# Qualifying Exams: Partial Differential Equations

## First Order Equations

The linear equation

## Second Order Equations

Classification of equations; Canonical forms of the hyperbolic, parabolic and elliptic equations; Canonical forms and equations of mathematical physics: wave equation, heat equation and Laplace's equations; The second order Cauchy problems

## Elements of Fourier Analysis

The Fourier series of a function; Convergence of Fourier series; Sine and Cosine expansions; Bessel's inequality and uniform convergence; Fourier Sine and Cosine transforms

## The Wave Equation

The Cauchy problem and d'Alembert's solution; d'Alembert's solution as a sum of forward and backward waves; Domain of dependence and the characteristic triangle; A nonhomogeneous wave equation; The wave equation on a bounded interval; Fourier series solution on a bounded interval; A nonhomogeneous problem on a bounded interval

## The Heat Equation

Cauchy problem for the heat equation; Initial and boundary conditions; Solutions on boundary conditions; Solutions on unbounded domains; The nonhomogeneous heat equation

## Dirichlet and Neumann Problems

Cauchy problem for Laplace's equation; Dirichlet problem for a rectangle; Dirichlet problem for a disk; Other coordinate systems; Circular membrane; Cylindrical coordinates; Spherical coordinates

## References

PV. O'Neil, Beginning Partial Differential Equations, Wiley, New York, 1999.