MATH 4347A – Introduction to PDE (Harrell)

Copyright 2011 by Evans M. Harrell II.

Playing power games

About the test...

Still not done grading....

Pitfalls of power games

A Taylor series might not exist (function not differentiable n times)

Or it might exist but have radius of convergence 0. (It still might be useful as an ‘asympotic series’)

A paradox. A function can have a Taylor series that converges even for all real x

to the wrong answer!

Pitfalls of power games

F(x) = exp(-1/x2), x ≠ 0, f(0) = 0

L’Hôpital tells us the derivatives at 0 can all be calculated, and….. and …..

They are all 0. All of them!

 The initial-value problem ut(x,t) = F(x,t,u,Du), u(x,0) = φ(x), where φ is analytic in a neighborhood of x=0

and F is analytic (in all variables) in a neighborhood of

(0,0,φ(0),Dφ(0)) exists, and the solution is unique and

analytic in a (possibly smaller) neighborhood of (x,t) = (0,0).

The Cauchy-Kovalevskaya (or Kowalevsky)

Theorem

How to determine utt? You can’t!

Nonlinear example - compressible fluid flow

ut + u ux = 0 u(x,0) given; say = 1 + x