Math 447: Intro to Partial Differential Equations

Catalog Information

Title

Introduction to Partial Differential Equations.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

W, Su

Prerequisite

Math 303; or 314 and 334.

Description

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

Chris Grant's Proposed Core Topics

1. Basic classification of PDEs
   • As nonlinear, linear homogeneous, linear inhomogeneous
   • By order
   • Of second-order linear PDEs in 2 variables as elliptic, parabolic, or hyperbolic
2. Basic Modeling
   • Derivation of the heat equation
   • Derivation of the wave equation
   • Derivation of Dirichlet, Neumann, and mixed boundary conditions for the heat 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