MAT 397A: Partial Differential Equations (Pdes)

Syllabus

AUBG, Spring 2011 MAT 397a: Partial Differential Equations (PDEs) (room: BAC 206: time: W 14:15-15:30, F 12:30-13:45) (tutorials: Tuesdays from 19:30 in room BAC 326) Prerequisites MAT 103, 104, 212 (Calculus I, II, III) and MAT 105 (Linear Algebra), This course is a Mathematics Major elective.

Alexander GANCHEV [email protected], http://home.aubg.bg/faculty/aganchev/ , office 304 (new bulding), phone: 480, office hours: TBA

Course description: The course offers an introduction to partial differential equations – the shortest route between mathematics and its applications. The three basic linear PDE’s, i.e., the diffusion, wave, and Laplace equations, will be considered as well as the three main methods for their analysis: separation of variables and Fourier series (i.e., Hilbert space methods), the method of Green functions and fundamental solutions, and variational methods. Some important non-linear equations will be considered briefly. Applications could include problems in physics, engineering, finance, etc.

Expected Outcomes: Upon completion of this course, it is expected that students will be familiar with the heat equation, Laplace's equation, and the wave equation, their derivation and properties, and will be able to solve elementary boundary and initial value problems for PDEs. It is also expected that they will have an understanding of trigonometric Fourier series and other orthogonal expansions (i.e., Sturm-Liouville eigenvalue problems). The students should be able to derive Green's kernels for simple problems and be acquainted with variational methods. The course should also give a glimpse of the unity of mathematics (in the case of linear PDE's – the unity of Calculus and Linear Algebra) and serve as a motivation for further explorations in real and functional analysis and differential equations.

Besides the above we want to learn to think creatively, be able to attack a problem you have not seen before, develop tools for that, develop a mathematical model for a given “real life” situation, develop mathematical modeling skills.

Prerequisites: This course is an introduction to PDEs and some nontrivial mathematical models and as such requires a certain dose of mathematical maturity. A necessary prerequisite is to have covered the Calculus sequence and a course in Linear Algebra Though not a prerequisit e, it will be helpful to be exposed to the multidimensional generalization of the Fundamental Th eorem of Calculus (e.g. MAT 212 and MAT 313) and to vector fields and basic facts about ordin ary differential equations (MAT 104 and MAT 213).

Textbooks: required: K.E. Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods (3rd edition, Dover, 1999)

Additional reading: M.Renardy, R.C. Rogers, An Introduction to Partial Differential Equations (2ed Springer 2004) QA374.R4244 2003 J.R. Hanna, J.H. Rowland, Fourier Series, Transforms, and Boundary Value Problems (2ed Wiley 1990) QA379.H36 1990 H.F. Weinberger. A First Course in Partial Differential Equations with complex variables and transform methods (Dover 1995) QA374.W43 1995 L.S. Evans, Partial Differential Equations (AMS 1997) I. Stavroulakis, S. Tersian. Partial Differential Equations – introduction with Mathematica & MAPLE (2nd edition, World Scientific, 2004) [available as AUBG Library electronic resource at the URL: http://site.ebrary.com/lib/aubg/docDetail.action?docID=10078524 ] A. Tveito, R. Winther. Introduction to Partial Differential Equations – computational approach (Springer 1998) [available as AUBG Library electronic resource at the URL: http://site.ebrary.com/lib/aubg/docDetail.action?docID=10015670 ] free books and lecture notes available online: D. Joyner. Introductory Differential Equations using SAGE (book 2007) wdjoyner.com/teach/DiffyQ/des-book.pdf M. Joshi, A. Wasserman. Partial Differential Equations (lecture notes) http://www.maths.cam.ac.uk P. Olver. Introduction to Partial Differential Equations (draft of book) http://www.math.umn.edu/~olver/pdn.html S. Mauch. Intro to Applied Mathematics (almost 3000 pages long) (http://www.cacr.caltech.edu/~sean/applied_math.pdf ) R. E. Showalter, Hilbert Space Methods for Partial Differential Equations (E.J.Diff.Eq Monograph 1994) http://ejde.math.txstate.edu/Monographs/01/abstr.html

Assessment: Your grade will be formed by

homework 15 % participation 5 % quizes 10 % project 10 % midterm exam 20 % final exam 40 %

______

total 100 %

percentage/Grade Map:

D > 45, D+ > 50, C- > 55, C > 60, C+ > 65, B- > 70, B > 75, B+ > 80, A- > 85, A > 90, A+ > 100

(For easier grading I will calculate everything in “grading points” with one grading point being equal to three percentage points.)

Exam and assessment policies: During quizzes, exams and the final you should work strictly by yourself – you should not communicate in any way with your classmates – violation of this will be considered cheating with all the ensuing consequences (see the AUBG documentation for the consequences of cheating). Cheating is not only talking to the person next to you (talking about anything: math, the problems, the weather, last nights party …) but also intentionally making your work available to others during the exam.

Projects: A typical project would be a paper describing a problem, a PDE model for the problem, a strategy for solving the PDE numerically, and a summary of your results. You will give a 10 minute presentation to the class.

Homework: Discussion of the homework is encouraged. However, you should write down your answer independently. No late homework will be accepted. The lowest grade will be dropped in the final grade calculation. There will be a lot of homework exercises assigned in addition to those to be turned in, and you should do all of them. Practice is very important to get a good understanding of the material and to do well in the exams.

Attendance: Students are expected to attend classes regularly and should comply with the university attendance policies. I expect you to come to class prepared (having read the assigned text if there is such) and to show active participation during the lecture.

Office hours: If the “official” office hours above are not convenient for you please contact me to arrange some other time. Don’t be afraid to come and ask. There are no stupid questions.

Online Components of the Course: The Course Schedule will be updated as we go along on and placed on my web space. You will be able to find additional material there that I will add as we proceed.

Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any changes.

Tentative schedule and topics covered:

• 1-st order PDE: the method of characteristics (linear, quasi-linear, nonlinear). • 2-nd and higher order equations: classification; initial/boundary-value problem; well-posedness; stability; dissipation; dispersion. 3. Separation of variables and generalized Fourier-series expansions: • Sturm-Liuoville and multi-dimensional elliptic eigenvalue problems: orthogonality and completeness of eigenfunctions • Laplace, heat and wave equations in bounded regions. 4. Transform methods (Fourier; Laplace; Henkel); problems in unbounded regions. 5. Green's functions and fundamental solutions: • Green' identities; • Generalized functions • Fourier method: series and integral expansions of Green's functions • Symmetries and the Method of images: Green's functions for the Laplace, heat and wave equations in special regions (space; half-space; quadrant; slab; box; disk; ball; sphere) 6. Variational, Perturbation and Asymptotic methods: • Hamilton’s principle of minimal action and applications: vibrating strings, membranes, etc. Energy conservation and causality. • The mini-max principle and Rayleigh-Ritz method for eigenfunctions of differential operators; • Regular perturbations (eigenvalue, boundary value perturbations) • Equations with large parameter: Stationary phase and Geometrical optics methods; Helmholtz equation.