpret the result. The "questioner" will ask clarifying questions and help the presenter explain the solution. The class and I will also be welcome to participate in the discussion.
Presentation problems will ordinarily be announced after the homework deadline. The order of "presenters" will be announced in the first or second week of the semester. The order of "questioners" will not be announced in advance. You may use permitted resources while preparing, including AI tools, but you must be able to explain and defend every statement you present and respond to follow-up questions without outside assistance, other than the notes you've prepared. Briefly disclose any AI assistance used in preparing a presentation.
Presentations will be assessed primarily on correct mathematical setup, clarity of reasoning, verification or interpretation of the result, and engagement with questions. A correctable computational error is less serious than an inability to explain the underlying method. PDE Problem Seminars count for 20% of the course grade, with 10% for presentations and 10% for questioning. I plan for each student to give two presentations each over the semester
I will give an example presentation in each class in the second week of the semester.
Working in pairs, you will complete a semester-long case study that connects a concrete application to the analysis and numerical investigation of a partial differential equation. You may choose a topic from an instructor-provided menu or propose a different topic for instructor approval.
Your project must clearly state the application, domain, unknown, parameters, initial and boundary conditions, and modeling assumptions. It must include both an analytic or qualitative component—for example, characteristics, separation of variables, Fourier analysis, a maximum principle, conservation, asymptotics, or a Green’s-function argument—and a numerical component. The numerical work must be validated using an exact solution, manufactured solution, theoretical property, refinement study, or another appropriate benchmark.
The final report should also discuss reliability and limitations: what the model does and does not claim, limitations of the data or numerical method, and why the conclusions should be trusted. You may use AI tools for brainstorming, programming, and checking work, provided that you disclose this use and understand, document, and can defend all mathematics and code that you submit.
The rubric for the PDE Case Study will be posted on Blackboard.
The PDE Case Study is worth 20% of the course grade: proposal (10%), progress memo (15%), final report and mathematical/numerical work (60%), and presentation and responses to questions (15%).
A list of possible topics will be posted on Blackboard. Representative topics include heat flow in a rod or composite material; string vibration or reflection at an interface; traffic-flow shocks; pollutant transport by advection-diffusion; drug or population diffusion; electrostatic potential; soliton dynamics in a nonlinear Schrödinger equation; pattern formation in a reaction-diffusion system; phase separation in the Cahn–Hilliard equation; image blurring as heat flow; and data-driven inverse problems, such as recovering a diffusion coefficient from noisy temperature measurements. These examples are not restrictive; students are welcome to propose other suitable applications. To coordinate topics and avoid duplication, we will use a shared Google Doc where students can sign up for topics and find others who are interested in the same topic.
There will be one in-class written midterm on Thursday, October 15, during the regular lecture period, and an in-class final exam. For the midterm, students may use one handwritten 8.5-by-11-inch page of notes. After the midterm has been graded and returned, students may regain some points by coming to office hours to discuss problems they answered incorrectly and how to correct them in a short oral examination. Details will be provided before the exam.
The final exam’s date and time will be announced after the final exam schedule is posted by the Registrar.
Assignment records will be maintained on the course Blackboard site.
Please let me know about all religious observances at the beginning of the semester. As mentioned above, there will be no make-ups for homework. In extreme circumstances—such as religious observance, a death in the family, or an emergency—make-up exams may be possible. Please tell me during the first week of school if you will need a make-up exam.
Please contact me as soon as possible. I am happy to work with you and BU’s Disability and Access Services.
Please do not cheat. Copying answers from a friend, solution manual, or online solution set is detrimental to your learning. You will be held to the BU Academic Conduct Code.
Mathematics is learned by doing. Read the corresponding sections of the textbook before and after they are covered in class, and work through the examples yourself as you go. The text is quite readable and will help you absorb the material. Start the homework early, discuss it with others, and attend office hours often.
If you feel that you are falling behind in the course, please do not hesitate to contact me. It is always easier to address misunderstandings sooner rather than later. In addition to working with me, I can put you in touch with TAs in the tutoring room who are knowledgeable about partial differential equations.