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18.152

Introduction to Partial Differential Equations

3-0-9 units · Undergraduate

Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.

Fall 2011

Topics

  • Introduction to PDEs
  • Introduction to the heat equation
  • The heat equation: Uniqueness
  • The heat equation: Weak maximum principle and introduction to the fundamental solution
  • The heat equation: Fundamental solution and the global Cauchy problem
  • Laplace’s and Poisson’s equations
  • Poisson’s equation: Fundamental solution
  • Poisson’s equation: Green functions
  • Poisson’s equation: Poisson’s formula, Harnack’s inequality, and Liouville’s theorem
  • Introduction to the wave equation
  • The wave equation: The method of spherical means
  • The wave equation: Kirchhoff’s formula and Minkowskian geometry
  • The wave equation: Geometric energy estimates
  • The wave equation: Geometric energy estimates (cont.)
  • Classification of second order equations
  • Introduction to the Fourier transform
  • Introduction to the Fourier transform (cont.)
  • Fourier inversion and Plancherel’s theorem
  • Introduction to Schrödinger’s equation
  • Introduction to Schrödinger’s equation (cont.)
  • Introduction to Lagrangian field theories
  • Introduction to Lagrangian field theories (cont.)
  • Transport equations and Burger’s equation

Readings

Fall 2005

Topics

  • Harmonic Functions and the Harnack Inequality
  • The Gradient Estimate
  • The Hopf Maximum Principle
  • The Poincare Inequalities
  • The Cacciopolli Inequality
  • More General Operators
  • Consequences of Cacciopolli
  • Maximum Principles and Gradient Estimates
  • Hopf and Harnack for L-harmonic Functions
  • An Improved Gradient Estimate for Harmonic Functions
  • More on Harmonic Functions on a Ball
  • Solving the Laplace Equation in R 2 : The Dirichlet Problem
  • The Heat Equation
  • A Gradient Estimate for the Heat Equation on a Ball
  • Campanato’s Lemma and Morrey’s Lemma
  • Five Inequalities for Harmonic Functions
  • Regularity of L-harmonic Functions Part I
  • Regularity of L-harmonic Functions Part II
  • Regularity of L-harmonic Functions Part III
  • Smoothness of L-harmonic Functions
  • The Mean Value Inequality Revisited Part I
  • The Mean Value Inequality Revisited Part II
  • Moser’s Approach

Prerequisites & course information

Subject meets with 18.1521 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Sources