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
4 units; lecture/term paper or discussion
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
At choice of instructor. See below.
Graduate standing or consent of instructor
Variational principles; Eigenfunctions; Integral equations and Green's functions; Laplace's equation; Diffusion equations; Wave phenomena.
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.
|Lectures 1, 2||Sec. 1.1 (Sagan)||Variational Problems in One Independent Variable|
|Lecture 3||Sec. 1.2 (Sagan)||Variational Problems in Two and More Independent Variables|
|Lecture 4||Sec. 1.3 (Sagan)||The Isoperimetric Problem|
|Lectures 5, 6||Sec. 1.4 (Sagan)||Natural Boundary Conditions; Applications of Calculus of Variations (choice of an instructor)|
|Lecture 7||Sec. 2.1 (Sagan)||Representation of Some Physical Phenomena by PDE: The Vibrating String|
|Lectures 8, 9||Sec. 2.3 (Sagan)||The Equation of Heat Conduction and the Potential Equation|
|Lectures 10, 11||Sec. 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, 15||Sec. 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, 18||Sec. 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, 20||Chap. 9 (Young) / Sec. 3.5 (Folland:Fourier)||Sturm-Liouville Theory|
|Lectures 21, 22, 23||Chap. 10, 8 (Young) / Sec. 10.1 (Folland:Fourier)||Green's Functions; Compact Operators; The Spectral Theorem|
|Lectures 24||Chap. 11 (Young) / Sec. 3.5 (Folland:Fourier)||Eigenfunction Expansions|
|Lectures 25, 26, 27||Sec. 10.2 (Folland: Fourier) / Chap. 3 (Folland: PDE)||Green's Functions in Potential Theory|
Total 27 Lectures
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);
The recommendation is one in-class midterm exam and a 2 hour, timed final exam during final exam week.