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Math 553 Partial Differential Equations
Prof. Robert G. Muncaster

Checklist for the Final Examination

The final exam will be 3 hrs in class, comprehensive, and closed book.

First Order Equations

  • integral surfaces and the method of characteristics for quasi-linear equations
    • practical use of the method
    • the Cauchy problem for a first order equation and existence and uniqueness results that go along with it (via the method of characteristics)
  • projected characteristics for semi-linear equations
  • weak solutions for conservation laws
    • the general definition of weak solutions for conservation laws and "weak solution + smoothness => classical solution"
    • the notion of shock lines or shock solutions and how to find them
    • fan or rarefaction solutions and how they fit in with the method of characteristics
  • connection between (projected) characteristics and being able to solve for all 1st derivatives for semi-linear equations

General Properties of Higher Order Equations

  • connection between (projected) characteristics and being able to solve for all 2nd derivatives for semi-linear equations
  • classification of 2nd order equations by characteristics
  • switching to canonical variables, and the canonical forms of hyperbolic, elliptic, and parabolic equations
  • the Cauchy problem for a higher-order equations and the Cauchy-Kowalevski Theorem
  • the concept of a well posed problem (in the sense of Hadamard)
  • transforming a higher order equation into a first order system
  • characteristics for a 1st order linear system and how they are related to the eigenvalue problem for the matrices in the equation
  • diagonalizing a system
  • Duhamel's principle and how is it used to solve non-homogeneous 1st and 2nd order equations

Theory of Weak Solutions

  • be familiar with multi-index notation
  • know what the adjoint operator L' is for an operator L and how it comes into the definition that an L1loc function u is a weak solution of a PDE
  • be familiar with some practical examples of verifying that a given function u is a weak solution of a given equation
  • know the theorem that says that a smooth function that is a weak solution is also a classical solution
  • know what is meant by "transmission conditions" for weak solutions of a PDE

Theory of Distributions

  • definition of a test function and a distribution plus related concepts such as compact support
  • basic examples of distributions
  • definitions of properties of distributions (mult'n by a function, derivatives, convolutions) and how they are motivated using distributions that come from L1loc functions
  • deriving properties of distributions (i.e. product rule for derivatives, etc.) from the definitions and corresponding properties for test functions
  • methods for finding fundamental solutions of ODEs
  • finding solutions of non-homogeneous equations using fundamental solutions
  • the connection between distributional solutions and weak solutions
  • finding distributional solutions, or verifying that a distributions satisfies a PDE

The Wave Equation

  • D'Alembert's solution and its derivation
  • the parallelogram rule and how it arises and how it is used in solving boundary value problems
  • spherical means and their use in transforming the wave equation in n spatial dimensions into an equation in 1 spatial dimension (in the variable r)
  • domain of dependence of a point (x,t), range of influence of a point x, and how these relate to the concepts of sharp and non-sharp signals
  • conservation of energy for a wave problem, and how this is related to proofs of uniqueness for wave problems

Laplace's Equation

  • compatibility condition for Neumann data
  • uniqueness results for smooth solutions of the Dirichlet and Neumann problems (via Green's identities)
  • the mean value property
  • the Maximum Principle and its use in establishing uniqueness and continuous dependence
  • the fundamental integral identity for C2 functions and how it can be used to solve Dirichlet and Neumann problems for a region in terms of a Green's function.
  • what is the fundamental solution for the Laplacian with Dirichlet data
  • the method of reflection points for finding a Green's function
  • eigenvalues and eigenfunctions for the Laplacian (with Dirichlet data) and how they can be used to solve initial boundary value problems on bounded domains

Heat Equation and Fourier Transforms

  • definition and properties of the Fourier transform and definition of the Schwartz space of functions
  • theorems about convolutions and the inversion theorem, as they relate to the Schwartz space
  • use of Fourier transforms in solving problems with spatial variables in Rn
  • derivation of the heat kernel representation of the solution of the heat equation in Rn
  • maximum principle for the heat equation and its use in establishing uniqueness for problems with data on a parabolic boundary
  • statement of the theorem on smoothness for solutions of the heat equation