Professor of Mathematics
May 4 2015:
This week: Classes and exams are over! Have an enjoyable and productive summer!
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 15 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.
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 1.21.4.  
Lecture 3  Thursday, January 15  "Summability" of asymptotic series. Dominant balances and root finding. Covering Chapter 1, sections 1.51.6.  
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 3.13.5.  
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 4.14.3. 
Problem Set 1 Due: 

Week 5  Lecture 8  Tuesday, February 3  Saddle points. Longtime asymptotics of diffusion. Covering Chapter 4, sections 4.44.6.  
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 5.15.4.  AiryContour.nb 
Lecture 11  Thursday, February 12  Longtime 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 secondorder 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: 

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 secondorder ODEs with a parameter. Regular perturbation theory. Covering Chapter 6, section 6.3.2, as well as Chapter 7, sections 7.1.17.1.2. 

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 LiouvilleGreen and Langer transformations. Covering Chapter 7, section 7.2.5. 
Problem Set 3 Due: 

Week 11  Lecture 18  Tuesday, March 17  Asymptotic construction of eigenfunctions and the BohrSommerfeld quantization rule. Introduction to boundaryvalue 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 boundaryvalue problems. Outer expansions, inner expansions, and boundary layers. Matching of inner and outer expansions and uniformly valid approximations. Covering Chapter 8, sections 8.28.5, with examples from section 8.6.  
Week 12  Lecture 20  Tuesday, March 24  Rigorous justification of matched asymptotics for singularly perturbed boundaryvalue 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 4:005:30 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  PoincaréLindstedt 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: Makeup 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: 
Lecture 25  Thursday, April 9  Dynamics of molecular chains. FermiPastaUlam models. Longwave 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 Kortewegde Vries equation in the smalldispersion limit. 
Bonus Lecture  Wednesday, April 15 1:002:30 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: 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. 