Math 442. Intro to Partial Differential Equations
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
5.5 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
- 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.