Math 447: Intro to Partial Differential Equations

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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

Desired Learning Outcomes

Prerequisites

Minimal learning outcomes

Additional topics

Courses for which this course is prerequisite