Chapter 1: Where PDEs Come From
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
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations
Chapter 2: Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions
Chapter 3: Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited
Chapter 4: Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition
Chapter 5: Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and the General Fourier Series
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions
Chapter 6: Harmonic Functions
6.1 Laplace’s Equation
6.2 Rectangles and Cubes
6.3 Poisson’s Formula
6.4 Circles, Wedges, and Annuli
Chapter 7: Green’s Identities and Green’s Functions
7.1 Green’s First Identity
7.2 Green’s Second Identity
7.3 Green’s Functions
7.4 Half-Space and Sphere
Chapter 8: Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace’s Equation
8.5 Finite Element Method
Chapter 9: Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schrodinger Equations
9.5 The Hydrogen Atom
Chapter 10: Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics
Chapter 11: General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues
Chapter 12: Distributions and Transforms
12.2 Green’s Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques
Chapter 13: PDE Problems for Physics
13.2 Fluids and Acoustics
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles
Chapter 14: Nonlinear PDEs
14.1 Shock Waves
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves
A.1 Continuous and Differentiable Functions
A.2 Infinite Sets of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function
Answers and Hints to Selected Exercises
- Eigenvalue problems (Chapters 10 and 11) are covered in appropriate depth.
- Treatment of distributions and Green’s functions eliminates student confusion by giving instructors the option of going directly from Green's functions in Chapter 7 to distributions and Fourier transforms in Chapter 12.
- Numerous exercises, varying in difficulty.
- Frequent mention of wave propagation, heat and diffusion, electrostatics, and quantum mechanics puts PDE into context, which is especially important for engineering and science majors.
- Rational organization of material: from science to mathematics, from one dimension to multidimensions, from full-line to half-line to finite interval, etc.
- Companion solutions manual allows students to see detailed worked out solutions.
- Introduction to nonlinear PDEs (Chapter 14) provides the student with a taste of the most important problems studied today by researchers in mathematics and science.
- Provides appropriate introduction to numerical analysis (Chapter 8).
- Organization gives instructors flexibility in chapter coverage; for example, one can go from Chapter 6 to Chapters 7 and 12 (Green’s functions and distributions), Chapters 13 and 14 (science and nonlinear PDEs), Chapter 8 (numerical), Chapter 9 (waves), or to Chapter 10 (disks, spheres).
|Faculty: Jerry L. Kazdan |
Telephone: (215) 898-5109
Office Hours: Wed. 10:30-11:30 (and also by appointment) in DRL 4E15
|TA: Howard Levinson |
Telephone: (412) 841-8097
Office Hours: ??? (and by appointment) in DRL 1N1
Partial Differential Equations, Spring 2015
Text: Walter A. Strauss, Partial Differential Equations: An Introduction, 2nd Edition, John Wiley (2007), ISBN-13: 9780470054567
As usual, since prices vary considerably, it is wise to search online for less expensive textbook sources.
Note that the first edition had many typos. For a list for both the first and second editions, see the author's web page
Content: The heart of this course is to achieve some real understanding of the wave, heat, and Laplace equations. The emphesis will be on mathematical and physical insight and ideas, not complicated formulas.
I taught this in Spring 2011. Although the course will be somewhat different, much of the material will be identical. You might find the homework, exams, and notes from that course useful: Math 425 Spring 2011
Prerequisites & Review Material
Course and Homework Grading
Some References: books, articles, web pages
LaTeX: If you will be writing many documents that contain equations, it is wise to learn (and use) LaTeX. It is available on Windows, Macs, and Linux -- and is free. See TeX Stuff. For some students, this might be the most useful item you learn in this course.
Some Classical PDEs
Striking a Match: Turbulence
Tacoma Narrows Bridge
Some notes (1965!) from a course like our Math 240 (there are typos.)
ODE's: Generalities on Linear ODE's DeTurck Notes
DeTurck's Math 425 for 2010
DeTurck notes on first order PDE's
Derivation of the heat equation
PDE: Change Variables: print version (display version)
Orthogonal Vectors and Fourier Series
Uniqueness for the initial value problem for the heat equation. The proof in Petrovsky assumes that the solution u(x,t) is bounded while the proof in John allows for the solution to grow at infinity as long as for any T there is a constant c so that |u(x,t)|< e(c|x|2) for all 0 ≤ t ≤ T. Note that if u(x,t) is allowed to grow too quickly at infinity, there are examples where uniqueness fails.
Standing Waves Music
Sines and Cosines using ODE
Spherical Harmonics Spherical Harmonics (Wikipedia)
Spherical Harmonics: Strauss
Completeness of Eigenfunctions of the Laplacian. A more "geometric" version of the proof in Strauss, Sec 11.3.
Notes on Convolution
Hear the Shape of a Drum Google search on: Gordon "Shape of a drum"
JLK Australia 2008 Notes (for a slightly more advanced course)
- Set 0: Rust Remover (LaTeX source ). Due: Never. This will not be collected.
- Set 1 (LaTeX source ). Due: Thurs., Jan. 22 in class
- Set 2 (LaTeX source ). Due: Thurs., Jan. 29 in class [solutions]
- Set 3 (LaTeX source ). Due: Thurs., Feb. 5 in class [solutions]
- Set 4 (LaTeX source ). Due: Thurs., Feb. 12 in class [solutions]
- Set 5 (LaTeX source ). Due: Thurs., Feb. 19 in class [solutions]
- Set 6 (LaTeX source ). Due: Thurs. Feb 26 in class [solutions]
- Set 7 (LaTeX source ). Due: Thurs., Mar. 5 in class [solutions]
- Set 8 (LaTeX source ). Due: Thurs., Mar. 26 in class [solutions]
- Set 9 (LaTeX source ). Due: Thurs., April 2 in class [solutions]
- Set 10 (LaTeX source ). Due: Thurs., April 9 in class [solutions]
- Set 11 (LaTeX source ). Due: Thurs., April 16 in class [solutions]
- Set 12 (LaTeX source ). Due: Thurs., April 23 in class [solutions]
Old Exams: (you may always use one 3"x5" card with notes on both sides)
Spring 2011 Exam 1, condensed, (solutions).
Spring 2011 Exam 2, condensed (solutions)