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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
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