MATH 720
Meeting times (lectures): MWF 1:25 – 2:15 PM at LeConte (LC) 310.
Instructor: Dr. Peter G. Binev http://www.math.sc.edu/~binev/
email: binev@math.sc.edu)
phones: 5766269 (at LC 425) or 5766304 (at SUM 206H)
Office hours: MWF 11:00 – 12:00 AM at LeConte 425/Sumwalt 204, or by appointment.
Prerequisites: MATH 555, or equivalent.
From the book: …a good background in calculus up to vector calculus (grad, div, curl) and the elementary mechanics of particles … an introductory (inviscid) fluid mechanics course and a first course in partial differential equations, enough to know the basics of the heat, wave and Laplace equations … linear algebra, complex analysis and probability put in an occasional appearance. Highschool physics is an advantage. But the most important prerequisite is an attitude: to go out and apply your mathematics, to see it in action in the world around you, and not to worry too much about the technical aspects, focusing instead on the big picture.
Cell Phones: All cell phones must be turned off during the class.
Homework: A few homework problems will be assigned each class. Be sure to do these problems before the next class. Some solutions will be collected.
Projects: Chapters
4, 5, 6, 11, 14, 15, 19, and 21 from the book are case studies. Each student
has to choose a project motivated by a problem described in one of these
chapters. Although the exact formulation of the project can be changed later in
the semester, the general area should be decided by September 12. Some of the
classes will be (partly) devoted to discussion of the projects which illustrate
the techniques currently studied. Every student has to prepare a paper (no more
than 8 pages) based on the project by the end of November. The file should be
send to binev@math.sc.edu no later than December
13, 2007. The project is part of the Final Exam.
Discussions: The homework and the projects will be discussed in class. The participation in the discussions is important part of the course.
Midterm Exam: There will be a midterm exam in a form of a test on October 19. The problem on the test will be similar to the ones from the homework. The test should give a general idea about the problems on the Comprehensive Exam to students that take this course as part of a comprehensive sequence.
Final Exam: The final exam will be a combination of the project and a test with problems similar to the ones from the homework. The date of the final is Friday, December 14  2:00 p.m.
Grading: The final grade will be determined from the homework and the participation in the discussions, the midterm exam (20%), and the project/final (50%).
Academic Dishonesty: Cheating and plagiarism will not be allowed (see http://www.jour.sc.edu/pages/academicintegrity/policies.html).
Web Materials: Here is a link to OnLine Materials for Math Courses at USC (look for MATH 720).
Printable Versions: syllabus schedule
Preliminary Schedule of Classes
[MATH 720 – Fall 2007]
Date 
Section 
Subject 
Homework Problems 
Aug. 24 
1.3 
Principles of modelling 
p. 12 / 1, p.13 / 2 
Aug. 27 
1.3, 1.4 
Conservation laws 
p.13 / 3, 4 
Aug. 29 
2.2 
Units and dimensions 
p.21/1, p.25/10, p.26/12 
Aug. 31 
2.3 
Electric fields and electrostatics 
p.21/2, p.22/4, p.23/5 
Sep. 5 
3.1 
Nondimensionalisation 
p.42 / 1,
p.43 / 2 
Sep. 7 
3.2 
Navier–Stokes
equations and Reynolds numbers 
Subsection 3.1.3 (p. 34) 
Sep. 10 
3.3 
Buckingham’s Pitheorem 

Sep. 12 
Case Studies 
hair modelling
and cable laying 

Sep. 14 
Case Studies 
the thermistor electrostatic painting 

Sep. 17 
7.17.2 
Firstorder quasilinear
PDE 
p.97/1, p.98/3, p.99/4 
Sep. 19 
7.3 
Shocks 
p. 100 / 56,
p.101 / 7 
Sep. 21 
7.4 
Charpitt’s
method 
p.102 / 8, 10 
Sep. 24 
7.5 
Secondorder linear equations in two
variables 
p.102 / 11 or 12 
Sep. 26 
Case Studies 
traffic modelling 

Sep. 28 
9.19.3 
The delta and Heaviside functions 
p.134 / 12 
Oct. 1 
9.49.5 
Balancing singularities 

Oct. 3 
9.6 
Green’s functions 

Oct. 5 
10.110.5 
Theory of
distributions 

Oct. 8 
10.6 
Extensions of the theory of distributions 

Oct. 10 
Case Studies 
the pantograph 

Oct. 15 
12.112.2 
Asymptotic
expansions 

Oct. 17 
12.3 
Convergence and divergence 

Oct. 19 
Test 
Midterm Exam 

Oct. 22 
13.113.2 
Regular
perturbation expansions 

Oct. 24 
13.313.4 
Linear stability 

Oct. 26 
13.513.6 
Small perturbations of a boundary 

Oct. 29 
Case Studies 
electrostatic painting 

Oct. 31 
Case Studies 
piano tuning 

Nov. 2 
16.116.2 
Functions with boundary layers 

Nov. 5 
16.316.4 
Boundary layers: examples from ODEs 

Nov. 7 
16.5 
Boundary layers: examples from PDEs 

Nov. 9 
Case Studies 
the thermistor 

Nov. 12 
18 
Lubrication
theory’ analysis in long thin domains 

Nov. 14 
18 
Heat flows and
advection–diffusion in a long thin domain 

Nov. 16 
Case Studies 
continuous casting of steel 

Nov. 19 
20 
Thin fluid layers: classical lubrication
theory 

Nov. 26 
20 
Thin fluid sheets 

Nov. 28 
Case Studies 
turning of eggs during incubation 

Nov. 30 
22 
Multiple scales 

Dec. 3 
23 
Ray theory and the
WKB method 

Dec. 5 
Case Studies 
Presentations 

Dec. 7 
Case Studies 
Presentations 

Although the instructor will try to keep relatively close to this schedule, there could be changes both in the subject of the class and in the time it is presented. Check this webpage for the current version.
The homework assignments will be added as the course progresses.