• Solution to LCE 3.5, problem 7.
• Evans, Sections 4.1-4.4
• The section of the undergraduate on-line text on the heat equation in one dimension

Exercises: Due Friday, 8 December:

• Homework will not be specifically assigned. On Friday, one of the final questions will be announced. The other one will be given out next week, and both are to be solved in class at that time. (Possibly a small modification will be inserted into the problem you are given on Friday, when it reappears on the final exam.) During the final, you will have access to the book by LC Evans, but only to that source.

  For the final you will be expected to answer questions at the level of any of the homework we have already done, any of the problems from LCE Section 4.7 pertaining to Sections 4.1 or 4.3, or the exercises in the section of the undergraduate on-line text on the heat equation in one dimension.

Past homework assignments

• Evans, Sections 1, 2.1, 2.2
• Evans, Appendix on notation
• H&H (Harrell and Herod), derivations of some important PDEs

• Section 1.5, #1 (and optionally 2-4 to help you want to understand multiindices)
• (Refer to the derivation of the wave equation for this problem.) Derive a wave equation for the vibrating string if the x-axis is horizontal and we take gravity into account. The gravitational potential energy of the string is where g, the gravitational constant, is approximately 980 in the cgs system. Assume that the mass density is equal to 1.

• Evans, Sections 2.2, 2.3, Appendix C.4
• H&H Appendix on the mean-value theorem, etc.

• Evans, Section 2.5, #1-4
• H&H, appendix, #1-2

• Evans, Sections 2.2, 2.3
• If you wish to see some related material in the undergraduate text, the closest match is Chapter XIX, but you should be aware that the notation is different, and even the sign convention for Poisson's equation is different. (If I find time, I will go through and alter the sign convention later.)

• Evans, Section 2.5, #3,5,10

• The solution to Problem 3 assigned last week.
• Evans, Sections 2.2, 2.3
• If you wish to see some related material in the undergraduate text, the closest match is Chapter XIX, but you should be aware that the notation is different, and even the sign convention for Poisson's equation is different. (If I find time, I will go through and alter the sign convention later.)

• Evans, Section 2.5, #6
• H&H, Chapter XIX, #6

• Evans, Section 2.3
• An alternative derivation of the maximum principle
• Schwartz space, differentiating under the integral sign, and all that

Exercises: Due Friday, 29 September:

• Evans, Section 2.5, #12-14

Reading: Due Friday, 6 October

• Evans, Sections 2.3,2.4
• Solution to Problem 2 of the recent test
• about Clerk Maxwell's equations and the wave equation

Exercises: Due Friday, 6 October:

• Evans, Section 2.5, #14, 15, 16. Note: Problem 15 will be collected and graded.
• Using the class derivation of the maximum principle for the heat equation as a model, derive the analogous result for the Laplace equation. Discuss the relationship between this and the "strong maximum principle" which was a consequence of the mean value theorem. If they are not fully equivalent, explain.

• Evans, Sections 2.4
• The chapter on d'Alembert's solutio n in the on-line text.

Exercises: Due Friday, 13 October:

• Evans, Section 2.5, #17
• H&H, Chapter VII, #VII.1 and VII.2.

Reading: Due Friday, 20 October

• Evans, Sections 2.4, 3.1

Exercises: Due Friday, 20 October:

• Evans, Section 2.5, #18

• Evans, Sections 3.1-3.2
• Some class notes taken last week.

Exercises: Due Friday, 27 October:

• Evans, Section 3.5, #1
• Solve the following quasilinear equations (From Zachmanoglou and Thoe):
  1. (the others were deferred a week)

• Evans, Sections 3.2-3.3
• Some class notes from last week and this week.
• Zachmanoglou and Thoe, Chapter III (optional)

Exercises: Due Friday, 3 November:

• Evans, Section 3.5, #1-3
• THE FOLLOWING WILL BE COLLECTED AND GRADED:
  Solve the following quasilinear equations (From Zachmanoglou and Thoe):
  1. x² u_x + y² u_y = (x+y) u
  2. x(y-u) u_x + y (u-x) u_y = (x-y) u
  3. (will not be graded) Solve problem b above with the initial conditions u(t, 2t/(t^2-1) = t.

• Solutions to the recent homework problems
• Evans, Sections 3.2-3.3 (beginning only)
• Some class notes from this week.
• Zachmanoglou and Thoe, Chapter III (optional)

Exercises: Due Friday, 10 November:

• Evans, Section 3.5, #1-4

• Evans, Sections 3.3-3.4
• Review the derivation of the wave equation as an example of variational calculus.

Exercises: Due Friday, 17 November:

• Evans, Section 3.5, #5-8. Problem 5 will be collected and graded.

• Solutions to the recent test
• Some current class notes.
• Evans, Sections 3.3-3.4

Exercises: Due Friday, 1 December:

• Evans, Section 3.5, #7,8,10.

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