AMATH 568: Advanced Methods for Ordinary Differential Equations

Course Description

Learning objectives

Syllabus

• Weeks 1-3: Regular perturbation theory for equations with small parameters. method of multiple scales.
• Weeks 4-6: Singular perturbation theory: dominant balance, boundary layers and WKB theory.
• Weeks 7-10: Asymptotic solutions near regular and irregular singular points of linear ODEs, including infinity, using Frobenius theory and dominant balance. Method of stationary phase. Application to Bessel, Airy and error functions.

Special days

• Mo 21 Jan: No class - MLK Day
• We 9 Jan: No class; will make up
• Mo 11 Feb: In-class midterm (bring paper and one double-sided page of hand-written notes)
• Mo 18 Feb: No class - Presidents Day
• We 20 Feb: No class; will make up
• Mo-Fr 25 Feb-1 Mar: No class; will make up
• Fr 15 Mar: Last day of class
• Mo 18 Mar 2:30-4:20pm: In-class Final Exam; one double-sided page of handwritten notes.

Supplemental reading

Grading

• Assignments will alternate between solo (do without consulting without others) and discussable (discussion with your fellow students is encouraged). In either case you must write your own scripts and solutions in your own words. Assignments will involve algorithm implementation and plotting; you may use Matlab or Python for this, but my scripts are all in Matlab. Please submit assignments via Canvas. Late homework (after 11:59pm on due date) must be submitted by 11:59pm on the Monday after the due date. The first late homework will be accepted without penalty. Thereafter, there will be a 20% deduction on the grade for each late assignment.
  
• Midterm (20%), in class Mo 11 Feb, open book and note. Here and here are two midterms from previous years. This year's midterm may include very different problems including any of the class topics up through Lecture 9, including material covered in Homeworks 1-4.
• Final Exam (30%), in class Mo 18 Mar 2:30-4:20 p.m., one double-sided hand-written page of notes. Here is a list of topics that might appear on the final. In-class final and solutions from a previous year

  Please let instructor know well ahead of time if you will have a conflict with the midterm or final exam time so a makeup can be arranged.

  Grades will be recorded on the class Canvas page, which I will also use for assignments, class announcements and discussion forums.

Lecture Notes

IMPORTANT: We will use the class period for Q/A and working posted discussion problems in small groups. Except for Lecture 1, you should read the lecture notes BEFORE the class period and come prepared to answer simple comprehension questions about them. Note that some lectures cover multiple class periods. Dates are subject to mid-course correction.

• 1. Introduction (Mo 7 Jan)

Regular perturbations

• 2. Polynomial roots I (Read for We 9 Jan).
• 3. Polynomial roots II (Read for Fr 11 Jan).
• 4. Regularly perturbed ODE BVP (Read for Mo 14 Jan).
• 5. Slightly nonlinear pendulum I (Read for Mo 14 Jan).
• 6. Slightly nonlinear pendulum II (Read for We 16 Jan).
• 6a. Slightly nonlinear pendulum III - Method of multiple scales (Read for Fr 18 Jan - We 23 Jan).
• 7. Parametric resonance (Read for Fr 25 Jan).

Singular perturbations

• 8. Intro and polynomial roots example (Read for Mo 28 Jan).
• 9. Boundary layers (Read for We 30 Jan).
• 10. WKB theory I (Read for Mo 4 Feb).
• 11. WKB theory II (Read for We 6 Feb).
• 12. WKB theory III: matching at turning points (Fr 8 Feb)
• 13. WKB theory IV: Quantum harmonic oscillator: a Sturm-Liouville eigenvalue problem (We 13 Feb)

MIDTERM: Regular perturbations (Mo 11 Feb, open book/note)

Approximate solutions near singular points of ODEs

• 14. Taylor series about regular points (Read for Fr 15 Feb) .
• 15. Regular and irregular singular points; Frobenius theory (Read for We 20 Feb).
• 16. Frobenius example: Bessel's equation at small x (Read for Fr 22 Feb).
• 17. Dominant balance and asymptotic behavior at irregular singular points (Read for Mo 25 Feb).
• 18. Dominant balance II: Bessel's eqn at large x (Read for We 27 Feb).
• 19. Dominant balance III: An asymptotic but nonconvergent series for erfc(x) (Read for Fr 29 Feb).

Approximate methods for integrals

• 20. Two-scale analysis of dispersive wave packets (Read for Mo-We 4-6 Mar).
• 21. Method of stationary phase applied to Ai(x), x << 0 (Read for Fr 8 Mar).
• 22. Method of steepest descent applied to Ai(x), x >> 0 (Read for Mo 11 Mar).
• 23. Perturbation methods for the Van Der Pol Oscillator (Read for We-Fr 13-15 Mar).

Matlab scripts

Lectures

• 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 solution for refraction of a wave incident at 45 degrees on a erf ramp of refractive index from 1 to 2.
• WKB_reflect.m. Plot matched WKB/Airy solution for reflection of a wave incident at 45 degrees on a erf ramp of refractive index from 2 to 1.
• WKB_HarmOsc.m. Plot exact and WKB solutions to the quantum harmonic oscillator eigenfunction problem y''+(lambda-x^2)y=0, y->0 as |x|->infty.
• airy_ts.m: Function to calculate Taylor series for two solutions y0(x) and y1(x) of Airy equation y"-xy=0 about x=0, truncated after term P.
• airy_ts_plot.m: Plot TS y0(x) over -7 < x < 3 for several truncations P, and P=30 TS approx to Ai(x). 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 0 < x < 10 for several truncations P. Uses bessel_frobenius.m.
• bessel_large_x.m: Plot comparison of large-x asymptotic series with J0(x) and Y0(x) over 0 < x < 10.
• erfc_largex.m: Plot behavior of x->infty asymptotic series for erfc(x) at various x and truncations.
• GaussPacketAnim.m: Animation of a Gaussian deep water wave packet; uses function GaussPacket.m.