MAT 207B Methods of Applied Mathematics Syllabus Page

MAT 207B Methods of Applied Mathematics Syllabus Page (Winter 2018)

Synopsis:

Calculus of Variations and Applications

Basics of PDEs

Basics of Fourier Series

Sturm-Liouville Problems

Eigenvalue Problems via Variational Principle

Green's Functions

Linear Integral Equations

Prerequisite:

Graduate standing or consent of the instructor. Textbooks:

H. Sagan: Boundary and Eigenvalue Problems in Mathematical Physics, Dover Publications, Inc., 1989. This is a required textbook.
N. Young: An Introduction to Hilbert Space, Cambridge Univ. Press, 1988. This is an optional textbook and I will use some chapters of this book, in particular, Chap. 8-11.

R. Courant & D. Hilbert: Methods of Mathematical Physics, Vol.I, Wiley, 1953.
G. B. Folland: Fourier Analysis and Its Applications, AMS, 2009.
I. M. Gelfand & S. V. Fomin: Calculus of Variations, Dover Publications, Inc., 2000.
F. B. Hildebrand: Methods of Applied Mathematics, Dover Publications, Inc., 1992.
S. Hildebrandt & A. Tromba: The Parsimonious Universe: Shape and Form in the Natural World, Coperincus, Springer, 1996.
M. Kot: A First Course in the Calculus of Variations, Student Mathematical Library, Vol. 72, AMS, 2014.
I. Stakgold & M. J. Holst: Green's Functions and Boundary Value Problems, 3rd Ed., John Wiley & Sons, Inc., 2011.
J. L. Troutman & M. P. Bautista: Boundary Value Problems of Applied Mathematics, 2nd Ed., Dover Publications, Inc., 2017.
K. Yosida: Lectures on Differential and Integral Equations, Dover Publications, Inc., 1991.

the comments, handouts, and reference page

Topics: I plan to cover the following (subject to change):

Calculus of Variations (Chap. I)
- Euler-Lagrange Equations
- Various applications including isoperimetric problems and computer vision
Basics of PDEs (Chap. II)
- Derivation of basic PDEs via vectorial approaches
- Derivation of basic PDEs via variational principle
- Separation of variables & boundary conditions
- Importance of the boundary conditions
Basics of Fourier Series (Chap. IV; mainly Sec.3)
Sturm-Liouville Problems (Chap. V & Young Chap. 9)
Green's Functions (Chap. IX; Young Chap. 10)
Eigenfunction Expansion (Chap. IX; Young Chap. 11)
Linear Integral Equations (Supplement via Yosida or Hildebrand)

Class Web Pages:

the comments, handouts, and reference page

Lecture Notes Page

Homework:

our homework page.

Grading Scheme:

40% Homework
60% Final Exam

Minimal Canvas Usage:

For this course, the usage of Canvas is minimal. We will use the Canvas website mainly for communication purposes such as important announcements, posting questions or comments that would benefit the whole class. Also, the HW solutions will be posted on Canvas and the gradebook will be maintained on Canvas. All the other important information, such as lecture notes, HW problem sets, and reference info for each lecture, etc., will use our course website located in my own homepage.

Please email me if you have any comments or questions!
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