## Partial Differential Equations Homework Solutions

A Partial Differential Equation (PDE for short), is a differential equation involving derivatives with respect to more than one variable. These arise in numerous applications from various disciplines. A prototypical example is the `heat equation', governing the evolution of temperature in a conductor.

Usually finding explicit solutions for even the simplest (LINEAR) PDE's is a formidable task, which doesn't always have a tractable solution. The mathematical study of PDE's usually focuses on deducing properties of solutions, without use of an explicit solution formula. For instance, the fact that heat doesn't collect at hot points is a consequence of the "Maximum principle"; a fundamental theorem about solutions to the heat equation, which also applies to solutions of a more general class of equations.

This course will serve as a conceptual introduction to PDE's differential equations, focussing more on studying properties of solutions and less on finding explicit (and horrendously complicated) solutions. It is aimed at undergraduate Math majors, however is suitable for students from Physics, Engineering and other disciplines who want to develop a more conceptual understanding of the subject.

##### References

*Introduction to PDE*by Walter Strauss. (**Strongly recommended!**Homework problems will be assigned from here.)*Basic Partial Differential Equations*by Bleecker and Csordas.*An Introduction to Partial Differential Equations*by Pinchover and Rubinstein.

Faculty: Jerry L. Kazdan Telephone: (215) 898-5109 email: Office Hours: Wed. 10:30-11:30 (and also by appointment) in DRL 4E15 |

TA: Howard Levinson Telephone: (412) 841-8097 email: Office Hours: ??? (and by appointment) in DRL 1N1 |

### Partial Differential Equations, Spring 2015

**Text: **Walter A. Strauss, * Partial Differential Equations: An Introduction*, 2^{nd} 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**Exams:**

*You may always use one 3"x5" card with*handwritten

*notes on both sides*

**Notes: **

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

Gibbs Phenomenon

Lorentz Transformations

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)

**Homework Assignments:**

- 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)

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