### MAT 207B Methods of Applied Mathematics Syllabus Page (Winter 2015)

Synopsis: We will cover the most important and fundamental concepts in Applied Mathematics, which are rather classical yet indispensable for understanding any modern applications, such as: Calculus of Variations and Applications; Basics of PDEs; Basics of Fourier Series; Sturm-Liouville Problems; Eigenvalue Problems via Variational Principle; Green's Functions; and 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.

The following are good references (some of them seem to be viewable at books.google.com): 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. I. Stakgold & M. J. Holst: Green's Functions and Boundary Value Problems, 3rd Ed., John Wiley & Sons, Inc., 2011. K. Yosida: Lectures on Differential and Integral Equations, Dover Publications, Inc., 1991.

In addition, some journal papers and book chapters will be discussed in the class and their links will be posted in 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 Page: I will keep updating the webpages for this course (one of which you are looking at now). In particular, please read the comments, handouts, and reference page often. After each class, I will put relevant comments and references as well as most of my handouts in class in this page that should serve as a guide to further understanding of the class material. Homework: Grading Scheme: 40% Homework 60% Final Exam

**Class Mailing List:**The class mailing list was created. Important announcement will be communicated through this mailing list. You can also submit your public comments, suggestions, and questions on HW, and/or some useful information related to the class to this mailing list. Once you send your email to this list, however, everyone will receive it. have any comments or questions!

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