# Fall 2019 Math 53: Partial Differential Equations

### Instructor: Yoonsang Lee

Office: Kemeny 206
Office hours: Monday 1-2pm, Wednesday 9:50-11 am, 1-2 pm

### 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)

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 30%,
• Midterm (Oct 20 Sunday, take-home) 30%,
• Final (Nov 24 Sunday 3 pm) 40%.
• ### 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