Fall 2020 Math 53: Partial Differential Equations

MWF: 1:10 pm to 2:15 pm

Instructor: Yoonsang Lee

Office: Kemeny 206
Office hours: TBA

Texts & Materials

Main textbook
  • W. A. Strauss, Partial Differential Equations: An Introduction
  • Other references include
  • J. D. Logan, Applied Mathematics (Chapters 6-8; easier than the main textbook)
  • L. C. Evans, Partial Differential Equations (graduate-level standard textbook)
  • V. I. Arnold, Lectures on Partial Differential Equations (for very serious students)
  • Assessment & Grading

    The homework and tests include problems to measure quantitative (calculation-style) and qualitative (proof-style) understanding of the materials. The grading will base on the following contributions:
  • Six homework assignments 25%,
  • Midterm 25%,
  • Final 40%.
  • Participation 10%
  • Course Schedule and Topics

    Introduction

  • Day 01: Introduction to PDEs (1.1)
  • Day 02: PDE modeling (1.2, 1.3)
  • Day 03: Review of ODEs, series solutions (Chapter 5 of Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems 10th ed.)
  • Classification of PDEs

  • Day 04: First-order equations (1.2)
  • Day 05: Initial and boundary conditions, well-posed problems (1.4, 1.5)
  • Day 06: Types of second-order equations (1.6)
  • Wave Equations

  • Day 07: Vibrations of a drum (1.3, 2.1); Homework 1 due
  • Day 08: Causality and energy (2.2)
  • Day 09: Reflections of waves (3.2), waves with a source (3.4)
  • Diffusion Equations

  • Day 10: Diffusion on the whole line (2.3, 2.4); Homework 2 due
  • Day 11: Diffusion on the half-line (3.1),
  • Day 12: Diffusion with a source (3.3); Homework 3 due
  • Boundary Value Problems

  • Day 13: Separation of variables, boundary conditions (4.1)
  • Day 14: Fourier transforms - orthogonality and completeness (5.1, 5.3, 5.4)
  • Day 15: Laplace's equation, Poisson's equation (6.1, 6.2, 6.3)
  • Eigenvalue Problems

  • Day 16: Computation of eigenvalues (11.2, 11.3)
  • Day 17: Symmetric differential operators (11.4)
  • Day 18: Asymptotics of eigenvalues (11.6)
  • Distributions and Weak Formulation

  • Day 19: Green's functions (7.1, 7.2, 7.3)
  • Day 20: Distributions (12.1, 12.2); Homework 4 due
  • Day 21: Weak solutions, FEM (8.5)
  • Function Spaces

  • Day 22: Hilbert space
  • Day 23: Lax-Milgram theorem
  • Day 24: Banach space; Homework 5 due
  • Abstract Formulation

  • Day 25: Second-order elliptic equations
  • Day 26: Second-order linear evolution equations
  • Day 27: Semigroup theory
  • Review

  • Day 28: Review; Homework 6 due