Math 4510-101/MSSC 5510-101 Partial Differential Equations- Spring 2021


  • Ahmed Kaffel
  • Email: ahmed.kaffel (at)
  • Office: Cudahy 360
  • Office hours: MW 3:00-4:00 pm or by appointment


  • MWF 11:00AM - 11:50AM at Johnston Hall 415

Course Description and Learning Objectives:

Partial Differential Equations (PDEs) model a wide variety of phenomena in the natural sciences, engineering, and economics. This course is an introduction to the theory of linear partial differential equations, with an emphasis on solution techniques and understanding the properties of the solutions thus obtained. Specific types of solution techniques that the student will acquire include separation of variables, Fourier methods, Green's functions, and the method of characteristics. Classification of PDEs and differences between properties of solutions of PDEs in various classes will also be an important theme in the course.


Math 2451 (Differential Equations) or equivalent, Math 3100 (Linear Algebra).


The required course text is
  • Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th edition, by Richard Haberman.
Please note that we will use the fifth edition. The fourth edition is also acceptable, but be aware that homework assignments will be made from the fifth edition. We will try to catch any changes in the assigned homework problems.

For supplementary and additional reading I recommend the following (non-required) textbooks:

  • Solution Techniques for Elementary Partial Differential Equations, by C. Constanda.
  • Partial Differential Equations, by W. Strauss
  • Partial Differential Equations - An Introduction to Theory and Applications, by M. Shearer and R. Levy
The textbook by C. Constanda has a strong problem-solving orientation (i.e. many worked examples), while the books by Strauss and Shearer-Levy are more theoretical (beautifully written) introductions to partial differential equations.


There will be two midterm exams and a final exam:

  • 1st midterm exam: Friday, March 5, 2021
  • 2nd midterm exam: Friday, April 9, 2021
  • Final exam: May 10 2021, 1:00PM - 3:00PM
The exam problems are based on the lectures, the textbook, homework problems, and activities. An absence from an exam is recorded as a score of 0. Makeup exams are generally allowed only for university-excused absences. See "Attendance Policy" below. If you feel that your situation warrants a makeup exam, please check with me as soon as possible to request one. Further documentation of your absence may be required. Make-up exams will not be given unless there are extreme circumstances and you inform me of the absence within 48 hours of the exam. You are responsible for scheduling your make-up exam.

Grade Policy:
Your final grade will be determined as follows:
  • Homework: 30%
  • Midterm exams: 20% each
  • Final exam: 30%
Your minimum final grade will be A, A-, B+, B, B-, C+, C, C-, D+, and D for course averages of 92%, 88%, 84%, 80%, 76%. 72%, 68%, 64%, 60% and 56%. Grades for assignments and exams will be posted on D2L. Please check your recorded grades regularly to monitor your progress in the course and to ensure accuracy of recorded grades. If you believe that an exam or a homework was graded or tallied incorrectly you may submit the exam for a regrade along with a separate sheet of paper explaining why you believe you deserve more points. Regrades will be accepted up to two class periods after the exam or the homework grade is posted. You must let me know (in writing; email is fine) within seven days of receiving the grade; otherwise I can't promise that I will consider the issue.

Homework assignments:
Homework assignments and due dates are posted in the homework section below and/or D2L, usually sometime on Friday. Unless otherwise noted, please submit your homeworks in the dropbox on the due date. Late homework will not be accepted except in cases of excused absences (see "Attendance" below). However, your lowest homework score will be dropped from the final grade calculation.

In order to master the material of the course, it is key that you do your homework. You should make every effort to solve the assigned problems using the concepts learned from the lectures and readings. You will be graded mostly on your ability to work problems on exams. If you have not practiced the techniques within the homework problems, you will have serious difficulties to work problems on exams. You are strongly encouraged to do your homework in groups. However, you must write up your solutions on your own. Copying is not acceptable. I strongly recommend neatly writing up solutions to the homework and saving these solutions. These solutions will be a valuable resource when you come to office hours or review for an Exam. Problems like the homework will appear on exams.

Please write your name at the top. Write legibly. Your solutions to the assigned problems should be detailed enough so that the reader can follow your thought process. Students asking for makeup exams or extensions of written homework due dates should let me know of any conflicts at least one week beforehand in the case of prescheduled absences and as soon as possible (but in no case more than two working days after the absence) in cases where the absence is not foreseeable.
Below you find the list of the homework assignments:

HW1 Assignment:
sec 1.2: 1.2.1 (a) (b)
sec 1.3: 1.3.1
sec 1.4: 1.4.1 (c)(e)(g), 1.4.7 (c)
Bonus problems: Read sec1.5 and do problems 1.5.3 and 1.5.5

HW2 Assignment:
sec2.2: 2.2.2(a), 2.2.3
sec2.3: 2.3.2(c)(e), 2.3.3(a)(b), 2.3.5

HW3 Assignment:
sec2.4: 2.4.1(b)(d), 2.4.3, 2.4.4
sec2.5: 2.5.1(b), 2.5.3(a), 2.5.5(a)(d)
Bonus problems: 2.4.1(a)

HW4 Assignment:
sec3.2: 3.2.1(c)(g), 3.2.2 (e), 3.2.4
sec3.3: 3.3.1(b)(c), 3.3.5(b)
Bonus problems: 3.3.15, 3.3.17

HW5 Assignment:
sec8.2: 8.2.1(b)(d), 8.2.2(b)(d)
sec8.3: 8.3.1(a), 8.3.6
Bonus problems: 8.2.3, 8.3.1(e)

HW6 Assignment:
sec4.4: 4.4.1, 4.4.6, 4.4.7, 4.4.9, 4.4.10, 4.4.12
Bonus problems: 4.4.3, 4.4.8

HW7 Assignment:
sec7.3: 7.3.1(a)(d), 7.3.3, 7.3.5
sec7.4: 7.4.1(b)(c)
Bonus problems: Read Section 7.8 and do problem 7.7.7

HW8 Assignment:
sec10.3: 10.3.1, 10.3.5, 10.3.6
sec10.4: 10.4.1, 10.4.3(a)

HW9 Assignment:
sec10.6: 10.6.18
sec12.2: 12.2.3, 12.2.5(a)(c)

HW10 Assignment:
sec12.2: 12.2.5(d)
sec12.3: 12.3.5, 12.3.6
sec12.4: 12.4.3
sec12.6: 12.6.7(b), 12.6.19(b)

Attendance Policy:
Attendance is required and will be recorded. Attendance is essential to succeed in this course. Missing class almost always results in poorer performance on exams and homework. By attending lectures you will get a sense of what I consider important and that should help you know what to focus on when you study for the exams. You are responsible for lecture notes, any course material handed out, and announcements made in class. You are expected to arrive in class on time with the appropriate lecture notes for the class, having completed the assigned reading for the current lecture and assigned problems for the previous lecture. You are expected to stay through the entire lecture, be prepared to ask questions, and be willing to learn mathematics. If you have a good reason to miss a class, please inform the instructor as soon as possible (preferably via email) to mitigate penalties. The university attendance policy considers more than 6 absences in a 3-credit course to be excessive. With 7 absences you may be dropped from the class. Note that with very few exceptions, the University does not make a distinction between excused and unexcused absences. If you miss a class, it is your responsibility to obtain and learn the material you missed. In the event that you miss a class you should get the material you missed (e.g., from another student) and make an attempt to learn it yourself (or with a tutor). I will also be happy to answer your questions about the missed class. The attendance policy of this course follows the College of Arts and Sciences policy in the Undergraduate Bulletin.

Days off:
- Tuesday, Feb. 9, 2021
- Wednesday, March 10, 2021
- Tuesday, April 20, 2021
- Wednesday, May 5, 2021

Course Schedule

Below is a tentative schedule for the topics to be covered and assignment due dates. Apart from the midterm exam dates and academic holidays, the other items in the schedule should be considered tentative until the day has passed. It is necessary to keep some flexibility in the schedule to account for adjustments in the pace of lectures. The schedule will be updated as we go with lecture notes. The lecture notes are meant to supplement your own note taking in class and your reading of the textbook. I strongly recommend to you to read the relevant sections of the textbook before and/or after each lecture.



Introduction, Derivation of the Heat Equation, Boundary Conditions

Notes-ch1-part1 Notes-ch1-part2 Notes-ch1-part3
Complete Notes-Chapter1 Slide-1 Slide2 Slide-3 Slide-4



Method of separation of variables

Notes-ch2-part1 Notes-ch2-part2 Notes-ch2-part3 Slide-5


HW1 on Feb 5


Method of separation of variables

Notes-ch2-part4 Notes-ch2-part5 Notes-ch2-part6 Slide-6
Complete Notes-Chapter2


HW2 on Feb 12


Fourier series

Notes-ch3-part1 Notes-ch3-part2Notes-ch3-part3 Slide-7


HW3 on Feb 19


Fourier series, Review

Notes-ch3-part4 Notes-ch3-part5 Review pde
Complete Notes-Chapter3


HW4 on Feb 26


Inhomogeneous problems

Notes-ch8-part1 Notes-ch8-part2
Complete Notes-Chapter8 Slide-8

First Midterm Exam on Friday, March 05



Wave equation

Notes-ch4-part1 Notes-ch4-part2Notes-ch4-part3
Complete Notes-Chapter4 Slide-9


HW5 on March 12


Higher dimensional PDEs

Notes-ch7-part1 Notes-ch7-part2Notes-ch7-part3 Slide-10 Slide-11 Slide-12
Complete Notes-Chapter7


HW6 on March 19


Fourier transform and infinite domain problems

Notes-ch10-part1 Notes-ch10-part2 Notes-ch10-part3 Slide-13 Slide-14


HW7 on March 26


Fourier transform and infinite domain problems

Notes-ch10-part4 Notes-ch10-part5Notes-ch10-part6 Slide-15
Complete Notes-Chapter10

§10.4, §10.6

HW8 on April2


Review, Schrödinger's equation, Sturn-Liouville problems

Review1-midterm2 Note-shrodinger Slide-16 Slide-17 Slide-18

Second Midterm Exam on Friday, April 9



Classification of 2nd order PDEs, Exam Return,
Method of characteristics for wave equations

Notes-ch12-part1 Notes-ch12-part2 Slide-19


HW9 on April 16


Method of characteristics for wave equations

Notes-ch12-part3 Notes-ch12-part4 Notes-ch12-part5
Complete Notes-Chapter12


HW10 on April 23


Method of characteristics for wave equations




Final Exam Review

Supplement notes

Notes: Reviews on ODEs, linear algebra and calculus: Review Notes-REVIEW-ODEs Review Notes-Linear algebra Review Notes-Calculus

Notes related to eigenfunctions: Notes-eig-part1 Notes-eig-part2 Notes-eig-part3

Notes on solving PDEs: Notes-eig-part4

Notes on MULTI-DIMENSIONAL PDES: SEPARABLE PROBLEMS: Separable1_PDEs_in_nd Separable2_special_functions Review_PDEs_nd_separable

Notes on Fourier series: Notes-Fourier Notes-Fourier Coef

Notes on Fredholm alternative and Green functions: Notes-Fredholm Notes-Green functions

Fourier Transform-Laplace Transform: Transforms1_Fourier Transforms2_Sine_and_more Transforms3_Laplace

Formula sheet to be used in exams: Formula Sheet2 Formula Sheet2

Summary Review: Review

Practice problems

Practice for Midterm exam1: Study_Guide_Midterm1

Practice for Midterm exam2:: Study_Guide_Midterm2

Practice for Final exam:: Study_Guide_Final


Mon, Jan 25: Welcome to Math 5510! I wish you all the best for this spring term.