Course Syllabus

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Partial differential equations (PDEs) play critical roles in wide areas of mathematics, science and engineering. This is an introductory course for PDEs, which is accessible to undergraduate and graduate students in mathematics and other scientific disciplines (for example, engineering, physics, and finance) who have completed the prerequisites. The course covers linear partial differential equations, including Poisson, wave and diffusion problems. The focus will be on learning analytic tools to solve these problems. This course does not cover numerical methods for PDEs.

This course is offered in the fall term and is cross-listed as Math 126 Partial differential equations.

Instructor: Yoonsang Lee
Office: 206 Kemeny
Office hours: Tuesday 3:30-4:30pm, Thursday 9-10am, Friday 10-11am.

Prerequisites
Math 22/24 and Math 23 (or equivalents)

Class hours
MWF 8:50 - 9:55 am

Main textbook

  • W. A. Strauss, Partial Differential Equations: An Introduction

Other references

  • R. Choksi, Partial Differential Equations A First Course (similar level)
  • 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) and qualitative (proof) understanding of the materials. The grading will base on the following contributions:

  • Six homework assignments 30%,
  • Midterm 30%,
  • Final 30% (Nov 18, 2022 Friday, 8:00-11:00 am)
  • 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)
  • 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)
  • Day 11: Diffusion on the half-line (3.1),
  • Day 12: Diffusion with a source (3.3)

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)
  • Day 21: Weak solutions, FEM (8.5)

Function Spaces

  • Day 22: Hilbert space
  • Day 23: Lax-Milgram theorem
  • Day 24: Banach space

Abstract Formulation

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

Student Accessibility and Accommodations

Students with disabilities who may need disability-related academic adjustments and services for this course are encouraged to see me privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office in Carson Hall 125 (phone: 646-9900 or Student.Accessibility.Services@Dartmouth.edu). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to me. As a first step, if you have questions about whether you qualify to receive academic adjustments and services, you should contact the SAS office. All inquiries and discussions will remain confidential.

Religious Observances

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me before the end of the second week of the term to discuss appropriate accommodations.

Course Summary:

Date Details Due