Partial Differential Equations I

Department: 
MATH
Course Number: 
6341
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Introduction to the mathematical theory of partial differential equations covering the basic linear models of science and exact solution techniques.

Prerequisites: 

MATH 4318

Course Text: 

No text

Topic Outline: 
  • Derivation of Basic Models
    • Constitutive relations, conservation principles
    • Wave, diffusion, and potential equations
    • Initial and boundary value problems
    • Existence and uniqueness
  • Types of Equations
    • Classification of first and second order equations
    • Laplace and Poisson equations
      • maximum principles
      • mean value properties
      • regularity
    • Heat equation
      • maximum principles
      • regularity
    • Transport and wave equations
      • characteristics
      • Huyghen's principle
    • Miscellaneous equations
      • Schrödinger's equation
      • conservation laws
      • reaction-diffusion equations
    • Methods of Exact Solution
      • Fourier series
      • Fourier transform
      • Fundamental solutions and Green's functions
      • Method of characteristics
      • Kirchoff's formula and the method of descent
      • Power series, Cauchy-Kowalevski and Holmgren theorems
    • Distributions
    • Miscellaneous Topics
      • Asymptotic expansion
      • Energy methods