## 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.

Lectures | Sections | Topics/Comments |
---|---|---|

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 L^{2} Theory; Orthonormal Bases; Convergence and Completeness; Best Approximations in L^{2} |

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

**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.