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

**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 operatorLand how it comes into the definition that anL^{1}_{loc}functionuis a weak solution of a PDE- be familiar with some practical examples of verifying that a given function
uis 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 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
nspatial dimensions into an equation in 1 spatial dimension (in the variabler)- domain of dependence of a point (
x,t), range of influence of a pointx, 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 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