#### 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