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May 4 2015:
This week: Classes and exams are over! Have an enjoyable and productive summer!

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The pdf files (and some auxiliary files) for all of the student presentations are now available for download on this website. You all gave excellent talks and I think you learned quite a bit.

Solutions to Homework Sets 1-5 can be downloaded from the CTools website for our course. These are for your personal use. Please do not distribute them or give them to other people.

A Mathematica notebook for exploring Burgers' equation with small diffusion/viscosity can be downloaded from the "Homework" column of the entry below for Lecture 6.

A Mathematica notebook for analyzing the steepest descent contours that arise in the study of the Airy function for large arguments can be downloaded from the "Homework" column of the entry below for Lecture 10.

Home Page for Math 557


Applied Asymptotic Analysis, by P. D. Miller, AMS Publications, Providence, Rhode Island, 2006. There is a web page for the book, including a current list of corrections.

Class Meetings:

Tuesdays and Thursdays 1:10 PM - 2:30 PM in 1866 East Hall.

Office hours:

Mondays 1-2:30 PM and Thursdays 2:30-4 PM in 5826 East Hall, or by appointment.

Grading and Course Policies:

Students will be evaluated on the basis of
Week Meeting Date In Class Homework
Week 1 Lecture 1 Thursday, January 8 Overview and Fundamentals. Covering Chapter 0 and Chapter 1, section 1.1.
Week 2 Lecture 2 Tuesday, January 13 Asymptotic series. Covering Chapter 1, sections 1.2-1.4.
Lecture 3 Thursday, January 15 "Summability" of asymptotic series. Dominant balances and root finding. Covering Chapter 1, sections 1.5-1.6.
Week 3 Lecture 4 Tuesday, January 20 Asymptotic expansions of integrals. Watson's Lemma. Covering Chapter 2.  
Lecture 5 Thursday, January 22 Laplace's Method. Covering Chapter 3, sections 3.1-3.5.
Week 4 Lecture 6 Tuesday, January 27 Stirling's series. Weakly diffusive shock waves. Covering Chapter 3, section 3.6.


Lecture 7 Thursday, January 29 Introduction to the method of steepest descents. Covering Chapter 4, sections 4.1-4.3.

Problem Set 1 Due:


LaTeX Source

Week 5 Lecture 8 Tuesday, February 3 Saddle points. Long-time asymptotics of diffusion. Covering Chapter 4, sections 4.4-4.6.
Lecture 9 Thursday, February 5 Airy's equation and Airy functions. Stokes' phenomenon. Steepest descent analysis with branch points. Covering Chapter 4, section 4.7 and part of 4.8.
Week 6 Lecture 10 Tuesday, February 10 Branch points continued. The method of stationary phase. Covering Chapter 4, more of section 4.8, and Chapter 5, sections 5.1-5.4. AiryContour.nb
Lecture 11 Thursday, February 12 Long-time asymptotics of dispersive waves. Semiclassical properties of free quantum particles. Covering Chapter 5, sections 5.5 and part of section 5.6.  
Week 7 Lecture 12 Tuesday, February 17 Linear second-order ODE with rational coefficients. Classification of singular points. Series expansions for ordinary points. Covering the rest of Chapter 5, section 5.6. Covering Chapter 6, section 6.1 and most of section 6.2.  
Lecture 13 Thursday, February 19 Frobenius series for regular singular points. Linear second order ODEs with irregular singular point at infinity. Formal solutions, and construction of true solutions approximated by the formal solutions. Covering the rest of Chapter 6, section 6.2, as well as section 6.3.1 and most of section 6.3.2.

Problem Set 2 Due:


LaTeX Source

Week 8 Lecture 14 Tuesday, February 24 Stokes' phenomenon for irregular singular points. Covering Chapter 6, section 6.3.2.  
Lecture 15 Thursday, February 26

Irregular singular points and Stokes' phenomenon (continued). Linear second-order ODEs with a parameter. Regular perturbation theory. Covering Chapter 6, section 6.3.2, as well as Chapter 7, sections 7.1.1-7.1.2.

Week 9   Tuesday, March 3 WINTER BREAK  
  Thursday, March 5 
Week 10 Lecture 16 Tuesday, March 10 Justification of regular perturbation theory. Singular perturbation theory, and WKB methods without turning points. Covering Chapter 7, sections 7.1.3, 7.2.1, 7.2.2, and parts of 7.2.3.  
Lecture 17 Thursday, March 12 Generalization of the WKB method. The Liouville-Green and Langer transformations. Covering Chapter 7, section 7.2.5.

Problem Set 3 Due:


LaTeX Source

Week 11 Lecture 18 Tuesday, March 17 Asymptotic construction of eigenfunctions and the Bohr-Sommerfeld quantization rule. Introduction to boundary-value problems for ODEs; asymptotic existence of solutions. Covering Chapter 7, section 7.2.4 and Chapter 8, section 8.1.  
Lecture 19 Thursday, March 19 Qualitative analysis of solutions to singularly perturbed boundary-value problems. Outer expansions, inner expansions, and boundary layers. Matching of inner and outer expansions and uniformly valid approximations. Covering Chapter 8, sections 8.2-8.5, with examples from section 8.6.
Week 12 Lecture 20 Tuesday, March 24 Rigorous justification of matched asymptotics for singularly perturbed boundary-value problems. Covering Chapter 8, section 8.7.
Lecture 21 Thursday, March 26 Perturbation theory in linear algebra. Mathieu's equation. Perturbation theory for periodic solutions. Covering Chapter 9, section 9.1 and most of section 9.2.  
Week 13 Lecture 22 Monday, March 30
4:00-5:30 PM in 2347 Mason Hall
Justification of expansions for periodic solutions of Mathieu's equation. Weakly nonlinear oscillations. Nonuniformity and secular terms. Covering the rest of Chapter 9, section 9.2, and sections 9.3.1 and 9.3.2.
Lecture 23 Tuesday, March 31 Poincaré-Lindstedt method for removal of secular terms. The method of multiple scales. Covering Chapter 9, sections 9.3.3 and 9.3.4.
  Thursday, April 2 CLASS CANCELLED: Make-up Monday, March 30
Week 14 Lecture 24 Tuesday, April 7 The nonlinear Schrödinger equation as an asymptotic model for weakly nonlinear waves. Covering most of Chapter 10, section 10.1.

Problem Set 4 Due:


LaTeX Source

Lecture 25 Thursday, April 9 Dynamics of molecular chains. Fermi-Pasta-Ulam models. Long-wave and wavepacket asymptotics. Covering most of Chapter 10, section 10.2.
Week 15 Lecture 26 Tuesday, April 14 Student Presentations I: Rometsch and Lu

Thomas's presentation on perturbation theory in quantum mechanics.

Luby's presentation on the Korteweg-de Vries equation in the small-dispersion limit.

Bonus Lecture Wednesday, April 15
1:00-2:30 PM in 1096 East Hall
The nonlinear Schrödinger equation and weakly nonlinear waves.
Lecture 27 Thursday, April 16 Student Presentations II: Wu, Olson, and Altin

A zip archive containing Bobbie's presentation on zeros of Taylor polynomials and the Mathematica program he used.

Matt's presentation on integral representations of solutions of hypergeometric equations.

Berk's presentation on the central limit theorem of probability.

Week 16 Lecture 28 Tuesday, April 21 Student Presentations III: Li, Gerlach, and Ibrahim

Problem Set 5 Due:


LaTeX Source

Harry's presentation on multidimensional Laplace integrals.

Andrew's presentation on the diffusion approximation in neutron transport theory.

A zip archive containing Amr's presentation (pdf form) on geometrical optics, and the movie he showed.