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1MA052 Homepage

A collection of links to various ODE resources

Maple and MatLab worksheets

Homework

Timetable





Course Info:

  • Course: Ordinary Differential Equations II, 1MA052

  • Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se

  • First Day Handout

    • Textbooks:

  • Text I (HSD): M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Academic Press (Elsevier) 2004.

  • Text II (NSS): R. Nagle, Edward Saff, Arthur Snider, Fundamentals of Differential Equations , 8th edition, (available from http://www.coursesmart.com as e-text).


    Things on the current agenda:

  • EXAMPLES OF FINAL EXAMS FROM PREVIOUS YEARS: here, here and here

  • Homework 3, the absolutely sharp deadline is 11AM, Monday Dec 12th!

  • Matrix exponentials, fundamentals matrices and generalized eigenvectors: handout I, Polytechnic Institute of New York, and handout II, by Xu-Yan Chen (GaTech)

  • General Linear Methofs for Ordinary Differential Equations.
    • page 16: Gronwall Lemma and Birkhoff-Rota Theorem on continous dependence.

  • Solutions of the hypergeometric equation through Frobenius, in Wiki

  • Green's function for the Sturm-Liouville problem, orthonormality of the eigenfunction set, and compact operators, lecture notes by John Erdos from King's College London

    •

    Course Objectives/Outcomes:

    1) Understand the matrix methods for first order linear systems. Be able to solve the relevant problems.

    2) Be able to state, prove and apply existence and uniqueness theorems.

    3) Be able to use the power series techniques to solve differential equations.

    4) Understand the non-linear systems and their stability properties; limit cycles and Poincare-Bendixson Theorem.

    5) Understand the basics of the Sturm-Liouville theory. Be able to apply the theory in boundary very problems.

    6) Understand and be able to approach first-order systems as continuous dynamical systems. Be able to describe the details of the dynamics of the Lorenz attractor and homoclinic phenomena.


    Grading:

    There will be 3 homeworks, which will account for 45% of the final grade.

    There will be also a final, 3 hours, 55% of the final grade.

    The purpose of this scheme is spread the risks over the length of the course.


    Schedule

    1. 2010-10-27
      • Introduction, course information.
      • Review of planar linear systems.

      Reading: "Current adgenda", 6.4 HSD.


    2. 2010-10-28
      • Existence and uniqueness of solutions.
      • Picard's method of successive approximations.

      Reading: 17.1 through 3. HSD.


    3. 2010-11-1
      • Picard's method continued.
      • Continuous dependence on initial condition and parameters.

      Reading: 17.3 and "General linear methods..." on the Current Agenda


    4. 2010-11-3
      • More on continuous dependence.
      • Matrix exponential.

      Reading: 17.3 and "General linear methods..." on the Current Agenda, 6.4 in HSD


    5. 2010-11-4
      • Generalized eigenvalues

      Reading: Nagle, Saff ans Snider (NSS), chaper 9.8.


    6. 2010-11-7
      • Generalized eigenvectors
      • Nonhomogeneous systems: undetermined coefficients and variation of parameter

      Reading: NSS 9.7 and 9.8


    7. 2010-11-8
      • Power-series solutions

      Reading: NSS, chapter 8


    8. 2010-11-10
      • Power-series solutions, continued. Method of Frobenius.
      • Finding the second linearly independent solution.

      Reading: NSS, chapter 8


    9. 2010-11-17
      • Power-series solutions. Finding the second linearly independent solution continued.
      • Laplace equation.
      • Bessel equation.

      Reading: NSS, chapter 8, separation of variables in cylindrical coordinates


    10. 2010-11-21
      • Legendre equation.
      • Orthogonal polynomials.
      • Sturm-Liouville theory.


    11. 2010-11-22
      • Sturm-Liouville theory.


    12. 2010-12-1
      • ODE's as continuous dynamical systems.
      • Linear vs non-linear.
      • Stability.
      • Bifurcations.
      • Stable/unstable manifold theorem.
      • Poincare-Bendixson Theorem and Hilbet's 16th problem.


    13. 2010-12-6
      • Poincare-Bendixson Theorem continued.