Hunters in the snow
Hunters in the snow by Pieter Bruegel



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Matematisa institutionen, UU

Schemasökning, UU




Calendar

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Textbooks:

  • Robert C. McOwen, Partial Differential Equations, Methods and Applications, Prentice Hall/Pearson Education, Inc., 2003



    Dates/Topics (preliminary schedule)

      Week 6

    1. Tue, 2009-02-03, 10:15-12:00
      • Introduction, course information
      • 1st order quasilinear PDE, classification of 1st order equations
      • Method of characteristics
      • Cauchy problem for a first order quasilinear equation

    2. Thu, 2009-02-05, 10:15-12:00
      • General solution of a quasilinear equation, Lagrange method
      • The inviscid Burger equation as a model for breaking surface waves
      • Method characteristics for Burger's equation

    3. Fri, 2009-02-06, 10:15-12:00
      • 1st order PDE: general nonlinear equations
      • The method of characteristic strips
      • The method of envelopes
      • * Hamilton-Jacobi equation

      Week 7
    4. Tue, 2009-02-10, 13:15-15:00
      • Exercise session I

    5. Thu, 2009-02-12, 10:15-12:00
      • Higher order PDE: general principles
      • Cauchy problem, normal form of a PDE
      • Analytic solutions and Cauchy-Kowalevski theorem

    6. Fri, 2009-02-13, 10:15-12:00
      • Linear equations of the second order, principal symbol
      • Three types of linear equations of the second order
      • Examples of reducing to the canonical form

      Week 8
    7. Tue, 2009-02-17, 15:15-17:00
      • The wave equation
      • One-dimensional case: d'Alembert formula and propagation of one-dimensional waves
      • Weak solutions
      • Eigenfunctions approach

    8. Thu, 2009-02-19, 10:15-12:00
      • Nonhomogeneous case. Duhamel's principle
      • Higher-dimensional case. Co-area formula
      • Kirchhoff's formula in 3D
      • 2D solutions. Hygens' principle

    9. Fri, 2009-02-20, 10:15-12:00
      • Exercise session II

      Week 9
    10. Tue, 2009-02-24, 10:15-12:00
      • The Laplace and Poisson equation: derivation and physical interpretations
      • Separation of variables
      • The Dirichlet and Neumann problems for the Laplace equation
      • Green's formulas and uniqueness of solution of the Dirichlet problem

    11. Thu, 2009-02-26, 15:15-17:00
      • The fundamental solution to the Laplacian
      • Harmonic functions: Mean Value Property and Maximum Principle
      • Green's function and basic facts in potential theory

    12. Thu, 2009-02-27, 10:15-12:00
      • Poisson kernel
      • Method of reflection, the Dirichlet problem on a half-space and ball
      • Subharminic functions

      Week 10
    13. Tue, 2009-03-03, 10:15-12:00
      • The heat equation. Derivation and physical interpretations.
      • Eigenfunction expansion
      • Maximum principle and uniqueness

    14. Fri, 2009-03-05, 10:15-12:00
      • Fourier transform
      • The pure initial value problem
      • The fundamental solution, the heat kernel

      Week 11
    15. Tue, 2009-03-10, 10:15-12:00
      • Exercise session III

    •