Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 207B: Methods of Applied Mathematics
Approved: 2015-10-01, John Hunter and Naoki Saito

Units/Lecture:

4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
At choice of instructor. See below.

Prerequisites:

Graduate standing or consent of instructor

Course Description:

Variational principles; Eigenfunctions; Integral equations and Green's functions; Laplace's equation; Diffusion equations; Wave phenomena.

Suggested Schedule:

Instructor will choose pieces from the following textbooks:

  • H. Sagan: Boundary and Eigenvalue Problems in Mathematical Physics, Dover Publications, Inc., 1989.
  • G. B. Folland: Fourier Analysis and Its Applications, Amer. Math. Soc., 1992.
  • G. B. Folland: Introduction to Partial Differential Equations, 2nd Ed., Princeton Univ. Press, 1995.
  • N. Young: An Introduction to Hilbert Space, Cambridge Univ. Press, 1988.

Instructor will also supply info about some key articles to read during the course.

LecturesSections Topics/Comments
Lectures 1, 2Sec. 1.1 (Sagan) Variational Problems in One Independent Variable
Lecture 3Sec. 1.2 (Sagan) Variational Problems in Two and More Independent Variables
Lecture 4Sec. 1.3 (Sagan) The Isoperimetric Problem
Lectures 5, 6Sec. 1.4 (Sagan) Natural Boundary Conditions; Applications of Calculus of Variations (choice of an instructor)
Lecture 7Sec. 2.1 (Sagan) Representation of Some Physical Phenomena by PDE: The Vibrating String
Lectures 8, 9Sec. 2.3 (Sagan) The Equation of Heat Conduction and the Potential Equation
Lectures 10, 11Sec. 2.B, 2.C, 2.D (Folland:PDE) Basic Properties of Harmonic Functions; The Fundamental Solution of Laplacian; Dirichlet and Neumann Problems in Laplace/Poisson Equations
Lectures 12, 13, 14, 15Sec. 4.1, 4.2 (Sagan) / Sec. 2.1, 2.2, 2.3 (Folland:Fourier) Fourier Series of a Periodic Function; Convergence Theorems; Derivatives, Integrals, and Uniform Convergence; Fourier Series on Intervals
Lectures 16, 17, 18Sec. 3.3, 3.4 (Folland:Fourier) / Chap. 1, 2, 3 (Young) Basics of L2 Theory; Orthonormal Bases; Convergence and Completeness; Best Approximations in L2
Lectures 19, 20Chap. 9 (Young) / Sec. 3.5 (Folland:Fourier) Sturm-Liouville Theory
Lectures 21, 22, 23Chap. 10, 8 (Young) / Sec. 10.1 (Folland:Fourier) Green's Functions; Compact Operators; The Spectral Theorem
Lectures 24Chap. 11 (Young) / Sec. 3.5 (Folland:Fourier) Eigenfunction Expansions
Lectures 25, 26, 27Sec. 10.2 (Folland: Fourier) / Chap. 3 (Folland: PDE) Green's Functions in Potential Theory

Total 27 Lectures

Additional Notes:

Optional reference books: D. Colton: Partial Differential Equations: An Introduction, Dover Publications, Inc., 2004. B. Friedman: Principles and Techniques of Applied Mathematics, Dover Publications, Inc., 2011. J. P. Keener: Principles of Applied Mathematics: Transformation and Approximation, Revised Ed., Westview Press, 2000. J. D. Logan: Applied Mathematics, Wiley-Interscience, 4th Ed., 2013. I. Stakgold & M. Holst: Green's Functions and Boundary Value Problems, 3rd Ed., John Wiley & Sons, Inc., 2011 (available online for UCD users from Wiley Online Library, http://onlinelibrary.wiley.com/book/10.1002/9780470906538);

Assessment:

The recommendation is one in-class midterm exam and a 2 hour, timed final exam during final exam week.