2006-09-27

Test is open book/ open notes.

Things to know for the test

• chapter 1
  • first order ODEs and how to solved
  • You do not have to remember how to derive wave/heat eqns, but you shoudl be able to apply to description of physical situation
  • physical meanings of BC (section 1.4)

    			Dirichlet		Neumann
    	heat/diffusion		temp of conc. is 0		insulating inpermeable
    	wave		string has no displacement		free end

• chapter 2
  • wave eqn
    • IVP - write sketch
    • energy => uniqueness
    • Domain of dependence/influence
  • Diffusion
    • IVP - write sketch
    • energy => uniqueness
    • maximum principle

4 example problems:

• solve u_x + cos x * u_y = 0, u(0, y)=e^y^
  • soln: slove or characteristics and write u(x, y) in term s of auxilliary conditions
• u_tt = u_xx, u(x,0)= 2(1-|x|) for |x| < 1, 0 otherwise, u_t(x,0) = -2 |x|<1, 0 otherwise. Sketch soln at 3 times.
  • slon: linear combination of hammer blow and guitar pick problem. linear combination should be the solution
• u_t = k*laplace(u) on D with dirichlet or Neumann. Show d/dt 3int(u^2) = -k*int(norm(grad(u))). From this, show uniqueness.

2006-08-23

First PDE class.

class webpage: http://www.math.uiuc.edu/~abarreir/math442

Andrea Barreiro 227 Illini, 265-5063

abarreir@math.uiuc.edu

Final 40%

Exams

• 10/2 20%
• 11/13 20%

HW due friday first one due next friday. Will be posted Thusday.

course covers most important linear PDEs that arise in physics and classical techniques.

she studies oceanography!

• CFD,
• scattering through media (ultrasound, optical mapping)
• optics, quantium mechanics, elementary particles

PDE - diff eqn this more than one independent vars

order of PDE - order of highest derivative in eqn

linearity -

linear - eqn set to 0 can be expressed as a linear operator

{{{#latex F\(xy, u(x,y), u_t (x,y), ) = 0

F= \script L u }}}

u_x+ y*u_y = 0 ==> linear