## Math 442. Intro to Partial Differential Equations Instructor Syllabus

Text: Walter A. Stauss, Partial Differential Equations: An Introduction, John Wiley & Sons, 1992.

This course introduces students to partial differential equations, emphasizing the wave, diffusion and potential (Laplace) equations. The focus is on understanding the physical meaning and mathematical properties of solutions of partial differential equations. Methods include fundamental solutions and transform methods for problems on the line, and separation of variables using orthogonal series for problems in regions with boundary. Convergence of Fourier series is covered in detail.

Chapter 1 - Where PDEs Come From (5 days)

1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions

Chapter 2 - Waves and Diffusions (8 days)

2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation (Go lightly on stability)
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 3 - Reflections and Sources (3 days)

3.3 Diffusion with a Source
3.4 Waves with a Source (Just do p.69 and give a straightforward proof)

Chapter 4 - Boundary Problems (3 days)

4.1 Separation of Variables, the Dirichlet Condition
4.2 The Neumann Condition

Chapter 5 - Fourier Series (9 days)

5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions

Chapter 6 - Harmonic Functions (5 days)

6.1 Laplace's Equation
6.2 Rectangles and Cubes
6.3 Poisson's Formula

Chapter 10 - Boundaries in the Plane and in Space (3 days)

10.3 Solid Vibrations in a Ball
10.6 Legendre Functions

From the Instructor's favorite source - Transform Methods (3 days)

Properties of Fourier Transforms
Applications to Waves/Diffusion on the Line
Properties of Laplace Transforms
Applications to Waves/Diffusion on the Half-Line

Exams and Leeway (4 days)

TOTAL: 43 days

Notes:

• Well-posedness issues (stability) may be treated lightly.
• The instructor may wish to provide students with a table of Fourier series of common functions.
• Chapter 9 - Waves in Space: depending on student interests, the instructor might want to cover some of the material in Sections 9.4 (The Diffusion and Schrodinger Equations) and 9.5 (The Hydrogen Atom), perhaps on the homework.