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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 L¹_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 L¹_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 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