Professor of Mathematics
Apr 22 2013:
This week: Student Presentations.
Please fill out the online course evaluation form.
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 analysis of the Airy function for large arguments can be downloaded from the "Homework" column of the entry below for Lecture 9.
Solutions to HW Sets 1, 2, 3, 4, and 5 are now available online.
| Week | Meeting | Date | In Class | Homework |
|---|---|---|---|---|
| Week 1 | Lecture 1 | Thursday, January 10 | Overview and Fundamentals. Covering Chapter 0 and Chapter 1, section 1.1. | |
| Week 2 | Lecture 2 | Tuesday, January 15 | Asymptotic series. Covering Chapter 1, sections 1.2-1.4. | |
| Lecture 3 | Thursday, January 17 | "Summability" of asymptotic series. Dominant balances and root finding. Covering Chapter 1, sections 1.5-1.6. | ||
| Week 3 | Lecture 4 | Tuesday, January 22 | Asymptotic expansions of integrals. Watson's Lemma. Covering Chapter 2. | |
| Lecture 5 | Thursday, January 24 | Laplace's Method. Covering Chapter 3, sections 3.1-3.5. | ||
| Week 4 | Lecture 6 | Tuesday, January 29 | Stirling's series. Weakly diffusive shock waves. Covering Chapter 3, section 3.6. | |
| Lecture 7 | Thursday, January 31 | Introduction to the method of steepest descents. Covering Chapter 4, sections 4.1-4.3. |
Problem Set 1 Due: |
|
| Week 5 | Lecture 8 | Tuesday, February 5 | Saddle points. Long-time asymptotics of diffusion. Covering Chapter 4, sections 4.4-4.6. | |
| Lecture 9 | Thursday, February 7 | 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. |
A link to the Digital Library of Mathematical Functions. |
|
| Week 6 | Lecture 10 | Tuesday, February 12 | Branch points continued. The method of stationary phase. Covering Chapter 4, more of section 4.8, and Chapter 5, sections 5.1-5.4. | |
| Lecture 11 | Thursday, February 14 | 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 19 | 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 21 (AWAY) | 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 26 | Stokes' phenomenon for irregular singular points. Covering Chapter 6, section 6.3.2. | |
| Lecture 15 | Thursday, February 28 | 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 | SPRING BREAK | |||
| Week 10 | Lecture 16 | Tuesday, March 12 | 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 14 | Generalization of the WKB method. The Liouville-Green and Langer transformations. Covering Chapter 7, section 7.2.5. |
Problem Set 3 Due: |
|
| Week 11 | Lecture 18 | Tuesday, March 19 | 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 | Tuesday, March 19 5:00-6:30 PM |
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. | ||
| Lecture 20 | Thursday, March 21 | Rigorous justification of matched asymptotics for singularly perturbed boundary-value problems. Covering Chapter 8, section 8.7. | ||
| Week 12 | Tuesday, March 26 | CLASS RESCHEDULED | ||
| Thursday, March 28 | CLASS RESCHEDULED | |||
| Week 13 | Lecture 21 | Tuesday, April 2 | 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. | |
| Lecture 22 | Tuesday, April 2 5:00-6:30 PM |
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 | Thursday, April 4 | Poincaré-Lindstedt method for removal of secular terms. The method of multiple scales. Covering Chapter 9, sections 9.3.3 and 9.3.4. | ||
| Week 14 | Lecture 24 | Tuesday, April 9 | 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 11 | 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 16 | Student Presentations I (2 talks): Alex Golden and Dana Suttman. | Alex Golden's presentation on the Korteweg-de Vries equation as a model for long waves in cold plasmas. Dana Suttman's presentation on Laplace's method for multidimensional integrals. |
| Lecture 27 | Thursday, April 18 | Student Presentations II (3 talks): Jeremy Hoskins, Daniel Jónás, and Colin Tinsman. |
Daniel Jónás' presentation on the asymptotic behavior of zeros of partial sums of the Taylor series for the exponential function. Colin Tinsman's presentation on barrier tunneling in quantum mechanics and the WKB method. |
|
| Week 15 | Lecture 28 | Tuesday, April 23 | Student Presentations III (3 talks): Louis Ly, Jiah Song, and Jimmy Vogel. |
Problem Set 5 Due: Louis Ly's presentation on uniform asymptotic expansions of exponential integrals with coalescing saddle points. Jiah Song's presentation on the convergence of the expansions generated by the method of Frobenius. Jimmy Vogel's presentation on the cubic nonlinear Schrödinger equation in nonlinear optics. |