# Difference between revisions of "Math 447: Intro to Partial Differential Equations"

### Title

Introduction to Partial Differential Equations.

(3:3:0)

W, Su

### Prerequisite

Math 303; or 314 and 334.

### 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.

1. Basic classification of PDEs
• Linearity
• Homogeneity
• Order
• Elliptic, parabolic, or hyperbolic
2. Basic Modeling
• Derivation of the heat equation
• Derivation of the wave equation
3. 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
4. Special eigensystems
• Fourier series
• Computation of the Fourier series of a p-periodic function on an interval of length p
• Computation of Fourier sine and cosine series of a symmetric p-periodic function on an interval of length p
• Fourier series of modifications and combinations of functions
• Theorems on pointwise, uniform, and L2 convergence
• Bessel's Inequality and Parseval's Equation
• Fourier integral representations of functions on lines and half-lines
• Bessel's equation and Bessel functions
• Legendre's differential equation and Legendre polynomials
5. Representation of solutions to the canonical equations on simple domains
• Laplace's equation
• Eigenfunction expansion of solutions on rectangles
• Integral representation of solutions on rectangular strips, quarter-planes, and half-planes
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion on balls
• Wave equation
• Fourier expansion of solutions on bounded intervals
• D'Alembert's formula for solutions on lines and half-lines
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion on balls
• Heat equation
• Steady-state solutions for IBVPs on bounded intervals
• Fourier expansion of solutions on bounded intervals
• Integral representation of solutions on lines and half-lines
• Eigenfunction expansion of solutions on rectangles
• Bessel function expansion of solutions on disks
• Legendre polynomial expansion of solutions on balls