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Calendar

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Textbooks:

  • [Evans]      Lawrence C. Evans. Partial Differential Equations, AMS, Providence, RI. Series: Graduate Studies in Mathematics, Vol. 19, 1998.
  • [McOwen]  Robert C. McOwen, Partial Differential Equations, Methods and Applications, Prentice Hall/Pearson Education, Inc., 2003 (Second Edition)



    Dates/Topics

    1. 2008-09-01
      • Introduction, course information
      • Review of ordinary differential equations (existence, uniqueness and non-uniqueness)
      • Example of derivation of PDE
      • 1st order PDE, I: characteristic curves for a linear equation
      • Formulation of the Cauchy problem for a first order equation

    2. 2008-09-02
      • 1st order PDE, II: examples of quasilinear equations
      • The inviscid Burger equation as a model for breaking surface waves
      • Method characteristics for Burger's equation, shocks and entropy condition

    3. 2008-09-03
      • 1st order PDE, III: general nonlinear partial differential equations
      • The method of characteristic strips and the method of envelopes
      • The eikonal equation of geometric optics

    4. 2008-09-04   Exercise session:  Problem set 1

    5. 2008-09-05 (McOwen 49-54)
      • Second order equations
      • Linear equations of the second order: canonical forms
      • The Laplace and Poisson equations; the wave equation; the heat equation

    6. 2008-09-08
      • The wave equation, I: one-dimensional case
      • Initial value problem
      • Weak solutions

    7. 2008-09-09
      • The wave equation, II: higher dimensions
      • Solution of the wave equation in three space variables using radial functions

    8. 2008-09-10
      • The Laplace and Poisson equation, I: introduction

    9. 2008-09-17
      • The Dirichlet and Neumann problems for the Laplace equation
      • Fundamental solution to the Laplacian

    10. 2008-09-18
      • The Laplace and Poisson equation, II
      • Harmonic functions

    11. 2008-09-19   Exercise session:  Problem set 2

    12. 2008-09-22, 10:15 -12:00
      • The heat equation

    13. 2008-09-22, 15:15 -17:00
      • Functional spaces. Banach and Hilbert spaces.
      • Hölder spaces

    14. 2008-09-23
      • Weak derivative. Distributions.

    15. 2008-09-24   Exercise session:  Problem set 3

    16. 2008-09-25,
      • Sobolev spaces, I: introduction.

    17. 2008-09-30, 10:15 - 12:00
      • Sobolev spaces, II: Sobolev inequalities and embeddings (subcritical case)

    18. 2008-09-30, 15:15 - 17:00,
      • Sobolev spaces, III: Sobolev inequalities and embeddings (supercritical case)

    19. 2008-10-06, 13:15 - 15:00
      • Elementary elliptic equations, weak problems

    20. 2008-10-06, 15:15 - 17:00
      • General elliptic equations. Lax-Milgram theorem

    21. 2008-10-07   Exercise session
    22. 2008-10-09  
      • Maximum principles for elliptic equations

    23. 2008-10-10  
      • The heat equation, II. Presentations

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