## Partial Differential Equations Math 110, Fall 2014:

under construction

Office hours: M3-4, W2-3 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. I also put a copy on reserve at the science library.

Syllabus We plan to go over most of the material of the first six sections of the book. You may also want to have a look at old homework assignments. I am not planning to give the same assignments, but the new ones will likely be similar.

Teaching assistant: Pun Wai Tong, APM 6432, email: p1tong at ucsd.edu, office hours: Th 5-6, F 2-3

Computation of grade: (tentative) The grade is computed from your scores in the final (50%), 1 midterm (30%) and homework (20%). 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: November 14

Homework assignments Homework is to be turned in on Fridays. There will be a drop box in the basement of the APM building.

for 10/10: Section 1.1:3, 4, 12, Section 1.2: 1, 3, 6

for 10/17: Section 1.3: 1 (hint: use Newton's law, where you now also have to take into account the friction force = - \int gu_t dx, where g is a constant, and the integration goes over the whole string), 4, 9, Section 1.4: 1, 2, Section 2.1: 2, 5,

for 10/24: Section 2.2: 3, 5 Section 2.3: 3, 4, 8

Hints: For Section 2.3, 3(c) think of the temperature of a rod whose ends are kept at temperature 0 and use parts (a) and (b); for Section 2.3 4, you can use the strong maximum principle, stated below the maximum principle on page 42. Remark: For Section 2.3, 4(c) and 8 one needs the energy method' which is described in the "uniqueness" part of chapter 2.3, page 44 (this was done in class on Friday)

for 10/31: Section 2.4: 1 (see also the problem below), 9, 10, 12, 18, Section 2.5: 1 (find a solution of the wave equation which does not have its maximum at t=0) + problem below

Problem Let u be the solution of Problem 1 of Section 2.4. Show that u(x,t)>0 for all x as soon as t>0 (consider Eq (8) of Section 2.4).

Remark: The last problem shows that the fact u(x,0)>0 for |x| less than 1 immediately spreads to all x, arbitrarily far from [-1,1]. Compare this with the causality principle for solutions of the wave equation which implies that information only travels with finite speed c.

for 11/7: Section 3.1: 1, 4, 5(a) (only state by which function you have to replace the function f in Problem 4), Section 3.2: 2, 5 (u has a singularity at (x,t) if it is not differentiable at that point) Section 3.3: 2 (consider the function V(x,t)=v(x,t)-h(t) as described in the part SOURCE ON A HALF-Line in chapter 3.3). Section 3.4: 2

Problems relevant for midterm: Section 4.1: 2 Section 4.2: 2, Section 4.3: 2, 4 (do what we did in class now for \lambda \ <0 and/or read Section 4.3: NEGATIVE EIGENVALUE)

for 11/21: Section 5.1: 2, 8, 4, 5, 6a, Section 5.2: 4 (you do not have to prove (5)), 11, 12

for 11/26 (Wednesday, 5pm!): Section 5.3: 2 (two functions on the interval [a,b] are called orthogonal if the integral \int f(x)g(x) dx from a to b is equal to 0), 5(b)-(d), 12, 13, Section 5.4: 1(use the formula for the geometric sum), 5, 12, 16(hint: use Theorem 5 of Section 5.4; also observe that both the sine and the cosine series have been calculated for the function x in the book. You can get from this the full Fourier series for |x|).

for 12/5: Section 6.1: 2, 6, 9, Section 6.2: 2 (you have to show that \int\int sin(n\pi x)sin(m\pi y)sin(n'\pi x)sin(m'\pi y) dy dx is equal to 0 if n not equal n' or m not equal m'), 4 (use strategy outlined in class; for calculating sine and cosine series, you may use results of Section 5.1)

for 12/12: Section 6.3: 1, 3, Section 6.4: 5, 10

Midterm The midterm will take place on Friday, November 14, in class. The material will go until Chapter 4.3. You will be allowed to use a cheat sheet, regular paper size, both sides OK, but no calculators or other books. I have posted an old midterm and more relevant problems below. The problems in the midterm will be similar to homework problems or practice exam problems.

Problems relevant for midterm: Section 4.1: 2 Section 4.2: 2, Section 4.3: 2, 4 (do what we did in class now for \lambda \ <0 and/or read Section 4.3: NEGATIVE EIGENVALUE)

Office hours for midterm week: TA: W5-6, Th 5-7, Prof: Th 11-12, 1-2 (Th afternoon office hour had to be changed due to some committee rescheduling)

Some solutions Below we post solutions for some of the homework problems. Sometimes, you may not find the solution of the homework problem itself, but of a related problem (perhaps a problem before or after the assigned one), which may still be useful to look at. I also post solutions for the practice midterm. Try first, for a while, even if you do not know right away how to tackle the problem. Otherwise, these exercises and solutions do not really help you!

More homework solutions. These are from an earlier course. So instead of 4.3 Problem 4 you find the solution of the more difficult problem 4.3 Problem 8, and also one or two other problems are missing. We may or may not be able to type solutions for the missing problems in time. Please ask if you would like to see the solutions.

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Homework assignments Homework is to be turned in in class on the given date at the latest. The TA also made a drop box. In order to make it easier for you to print out the assigments, I put the newer assignments on a new page. Please click below for new assignments:

for 1/8: Section 1.1:3, 4, 12, Section 1.2: 1, 3. If you do not have the book, can also find the problems at the bottom of this web page.

other homework assignments can be found under the link above

First homework assignment, due 1/8: As the book is not available yet in reserve, here are the homework problems. Observe that u_x means u sub x' and x^2 means `x squared'. If in doubt, ask the TA or me.

Section 1.1, Prob. 3: For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous (i.e. of the form L(u)=g, g not equal to 0, with L linear), or linear inhomogeneous; provide reasons.

(a) u_t - u_xx +1 = 0

(b) u_t - u_xx +xu = 0

(c) u_t - u_xx + uu_x = 0

(d) u_tt - u_xxt + uu_x = 0

(e) iu_t - u_xx + u/x = 0

(f) u_x(1+(u_x)^2)^{-1/2} + u_y(1 + (u_y)^2)^{-1/2} = 0

(a^{-1/2} means 1 over the square root of a)

(g) u_x + e^yu_y = 0

(h) u_t + u_xxxx + square root{ 1 + u} = 0

Section 1.1, prob. 4: Show that the difference of two solutions of an inhomogeneous linear equation Lu=g with the same g is a solution of the homogeneous equation Lu=0.

Section 1.1, prob. 12. Verify by direct substitution that u(x,y)= sin(nx)sinh(ny) is a solution of u_xx + u_yy = 0 for every n>0. <\p>

Section 1.2, prob. 1: Solve the first-order equation 2u_t + 3u_x = 0 with the auxiliary condition u(x,0)= sin(x); here u depends on variables x and t.

Section 1.2, prob. 3: Solve the linear equation (1+x^2)u_x+u_y=0 (here x^2 = x squared, u_x = u sub x). Sketch some of the characteristic curves.