Comments on: How to differentiate a non-differentiable function

By: Daniel Lemire

Daniel Lemire — Fri, 22 Jan 2010 17:31:41 +0000

It is a great post, but it would be better with an example. For example, the derivative of the step function.

By: Carnival of Mathematics #59 « The Number Warrior

Carnival of Mathematics #59 « The Number Warrior — Fri, 06 Nov 2009 07:08:05 +0000

[...] John Cook at The Endeavour writes about how to differentiate a non-differentiable function. [...]

By: Michael Duffy

Michael Duffy — Fri, 30 Oct 2009 01:07:40 +0000

Right, this is the basis for finite elements: shape functions that apply only within the element.

The beauty of them is that weighted residual methods like this still apply even for those situations where there isn’t a calculus of variations functional at hand. It made it possible to extend FEA beyond linear conduction heat transfer and small strain elasticity, both of which can be tackled using calculus of variations, into general continuum mechanics.

Typically fantastic stuff, John.

By: John

John — Mon, 26 Oct 2009 18:37:01 +0000

John: Distributions are functionals on the entire space of test functions. So two distributions are equal if the have the same effect on all test functions. In practice, the test functions fade into the background. You never need to look at a particular test function.

By: John Moeller

John Moeller — Mon, 26 Oct 2009 18:14:29 +0000

Ah. So by sticking a shift by t into φ, you can define f(t), f’(t), f”(t) etc. Slick. But you’d have to say that the equivalence is w.r.t. a particular test function, correct?

By: John

John — Mon, 26 Oct 2009 14:58:21 +0000

Sue, here’s some more detail on the integration by parts. Let u = φ and v = f’ dx. ∫ f’ φ = f φ – ∫ f φ’. When these terms are evaluated at -∞ and ∞, the f φ term drops out since φ is zero at -∞ and ∞.

By: Sue VanHattum

Sue VanHattum — Mon, 26 Oct 2009 13:56:37 +0000

I got stuck right here:
The fact that φ is zero outside a finite interval mean the “uv” term from integration by parts is zero.

Why wouldn’t the uv term be determined by the parts of f and phi on the interval where phi is non-zero?

By: John

John — Mon, 26 Oct 2009 12:56:00 +0000

Jaime, I'm not certain about the history of distributions, but I believe there were many historical precedents before people like Sobolev formalized the theory. For example, mathematicians studied non-differentiable functions by looking at limits of a sequence of differentiable functions and I imagine that goes back decades before formal distribution theory. You can also use Fourier transforms to extend the idea of derivatives (see this post) and I imagine that also predates distribution theory by decades. As for Navier-Stokes, you might find these notes useful.

By: Iftikhar

Iftikhar — Mon, 26 Oct 2009 12:52:27 +0000

Jamie,

I believe it was Sergei Sobolev, the eminent Russian mathematician, who developed this field inititally. I remember doing this as part of my dissertation for my bachelors degree. Fascinating field developed by amazing mathematicians.

By: Jaime

Jaime — Mon, 26 Oct 2009 11:01:48 +0000

Great post John!

First heard of weak formulations when taught about the Finite Element Method, buth didn’t dive into them until a couple of years ago, when I started studying Jean Leray’s work on Navier-Stokes equations.

I seem to recall that it was actually Leray who first came up with this whole “weak solution” and “generalized derivative” thing. Do you have any historical insight?