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 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.
| 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: |
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| 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: |
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| 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. |
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| 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: |
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| 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: |
| 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. |
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| 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. |