## Partial Differential Equations Math 110B, Winter 2015:

under construction

Office hours: M2-3, W3-4 and by appointment (just talk to me after class or send me an email)

Office: APM 5256, tel. 534-2734

Email: hwenzl@ucsd.edu

Course book: Walter A. Strauss: Partial Differential Equations, Wiley & Sons, 2nd edition.

Teaching assistant: Donlapark Pornnopparath, email: dpornnopparath@ucsd.edu office: APM 6414, office hours: 12-2 pm on Thursday at APM 6414 or by appointments.

Computation of grade: The grade is computed from your scores in the final (50%), 1 midterm (30%) and homework (20%), with passing grades for exams required. Although homework counts comparatively little, it is extremely important that you do it, as most of the exam problems will be very similar to homework problems. It is OK to compare homework notes or to discuss problems with other students; just copying someone else's homework, however, will not count.

Dates of exams: No make-up exams!

Midterm: February 11

Homework assignments Homework is to be turned in on Fridays (unless noted otherwise) by 5pm. We have a dropbox in the basement of APM. Even though homework counts for comparatively little for the overall grade, it is extremely important that you do, or, at least, seriously try to do them. Most of the exam problems will be similar to homework or practice exam problems.

for 1/16: Section 9.1: 1, 3 Section 9.2: 3, 6(a)-(d) (you may assume that the center of the sphere S is of the form x_0=(0,0,z), if you wish. Also look at the hints at the end of the problem), 9

for 1/23 Section 9.3: 2, 4, 5 (this means you should check that formula (13) (or (11)) for the wave equation as on the bottom of p 245 with f = 2 produces the solution u(x,t)=t^2).

for 1/30 Section 9.4: 1, 3(see Section 3.1, p59-60), 6 (observe that H_k(x)exp(-x^2/2) is an eigenfunction of the differential operator L(v)=v''-x^2v. Show that (L(f),g)=(f,L(g)) for functions f,g, where (f,g)=\int f(x)g(x) dx, with the integration going over R). Also, please prove identity (6) on page 259.

for 2/6: Section 7.1: 3 (apply the maximum principle to the difference of two solutions), Section 7.2: 2 (use Green's second identity for a large ball, with a small ball around 0 cut out (similar to Fig. 1 on p 186) with u=1/|x| and v=\phi. Section 7.4: 6 (apply the same strategy as for the half space, now using Formula (5) on p 187)

for 2/20: Section 10.1: 1, 3, Section 10.2: 1, 2, 3, 4 (for problem 10.2.1 you might want to wait until the lecture on Wednesday. Or read chapter 10.2 or the notes below).

for 2/27: Section 10.5: 3, 4, 6, 13 (use identity (17) of Section 10.5), 14 (hint: write u in polar coordinates as u(r,\theta)=R(r)\Theta(\theta) and proceed similarly as in Section 10.2, page 265; you need not worry about the variable t here). Please also derive Equation (5) on page 272 from Equation (2) on page 271, using the substitution (4) on page 272

for 3/6: Section 10.3: 2, 4, 6, Section 10.6: 4, 5, 6 (use Lemma 1 of the notes for 10.3 below for \lambda = 0, and the fact that P^m_l is the Legendre polynomial P_l for \lambda = 0). Also prove identity (5) on page 290 (if you were not in class: use the fact that the Legendre polynomial P_l is an eigenfunction of the differential operator L[u]= ((1-z^2)u')' with eigenvalue -l(l+1)).

for 3/13: Section 11.1: 1 Section 11.2: 6. 7

Final The material for the final will go over what we covered from Sections 7, 10 and 11. So e.g. only 11.1 and 11.2 from Section 11, and you need not worry about Section 10.4 and 10.7 from Section 10. You are allowed to use up to two cheat sheets. However, I will not expect you to copy lots of specialized formulas such as explicit power series for Bessel functions, explicit spherical harmonics (table on page 276), identities on page 284, explicit formula for Legendre polynomials and associated Legendre functions and normalizing constants on page 290 etc. If needed, I will put this on the problem sheet.

office hours for finals week: Monday 2-3, Tuesday 4-5.

Below are solutions of homework problems from Chapter 10.3 and 10.6. While exam problems will certainly not be as long and tricky as some of those problems, it is still useful that you understand how to solve them. I have also added one more practice problem as of Saturday evening.

Midterm 2/11 You will be allowed up to two cheat sheets, but no book, calculators or other notes. The material will go over what we have covered from Sections 7 and 9. Below are some solutions of homework problems, and a practice exam (so far only with problems from Section 7). I plan to put up more practice problems later, so please get started soon.