# Math 447: Intro to Partial Differential Equations

## Contents

## Catalog Information

### Title

Introduction to Partial Differential Equations.

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

W, Su

### Prerequisite

### Description

Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.

## Desired Learning Outcomes

The main purpose of this course is to teach students how to solve the canonical linear second-order partial differential equations on simple domains. Secondarily, students should be introduced to the theory concerning the validity of such solutions.

### Prerequisites

Current prerequisites ensure that students have had instruction in multivariable calculus and ordinary differential equations.

### Minimal learning outcomes

Primarily, students should be able to use the solution techniques described below. Students should gain a basic understanding of issues concerning solvability and convergence, but the current prerequisites don't guarantee that incoming students will have had any prior exposure to the theory of the convergence of sequences of functions, so expectations in that area are modest.

- Basic classification of PDEs
- Linearity
- Homogeneity
- Order
- Elliptic, parabolic, or hyperbolic

- Basic Modeling
- Derivation of the heat equation
- Derivation of the wave equation

- Basic principles, techniques, and theory
- Principle of superposition
- Method of separation of variables
- Definition of eigenvalues and eigenfunctions corresponding to two-point BVPs
- Basic Sturm-Liouville theory

- Special eigensystems
- Fourier
- Series representations
- Effect of symmetry and modifications and combinations of functions
- Theorems on pointwise, uniform, and
*L*^{2}convergence- Bessel's Inequality and Parseval's Equation

- Integral representations

- Series representations
- Bessel
- Legendre

- Fourier
- Representation of solutions to the canonical equations on simple domains
- Laplace's equation on rectangles, rectangular strips, quarter-planes, half-planes, disks, and balls
- Wave equation on bounded intervals, half-lines, lines, disks, and balls
- Heat equation on bounded intervals, half-lines, lines, rectangles, disks, and balls

### Additional topics

### Courses for which this course is prerequisite

Students taking Math 511 are supposed to have had either Math 447 or Math 303. It is proposed that Math 447 become a prerequisite (or at least recommended) for Math 547, so that there will be less duplication of material in the PDE curriculum.