124B Final Examination, Dr D. Labutin
Rules of the exam:
1. This is a work-alone exam without a strict time limit.
2. You may use the lecture notes, the textbook, and your homework.
3. Write your solutions in a clean (large size, please) blue book, putting the problems in order, and leaving space
between them, or turn in the TEX printout.
4. The test must be turned in according to the deadline in the syllabus.
Your signature on the front of the blue book is your a rmation that you have followed these rules. Unsigned bluebooks
will not be accepted. Exam has 6 problems. The perfect score is 30.
1. (4 pts) Find u(x;y) solution of the Dirichlet-Neumann problem
Give the explicit answer computing all integrals down to a number. Use the fact from HW: for any complex B2C
MORE PROBLEMS ARE ON THE NEXT PAGE !
4. (6pts) Use the previous problem to write down the solution integral formula for the problem (2 pts)
(
ut = uxx +cux +f(t;x) t> 0; x2R
ujt=0 = (x) :
Use your formula to compute the solution of (4 pts)
(
vt = vxx +cvx + (t 1) (x+ 1) t> 0; x2R
vjt=0 = 0
explicitly (the answer should contain no R).
5. (6pts) In the triangular domain D =f(x;y) jy> 0; x+y< 1;y x< 1g consider the Dirichlet problem
Show explaining your arguments in details that:
1. u 0 in D (2 pts);
2. u x2 +y2 in D (2 pts);
3. u x x24 +y y24 in D (2 pts).
6. (4 pts) The temperature u of the disk D of radius 2 centered at the origin does not change in time, u = u(x;y). The
heat capacity density of the disk is constant (may think = 1) so that the heat energy of the disk is
The temperature of the boundary of the disk is given by