Fractional Calculus

[Resha Caner]

I once gave fractional calculus a brief look to see if it could help solve a problem of mine. Since it didn't apply (as far as I could tell), I moved on.

 

Still, it's a cool idea. So, does anyone know of any "practical" applications of fractional calculus, or is it only a mathematical curiosity?

[Bignose]

There are some interesting applications. Non-Fickian diffusion can be described using fractional calculus.

 

Normal Fickian diffusion can be described by [math]\frac{dc}{dt} = \frac{d^2 c}{dx^2}[/math] where c is a concentration, t time, and x position.

 

Non-Fickian diffusion obeys [math]\frac{dc}{dt} = \frac{d^n c}{dx^n}[/math] where

if 2>n>1 the diffusion is called "accelerated" or "fast"

if n>2 the diffusion is called "retarded"

if n<=1 the diffusion is sometimes called "bombastic"

 

The viscoelasticity of polymers can sometimes be described by fractional calculus as well. In a fluid, the stress tensor is a function of the gradient of velocity, velocity of course being the derivative of position with respect to time. In a solid, the stress tensor is a function of the gradient of the displacement or position -- in a way the zeroth derivative of position is position. In a viscoelastic material, there are both "fluid-like" and "solid-like" components to it. Sometimes, an appropriate stress tensor can be made by a weighted sum of the fluid stress tensor and the solid stress tensor. But, there are other models out there that make the stress tensor a function of the gradient of a fractional derivative of position.

 

[math]\mathbf{T} = f(\nabla \frac{d^n \mathbf{x}}{d t^n} ) [/math] where 0<n<1.

 

While there are a few applications out there, I suspect that it will remain a curiosity more than anything. They are very hard to work with -- almost always requiring the use of Fourier or Laplace transforms. And usually a more simple model (made of derivatives of integral order) can be generated that can mimic the results of the fractional derivatives. Lastly, in today's world of computational models, I'm not very sure how to model a fractional derivative via discrete methods. It is pretty straightforward to model first and second derivatives on a grid, but I don't really know how I'd go about modeling the 1.5th derivative. This really hampers it's usefulness as a model. However, I would bet good money that someone has figured out how to discretize fractional derivatives.

[Resha Caner]

There are some interesting applications. Non-Fickian diffusion can be described using fractional calculus.

 

I have no idea what this is.

 

The viscoelasticity of polymers can sometimes be described by fractional calculus as well.

 

This, though, sounds intriguing. Any papers you can point me to? One mechanical device that I have done some development on uses a viscoelastic fluid. It's not what I tried to apply fractional calculus too, but I'm always looking for new ideas.

 

While there are a few applications out there, I suspect that it will remain a curiosity more than anything. They are very hard to work with

 

Yeah, that was my experience too. But let's not give up yet.

 

P.S.: I haven't tried coding in an equation yet. Do you have to do it by hand, or is there a WYSIWYG editor? I remember coding equations for my thesis in LaTex. Yuck.

[Cap'n Refsmmat]

By hand, in [math][/math] tags. You don't need to worry about $$ and \[ sort of things.

[Resha Caner]

Is there a tutorial for that somewhere?

[iNow]

http://www.math.harvard.edu/texman/

 

You just put your text between math tags.

 

So this:

 

[math]x = \frac{1}{2} + \sqrt{56/2}+86^3[/math]

 

 

... renders as:

 

[math]x = \frac{1}{2} + \sqrt{56/2}+86^3[/math]

[Cap'n Refsmmat]

And we do have a tutorial.