LEC 02 (Prof. Xiuxiong Chen) has a different schedule, different homework etc, and a separate web page.
This course is an introduction to Fourier series and to their use in solving partial differential equations (PDEs). We will discuss in detail the three fundamental types of PDEs: the heat equation, the wave equation and Laplace's equation. These equations are important in many applications from various fields (mathematics, physics, engineering, economics, etc.) and illustrate important properties of PDEs in general.
This is a required text.
Final Exam: Thursday, Dec 19, 8:00-10:45am, in Javits 101. Please
note the location: the exam is not in our usual classroom!
The final exam is cumulative. A checklist of topics is here.
The exam will include a reference page with a few formulas. This reference page is here.
The second midterm covered sections 3.2, 3.3. 3.4, 4.1, 4.2, 4.3, 4.5. A checklist of topics (with some practice questions from textbook) is here.
The first midterm was given on Oct 3 and covered sections 1.1-1.5, 2.1-2.8. A checklist of topics for the first midterm is here.
No make-up exams will be given for midterms. If a student misses a midterm exam for a well-documented medical reason or other similar circumstances beyond the student's control, the student may be excused from the exam, with the final grade determined from the other exams and homework. For the final exam, make-ups will be given ONLY in cases of properly documented medical reasons or other similar circumstances, at the instructor's discretion.
Important: Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed. Homework is due at the beginning of class on the due date. Late homework will not be accepted.
Week 1 (08/26 – 08/30)
Read sections 1.1, 1.2. Really, really read, not just take a look, and try to follow the calculations!
Homework 1, due Sept 5:
1.1: 1bd, 2bd, 7b, 8 (in 8, "verify" means check the condition stated at the very beginning of 1.1).
1.2: 1b, 5, 10acd, 11 for functions from 10ac.
You will quickly discover that the textbook has answers for odd-numbered exercises. You are welcome to check your answer with the book, but please show all work (no points will be given for correct answers if the work not shown or completely wrong).
Week 2 (09/3 – 09/6)
Read sections 1.3, 1.4, 1.5 . There will be some unfamiliar concepts but please try to read the sections before class.
Homework 2, due Sept 12:
1.3: 1abd, 2ad, 5, 6
1.4: 1aef, 3bd
1.5: 2, 5, 8
Week 3 (09/9 – 09/13)
Read sections 2.1-2.3.
Homework 3, due Sept 19:
2.1: 2 ("verify" means check directly, by differentiating, that the functions satisfy the equation; note that there are no boundary/initial conditions here)
2.2: 1, 2, 6ab, 7, 4
2.3: 1 (write 3 terms), 4
Week 4 (09/16 – 09/20)
Read sections 2.3 -2.5.
Homework 4, due Sept 26:
2.3: 7; Project 2.3
2.4: 4, 5, 8
This homework has fewer questions but they are longer. Some require going through a full heat equation procedure. A good summary of steps is at the end of section 2.5.
Week 5 (09/23 – 10/4)
Read sections 2.7, 2.8, 2.6. Read all the previous sections we covered, to review for the midterm. .
Homework 5, due TUESDAY, Oct 1:
2.7 question 3acd. Sketch two eigenfunctions for each part. State the orthogonality relation for each part (your statements should something about certain integrals involving specific functions from your questions; no need to compute the integrals)
question 28 p. 209
Weeks 6-7 (10/7 – 10/18)
Read sections 3.1 -3.4.
Homework 6, due Oct 17:
3.2: 1, 4, 5, 9, 17
Week 8 (10/21 – 10/25)
Read sections 3.3 -3.4, begin Chapter 4 .
Homework 7, due Oct 24:
3.3: 1, 2, 3, 4, 10
Week 9 (10/28 – 11/1)
Read sections 4.1-4.3 .
Homework 8, due Oct 31:
4.2: 6, 7a, 8
Week 10 (11/4 – 11/8)
Read sections 4.3, 4.5 .
Homework 9, due Nov 7:
4.3: questions 1, 2 + finish example 2 and example 4 from the text. In example 2, solve the boundary value problems for functions u1 and u2. There are some formulas given in the book but please work from scratch: use separation of variables, set up the eigenvalue problem, find the eigenvalues and eigenfunctions. Find the remaining parameters from the boundary conditions. Assemble the solution u(x,y) from u1 and u2. Do similar work in example 4 for the function w. Examples 2 and 4 solve the same problem by different methods: do you get the same answer? (Please compute all the required Fourier series as well.)
Week 11 (11/11 – 11/15)
Read sections 4.5, 1.9 (1.9 not on the test) .
Homework 10, due TUESDAY, Nov 12:
4.5: questions 4, 11, 12. Note that the homework is due on Tuesday!
Week 12 (11/18 – 11/22)
Read sections 1.9, 2.10, 2.11, 4.4.
Homework 11, due TUESDAY, Nov 26:
1.9: question 1b,
2.10: questions 1, 4,
4.4: questions 12, 17.
Note that the homework is due on Tuesday!
Week 13 (12/02 – 12/06)
Read section 3.6 (only the first part, no time-dependent boundary conditions).
Homework 12, due Thursday, Dec 5:
3.6: question 6 (assume c=1).
4.4: questions 5c, 18.
Note that the homework is due this Thursday, at our last class!
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