Description

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

Prerequisites

1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341, ENEE290, MATH243).

Level of Rigor

Standard

Sample Textbooks

Introduction to Partial Differential Equations, by Walter Strauss

First Course in Partial Differential Equations, by Weinberger.

Applications

Economics, business, engineering, physics, astronomy, computer science, chemistry

If you like this course, you might also consider the following courses

MATH 463

Additional Notes

Students interested in grad school in AMSC should strongly consider this course

Students interested in grad school in MATH should consider this course

Topics

First order equations (linear and nonlinear): Method of characteristics.

Second order equations: Diffusion (heat) equation, Wave equation and Laplace equation 

Homogeneous and inhomogeneous problems

Qualitative properties of solutions (Energy, mean value property and maximum principle for harmonic functions)

Initial boundary value problems on the whole line and on the half line (Dirichlet and Neumann boundary conditions).

Initial boundary value problems on a finite interval: Method of separation of variables

Fourier Series

Laplace equation in the rectangle and disk

Heat and wave equations in higher dimensions (eigenfunction expansions)

Additional topics:

Nonlinear conservation laws, derivations, shock waves

Linearized equations

Numerical methods, CFL condition, Crank-Nicolson scheme

Finite element method

Derivation from equations of gas dynamics or from equations of the vibrating string

Derivation from Fourier's Law of cooling or Fick's law of diffusion

Traveling wave solutions to a nonlinear heat equation

Series solution and Poisson kernel representation of solution of the Dirichlet problem in the disk

Harnack inequality and Liouville's theorem

Green's function for the Poisson equation in R2 and R3