COURSE DESCRIPTION

This course covers the main families of linear partial differential equations: transport, wave, elliptic, and parabolic. For each one of them we present methods to compute exact solutions based on the linear structure of the problem. We will also analyze qualitative properties of the solutions, fundamental to understanding the challenges encountered in their corresponding numerical models.

Prerequisites

Common requirements for the summer program
COURSE GOALS

On completion of this course, students will

  • Understand some of the basic techniques for analyzing the major families of partial differential equations: elliptic, parabolic, and hyperbolic.
  • Be fluent in linear methods such as the Fourier transform and the fundamental solution.
  • Have an overview of some the major applications in engineering, physics, and finance.
COURSE CONTENTS
  1. Introduction (1 week)
  1. First order linear equation and the method of characteristics.
  2. Geometric interpretation of the gradient, divergence and the Laplacian.
  1. The Fourier transform (2 weeks)
  1. One dimensional heat equation.
  2. One dimensional wave equation.
  3. Separation of variables.
  1. Wave equation (1 week)
  1. d’Alembert's formula.
  2. Non-homogeneous problem.
  3. Maxwell equations.
  1. Laplace equation (2 weeks)
  1. Fundamental solution.
  2. Mean value formula.
  3. Green's function and Poisson’s kernel.
  4. Maximum principle.
  5. Dirichlet principle.
  6. Relation with Brownian motion.
  1. Heat equation (2 week)
  1. Fundamental solution.
  2. Mean value formula.
  3. Green's function and Poisson’s kernel.
  4. Maximum principle.
  5. Dirichlet principle.
  6. Black-Scholes equation.

Bibliography

  1. Myint-U, Tyn; Debnath, Lokenath (2007). Linear Partial Differential Equations for Scientists and Engineers (eBook), Boston: Birkhäuser.
  2. Evans, Lawrence C. (2010). Partial differential equations, Second Edition, Graduate Studies in Mathematics, 19. Providence, RI: American Mathematical Society.
  3. Cooper, Jeffery M. (2000). Introduction to Partial Differential Equations with MatLab, Boston: Birkhäuser.
  4. Haberman, Richard (2013). Applied partial differential equations with Fourier series and boundary value problems, Fifth Edition, Upper Saddle River, NJ: Pearson Education, Inc.

Grading

Homework (30%), midterm (30%) just before the drop deadline, final exam (40%)