This is a list of all courses offered by the Math Department.  Not all courses are offered each year.  What is provided is a general description of the courses and the prerequisites.  The actual content may vary.

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Description

Introduction to the subject of partial differential equations: first order equations (linear and nonlinear), heat equation, wave equation, and Laplace equation. Examples of nonlinear equations of each type. Qualitative properties of solutions. Method of characteristics for hyperbolic problems. Solution of initial boundary value problems using separation of variables and eigenfunction expansions. Some numerical methods.

Prerequisites

MATH 241 and MATH 246; or MATH 340 and MATH 341

Topics

First order equations

First order linear equations (with method of characteristics)
Weak solutions
Nonlinear conservation laws, derivations, shock waves
Linearized equations
Numerical methods, CFL condition.

Diffusion (heat equation) in one space variable

Derivation from Fourier's Law of cooling or Fick's law of diffusion
Maximum principle, Weierstrass kernel, qualitative properties of solutions
Traveling wave solutions to a nonlinear heat equation, Bergers' equation or reaction diffusion equations
Initial boundary value problems on the half line
Initial boundary value problems on a finite interval, method of separation of variables, linear operators and expansions of solutions in terms of orthogonal eigenfunctions
Inhomogeneous problems
Numerical methods, Crank-Nicolson scheme

The wave equation on the line

Derivation from equations of gas dynamics or from equations of the vibrating string
Characteristics, d'Alembert's formula, domains of influence and dependence
Half line problems, reflections of waves by Dirichlet and Neumann boundary conditions
Initial-boundary value problems using separation of variables
Numerical methods

Heat and wave equations in higher dimensions

Solutions of initial value problem on R2 and R3, Weierstrass kernel for heat equation and Kirchoff's formula for the qave equation
Boundary value problems in the rectangle and disk, eigenfunction expansions, Bessel functions

Laplace equation

Mean value property and maximum principle for harmonic fuctions
Series solution and Poisson kernel representation of solution of the Dirichlet problem in the disk
Harnack inequality and Liouville's theorem (used to prove the uniqueness of solutions of Poisson's equation)
Green's function for the Poisson equation in R2 and R3
Green's function for the disk, half plane, sphere
Numerical methods

Epilogue: classification of second order linear equations