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.

Desired Learning Outcomes

Prerequisites

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

Minimal learning outcomes

The focus of the course is on bringing students to the point that they can solve the canonical linear second-order partial differential equations on simple domains. Students should also gain some understanding of the theory about the reliability of these solution methods. With the current prerequisites, however, students can't be counted on to have had any exposure to the theory of the convergence of sequences of functions, so theoretical understanding probably needs to be a secondary goal.

  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

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.