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Department of Mathematics College of Liberal Arts and Sciences

MAP 4341/5345 Introduction to Partial Differential Equations, Syllabus

Class 2026 (Spring)

Reading Material

Course Textbook: R. Haberman, Applied Partial Differential Equations, 4th edition (optional)
Recommended Textbook: P.J. Oliver, Introduction to Partial Differential Equations, Springer, 2014 (optional)
S.V. Shabanov, Lecture Notes on Partial Differential Equations (PDEs)

The lecture notes will be posted in the course page. They are close to the classroom lectures and contain practice (homework) problems. Haberman’s textbook is an official textbook for this course, but Oliver’s textbook is a better reading, in my view. My lecture notes are self-sufficient.

Prerequisites

Students are expected to be familiar with ordinary differential equations and methods to solve them. Basic knowledge of differentiation and integration of functions of several variables is necessary (Calculus 3). Concepts of linear algebra are not mandatory but will be very helpful to comprehend  the content of the course. At the end of the course, the Bessel and Legendre equations from MAP4305 will be used. If you did not take this course, you have to get my permission to enroll.

Course Content

Part 1: Partial differential Equations (PDEs). A solution to a PDE. General methods for solving PDEs: Separation of variables, Change of variables, Ansatz, and their combinations. Examples: 2D Wave equation. 2D Heat equation. 2D Laplace equation. 2D Helmhotz equation. Separation of variables in a PDE. Principle of superposition for a linear PDE. Boundary conditions.

Part 2: First-order PDEs. The method of characteristics for first-order PDEs. The Cauchy problem for first-order PDEs.

Part 3: Classification of second-order PDEs. Hyperbolic, elliptic, and parabolic equations.  Initial and boundary value problems for basic second-order PDEs. The existence and uniqueness of the solution. Differential operators in the space of square integrable functions on a bounded region. The eigen-value problem for a differential operator. The Sturm-Liouville problem. Complete sets of functions. The Fourier method for hyperbolic, parabolic, and elliptic problems in two variables for rectangular regions. Separation of variables in polar and spherical coordinates. Harmonic functions and harmonic polynomials in two and three variables. Spherical harmonics. The Fourier method for hyperbolic, parabolic, and elliptic problems in circular, cylindrical, and spherical regions.

Goals: Learning basics techniques to solve first and second-order  PDEs with emphasis on the Fourier method for solving initial and boundary value problems for hyperbolic, parabolic, and elliptic second-order linear PDEs.

Class meetings and attendance

The class meets on MWF in-person. The class attendance is not mandatory. But participation in the class and taking notes are essential to avoid any backlog of material to study because the course is developing fast and contains plenty of difficult concepts needed for solving homework and test problems. These concepts might be hard to study on your own. A habit of missing class meetings and studying shortly before a test is a likely road to failure of the course or a low grade. Questions during the lectures are encouraged.

Office hours

There will be in-person office hours. The schedule will be posted after the first week of classes in the course page and Canvas.

Exams

There will be 4 graded assignments and the final exam. Two midterm exams are online via Canvas and the other two are in person. An online exam is followed by an in-person exam. The in-person exams are cumulative. The first in-person exam covers all the material discussed in class prior the exam date, and the second coves all the material discussed in class after the first in-person test. The online tests are not cumulative and cover only the material discussed in class after the previous test. The test schedule will be posted in the course page and in Canvas. All tests will be conducted during the UF exam periods (after 7 pm). Monday is a preliminary test day.   One formula sheet written by yourself is permitted on the in-class tests, and any kind of electronic devices are not permitted.  The online tests will be conducted via Canvas, and you can use any material to prepare your submission but you are not allowed to get help from any person or discuss the problems with any person during the online test (see the “student honor code” below).  Each online assignment is open for a specified period of time (1-2 hours) during which it must be completed and submitted via Canvas. The submission is free-response. Indicate the problem number, write your solution (do not omit technical details), box the answer, do the same for all problems, enumerate all pages as 1/n, 2/n, …, n/n, where n is the total number of pages, write and sign the academic honesty pledge at the bottom of the last page, write your name and your UFID number, scan all the pages in the above order into a single PDF file, and submit the file via Canvas. Make sure that you have a software or app to make such a PDF file. Other formats are not acceptable. Late submissions will not be accepted. The final exam is in-person. It is cumulative and cover roughly one month of lectures at the very end of the course. The official day and time for all finals can be found in the UF registrar. Students who score 90% and above on all mid-term tests can take a take-home final that will be posted at the last day of classes. It will be due in 2 days. Submission via Canvas. A make-up for any missed test (in-class or online) is only with a written official (e.g. medical) excuse.

Special accommodation: Students requesting special accommodation for exams must first register with the Dean of Student Office. The Dean of Student Office will provide documentation to the student who must then provide this documentation to me when requesting accommodation.

Student honor code: Each online submitted assignment must contain the signed academic honesty pledge: “Herewith I acknowledge that I did all of the above problems myself and did not receive any help from any person”. Submissions without the signed honesty pledge will not be accepted. You are NOT allowed to discuss any assignment during the time period the assignment is open on Canvas. A breach of this policy is considered as cheating. If caught cheating, the course grade is an F, no exception.

Homework

Homework assignments will be  posted in the course webpage. Homework  is not turned in. Some of the homework problems will be discussed in class or during the office hours (if asked). Solving these problems is essential for understanding the course and attaining a good grade. Test problems in this course are technically involved. So, practicing the use of the concepts and developing required algebraic skills to realize the concepts are a must-do for any successful student.

Grading

Each assignment contains some number of problems. Each problem is worth an indicated number of points if solved correctly. There is a partial credit for incomplete solutions if the idea (concept) for solving is correct. If M is the total number of earned points and N is the total number of regular points, then your current grade is determined by the average:

G = (M/N) 100%

The grade thresholds are:

A: G>90; A-: G>85; B+: G>80; B: G>75; B-: G>70; C+: G>65; C: G>60; C-: G>55; D+: G>50; D: G>45; D-: G>40; F: G<40

Here > means “greater or equal” and < means “strictly less”. A rough estimate for N is 50, give or take a few points.

Extra credit

There will be extra credit problems in the tests. They are not counted in the number N, but can increase your number M if solved correctly.  The perfect score can therefore exceed 100% when the extra credit questions are correctly answered. However, these problems require more creativity than the regular ones.