Math3(6)47; Fall, 1998 (Birtel)
 
  • Text: Introduction to Partial Differential Equations with Applications by Zachmanaglou and Thoe
  • 0. Introduction:
     
       characteristic directions, characteristic surfaces, integral
       surfaces for vector fields
     
    1. First Order Linear PDE's in two (or more)variables:

    a. Existence, Uniqueness.

    b. Canonical form

    c. Methods of Solution

    2. Second Order Linear PDE's in two or more variables:

    a. classification

    b. reduction to canonical form

    c. superposition

    3. Preliminaries to Section 4:

    a. Stokes' Theorem, Green's Identities

    b. Sturm-Liouville boundary value problem in ODE

    c. Fourier Series and Series of Orthogonal functions

    d. Gamma and Beta functions; volume of th~ n-ball

    e. Bessel's Equation, Legendre's Equation

    4. Equations of Mathematical Physics

    a. Existence, uniqueness and well-posedness for A,B,C

    b. Methods of solution for A,B,C

    c. Techniques for inhomogeneous equations and for inhomogeneous initial conditions for A,B,C

    Where

    A. Laplace's equation (including properties of harmonic functions)

    Boundary Value Problems. (Of Dirichlet and Neumann Type) for special bounded and unbounded domains; homogeneous equations, homogeneous initial conditions

    B. Wave Equations

    1) Initial Value Problems

    2) Initial-Boundary Value Problems in domains with boundary; homogeneous equations, homogeneous initial conditions

    C. Heat Equations

    1) Initial Value Problems

    2) Initial-Boundary Value Problems in domains with boundary; homogeneous equations, homogeneous initial conditions
     

    5. Qualitative Summary: Contrast properties of elliptic, hyperbolic, parabolic equations.