## Partial Differential Equations Math 110, Winter 2008:

under construction

Office hours: M2-3, W4-5 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, 1st edition. The first edition is being replaced by the second edition, but it seems there are enough used copies around of the first edition. Let me know if you have difficulties getting the first edition. I also put a copy on reserve at the science library.

Teaching assistant: Patrick Driscoll, APM 6351, office hour M: 11-11:50, W 3-3:50, email: pdriscoll@math.ucsd.edu

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: 2/15 in class; for further information click at link for homework below

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:

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

for 1/11: (for first edition) Section 1.1:3, 4, 12, Section 1.2: 1, 3. The material for Problems 3,4 will be explained on Wednesday - or read about it in the book.

for 1/18: Section 1.3: 1, 3, 9, Section 1.4: 1, 3 (use your physical intuition to guess what the solution should look like for large t), Section 2.1: 2, 5

for 1/25: Section 2.2: 3, Section 2.3: 4ab (hint: you can use the strong maximum principle, stated below the maximum principle on page 41), Section 2.4: 1, 7 (please do the actual calculation, using the trick indicated in problem 6 and also in class), also do the problem below.

for 2/1: Section 2.3: 8 (try to express the derivative of the energy function in terms of values at the endpoints) Section 2.4: 8, 18 (hint: consider the function w(x,t)=u(x-Vt, t) which satisfies an easier PDE). Section 4.1: 2, 3 and see problem below:

for 2/8: Section 5.1: 2, 8, 9, 4 Section 5.2:2, 13. Look at separate webpage! Link above.

Additional homework problem, due 2/1: Calculate the integral of (2/l) sin(n\pi x/l) over the interval [0,l] (here l = the letter 'ell' and n is a nonnegative integer). Compare the values with the data given in Problem 2 of Section 4.1.

Additional homework problems, due 1/25: (mispint corrected on 1/23, 11:13pm) (a) Find the solution of the wave equation u_tt=c^2u_xx with initial conditions u(x,0)= sin nx and u_t(x,0)=0. Here n is an integer. (b) Solution of the wave equation which can be factorized as u(x,t)=X(x)T(t) for functions X and T are called standing waves (or separated solutions). Check that the solution in (a) is a standing wave. Find three more types of standing waves (enough to check wave equation for one of those types).

First homework assignment, due 1/11: 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 1.22, 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.