Math 425 - Notes and Homework Thursday January 24, 2008

Topics for this week -

1. Basic PDE concepts

Examples -

1. The "big three" (or four) PDEs - Laplace, heat, wave (and transport) equations.
2. Methods for finding solutions - first-order equations like 3u_x + 4u_y = Q, where Q is zero, or u, or u + f(x,y). Initial-value problems for this.
3. Initial, boundary and initial/boundary value problems for second order equations.

Second Homework Assignment - due Thursday, January 31

1. Reading: Read sections 1.2 through 1.4 and Appendix A.1 of the text (and the notes on vecctor fields)
2. 1. Find the general solution of u_xy = x²y for the function u(x,y).
   2. Ditto for: yu_xy + 2u_x = x (Hint: first integrate with respect to x)
   3. For the preceding PDE, find the solution that satisfies u(x,1) = 0 and u(0,y) = 0.
   4. Solve: u_x + 2u_y = 0, 5u_x + 6u_y = 0, cu_x + du_y = 0. (These are three separate problems)
3. Be prepared to discuss problems 1, 2, 5 and 6 on page 9 in class. (In the first edition, it's problems 1, 2, 5 on page 9 and "Solve the equation xu_x + yu_y = 0 ".)
4. Turn in problems 10 and 13 on page 10 next Thursday. (In the first edition, it's problems 8 and 11 on page 9).