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SOLVE NON HOMOGENEOUS FIRST ORDER PARTIAL DIFFERENTIAL EQUATION
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Math 553 Partial Differential Equations
Prof. Robert G.
Muncaster
Checklist for the Final Examination
First Order Equations
 integral surfaces and the method of characteristics for quasilinear 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 semilinear 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 semilinear equations
General Properties of Higher Order Equations
 connection between (projected) characteristics and being able to solve for all 2nd derivatives for semilinear 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 higherorder equations and the CauchyKowalevski 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 nonhomogeneous 1st and 2nd order equations
Theory of Weak Solutions
 be familiar with multiindex notation
 know what the adjoint operator L' is for an operator L and how it comes into the definition that an L^{1}_{loc} 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 L^{1}_{loc} 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 nonhomogeneous 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 nonsharp 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 C^{2} 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 R^{n}
 derivation of the heat kernel representation of the solution of the heat equation in R^{n}
 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