Advanced Methods for Ordinary Differential Equations

Instructor: Prof. Chris Bretherton breth@atmos.washington.edu ATG 710, x5-7414 Office hours: MF 1:30-2:20, or by appointment.

Teaching Assistant: Katie Oliveras oliveras@amath.washington.edu Condon 810b Office hours Mo 4-5, Tu 9:30-10:30

Course Description

Note: The Amath 568 catalog description also includes several topics now covered in Amath 402. These include linear systems of ODEs and phase-plane analysis and stability analysis of autonomous nonlinear systems. These are no longer covered in depth in Amath 568, though we will review them as necessary.

Syllabus

• Weeks 1-3: Solutions of ODEs near regular and irregular singular points. Review of Frobenius theory and application to Bessel's equation. Dominant balance.
• Weeks 4-6: Regular and singular perturbation theory for equations with small and large parameters. Asymptotic expansions. WKB theory. Airy functions.
• Weeks 7-8: Bifurcation theory for systems of nonlinear ODEs with a parameter.
• Weeks 9-10: Two weeks of student-desired topics (possibilities include Sturm-Liouville theory, Floquet stability analysis, stationary phase asymptotics, etc.)

Supplemental reading

Grading

• Homework (50%), posted to class web page and assigned each Wednesday, due the following Wednesday. Discussion with your fellow students is encouraged, but please write out your own solutions in your own words. Some assignments may involve computer plotting of solutions. Late homework (after 5pm on due date) must be handed in by the end of the next class after the due date. The first late homework will be accepted without penalty. Thereafter, there will be a 20% deduction on the grade for that assignment.
  
• Midterm (20%) and Final Exam (30%), open book and note, see schedule below. Please let instructor know well ahead of time if you will have a conflict with either date so a makeup can be scheduled.

Special days

• Mo-We 29-31 Jan: Bretherton at meeting in Boulder
• Mo 15 Jan: MLK Day
• Mo 12 Feb 8:30-9:20: Midterm (in class, open book)
• Mo 19 Feb: Presidents Day
• Fr 9 Mar: Last day of class
• Tu 13 Mar 8:30-10:20: Final Exam (in class, open book)

Homework and Exams

Lecture Notes

• We 3 Jan
• Fr 5 Jan
• Mo 8 Jan
• We 10 Jan
• Fr 12 Jan
• We 17 Jan
• Fr 19 Jan
• Mo 22 Jan
• We 24 Jan
• Fr 26 Jan
• Mo 29 Jan
• We 31 Jan
• Fr 2 Feb
• Mo 5 Feb
• We 7 Feb
• Fr 9 Feb
• We 14 Feb
• Fr 16 Feb
• We 21 Feb
• Fr 23 Feb will be devoted to discussing the midterm and other student questions
• Mo 26 Feb onward. pp. 1-8 are assumed to be review and will not be covered in depth. Please read these to be sure you are familiar with the words and concepts therein.

Matlab scripts

• airy_ts.m: Function to calculate Taylor series about x=0 truncated after term P for two solutions of Airy equation y"-xy=0.
• airy_plot.m: Plot TS y0(x) over -7airy_bvp.m: Compute and plot approximate soln of Airy eqn with y(0)=1, y(5) (approx y(infty)) = 0. Uses airy_ts.m
• bessel_frobenius.m: Function to calculate Frobenius series about x=0 truncated after term P for solutions J0(x) and Y0(x) of Bessel equation y"-y/x + y =0
• bessel_plot.m: Plot Frobenius series for J0(x) and Y0(x) over 0bessel_frobenius.m
• bessel_large_x.m: Plot comparison of large-x asymptotic series with J0(x) and Y0(x) over 0erfc_largex.m: Plot behavior of x->infty asymptotic series for erfc(x) at various x and truncations.
• pendulum.m: Plot exact (from ode45) and perturbation solutions to nonlinear pendulum eqn. y" + sin(y) = 0, y(0)=eps, y'(0)=0, 0 < t < 4*pi, for eps= 0,1. Uses pendulum_dydt.m.
• WKB_refract.m: Plot WKB approximate solution to refraction of a short-wavelength light beam coming in from below at 45 degrees to a refractive index gradient n(z) = 1 + 0.5*(1+erf(z)). Dashed lines are drawn at z = -1,1 to indicate the rough borders of the gradient layer. Wave bends toward the region of higher refractive index.
• WKB_reflect.m: Plot Approximate solution to refraction and total reflection of a short-wavelength light beam coming in from below at 45 degrees to a refractive index gradient n(z) = 2 - 0.5*(1+erf(z)). The WKB solution below a matching height (indicated with a chain-dashed line) is matched with a local Airy function solution above this line which is locally valid around the turning point (solid line). Note the smooth transition between the WKB and Airy solutions, the standing-wave pattern below the turning point, and the exponential decay above the turning point.
• vanderpol_tryme.m: Plot phase portrait (including a sample orbit) of van der Pol oscillator y" + alpha*(y^2-1)y' + y = 0 for alpha=0.2 and alpha=5. Uses vanderpol.m and PhasePortrait.m.
• swing.m: Function to plot phase portrait (including 3 sample orbits) and Poincare map for the pumped swing problem of HW6p2. Note required input arguments (see example in swing.m file). Uses PhasePortrait.m.
• swing_nonlin.m: Function to plot phase portrait (including 3 sample orbits) and Poincare map for the nonlinear pumped swing problem 2 of final. Note required input arguments (see example in swing_nonlin.m file). Uses PhasePortrait.m.
• tsunami.m: Plot large-omega asymptotic solution of the tsunami problem 3 of final, over the entire inner continental shelf (left) and near the shore (right).